A Unidirectional On-Chip Photonic Interface for Superconducting Circuits

We propose and analyze a passive architecture for realizing on-chip, scalable cascaded quantum devices. In contrast to standard approaches, our scheme does not rely on breaking Lorentz reciprocity. Rather, we engineer the interplay between pairs of superconducting transmon qubits and a microwave transmission line, in such a way that two delocalized orthogonal excitations emit (and absorb) photons propagating in opposite directions. We show how such cascaded quantum devices can be exploited to passively probe and measure complex many-body operators on quantum registers of stationary qubits, thus enabling the heralded transfer of quantum states between distant qubits, as well as the generation and manipulation of stabilizer codes for quantum error correction.


I. INTRODUCTION
Over the last two decades, superconducting circuit technologies have emerged among the most promising platforms for realizing quantum processors [1,2].One avenue consists in designing quantum networks in a modular approach, where distant stationary qubits interact by exchanging photons as "flying qubits" propagating in waveguides [3].As the size of experiments and number of qubits in quantum networks scale in complexity, controllable routing of quantum information between distinct components becomes a requirement [4].In most current experiments, this task is taken care of using ferrite junction circulators, which break Lorentz reciprocity via the Faraday effect [5,6].However, as these devices are bulky, lossy, and use large magnetic fields, they are not suitable for on-chip integration, and new, scalable alternatives must be developed.To address this challenge, several approaches were proposed in recent years.Most strategies require active devices [7][8][9][10][11][12][13][14][15][16], where reciprocity is broken by the interplay of several pump fields with precise phase relations, at the cost of adding energy to the system.On the other hand, passive devices have also been proposed based on superconducting junction rings, where circulation is obtained using a constant flux bias; these are however highly sensitive to charge noise [17,18].
In this work, we tackle the problem of quantum information routing from a different angle; rather than circulators, we design effective integrated qubits as composite objects coupled to a meandering 1D transmission line [see Fig. 1(a-c)], with the requirement that photons propagating in one direction are absorbed and reemitted along the same direction, without breaking reciprocity.Coherently driving several such unidirectional quantum emitters through the transmission line gives rise to an effective cascaded driven-dissipative dynamics, as repre-sented in Fig. 1(d), where photons radiated by each emitter coherently drives other emitters downstream; in the literature, this paradigm is sometimes referred to as "chiral quantum optics" [19], and features interesting steadystate properties, as will be discussed below.
In analogy to "giant" artificial atoms [20][21][22][23], which couple to a photonic or phononic waveguide at several points separated by distances comparable to the wavelength, our approach consists in designing a giant unidirectional emitter (GUE), here realized using two artificial atoms as anharmonic oscillators, as represented in Fig. 1(a).These atoms are coupled to a waveguide, at two points separated by a distance d ∼ λ 0 /4, with λ 0 the photon wavelength.By designing the interaction between artificial atoms, our composite object effectively admits a V -level structure with two delocalized excited states |L ∼ (i |1 1 0 2 + |0 1 1 2 )/ √ 2 and |R ∼ (|1 1 0 2 + i |0 1 1 2 )/ √ 2 (with |n k denoting Fock state n = 0, 1, . . . of atom k = 1, 2), with the remarkable property that their transitions to the ground state |0 1 0 2 couple respectively only to left-and right-propagating modes of the waveguide [see Fig. 1(b)], which is due to a destructive interference in the photon emission (and absorption).Below we will analyze an implementation of this model with superconducting transmon qubits coupled via a superconducting quantum interference device (SQUID) [see Fig. 1(c)].
As we will show later on, these composite emitters can be used as unidirectional photonic interfaces for additional long-lived stationary qubits (represented below in Fig. 3), which has immediate applications for quantum information processing and quantum computing.In our approach, quantum information is manipulated and directed passively, using an itinerant probe field as "flying qubit" propagating in the waveguide.This forms a naturally scalable architecture for quantum networking, which we will illustrate in particular with the realization of quantum state transfer between distant stationary qubits, and with the generation and manipulation of stabilizer codes for quantum error correction [24].Our x Y 3 H q y X q 2 3 e e u S V c w c w C 9 Y 7 1 / w m 4 2 1 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 Q p d I 4 K 1 n Z S F n D A M l 7 8 j q R o Y H 4 o = " > A A A B 6 3 i c b V D L S s N A F L 2 p r 1 p f V Z d u B o v g q i S 2 + N g V 3 b i s Y B / Q h j K Z T p q h M 5 M w M x F K 6 C + 4 c a G I W 3 / I n X 9 j k g Z R 6 4 E L h 3 P u 5 d 5 7 v I g z b W z 7 0 y q t r K 6 t b 5 Q 3 K 1 v b O 7 t 7 1 f 2 D r g 5 j R W i H h D x U f Q 9 r y p m k H c M M p / 1 I U S w 8 T n v e 9 C b z e w 9 U a R b K e z O L q C v w R D K f E W w y a U g C N q r W 7 L q d A y 0 T p y A 1 K N A e V T + G 4 5 D E g k p D O N Z 6 4 N i R c R O s D C O c z i v D W N M I k y m e 0 E F K J R Z U u 0 l + 6 x y d p M o Y + a F K S x q U q z 8 n E i y 0 n g k v 7 R T Y B P q v l 4 n / e Y P Y + J d u w m Q U G y r J Y p E f c 2 R C l D 2 O x k x R Y v g s J Z g o l t 6 K S I A V J i a N p 5 K H c J X h / P v l Z d I 9 q z u N e u O u W W t d F 3 G U 4 Q i O 4 R Q c u I A W 3 E I b O k A g g E d 4 h h d L W E / W q / W 2 a C 1 Z x c w h / I L 1 / g U Y g o 5 p < / l a t e x i t > |2 1 0 2 i < l a t e x i t s h a 1 _ b a s e 6 4 = " Y B p / G L v 8 g 4 L U Y h 7 X H f e i R 8 n c n i k = " > A A A B 9 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 B I v g q S R R 0 G P R i 8 c K 9 g P a G D b b S b t 0 s w m 7 G 6 X U / g 8 v H h T x 6 n / x 5 r 9 x 2 + a g r Q 8 G H u / N M D M v T D l T 2 n G + r c L K 6 t r 6 R n G z t L W 9 s 7 t X 3 j 9 o q i S T F B s 0 4 Y l s h 0 Q h Z w I b m m m O 7 V Q i i U O O r X B 4 P f V b D y g V S 8 S d H q X o x 6 Q v W M Q o 0 U a 6 f / I C 1 w m 8 r i S i z z E o V 5 y q M 4 O 9 T N y c V C B H P S h / d X s J z W I U m n K i V M d 1 U u 2 P i d S M c p y U u p n C l N A h 6 W P H U E F i V P 5 4 d v X E P j F K z 4 4 S a U p o e 6 b + n h i T W K l R H J r O m O i B W v S m 4 n 9 e J 9 P R p T 9 m I s 0 0 C j p f F G X c 1 o k 9 j c D u M Y l U 8 5 E h h E p m b r X p g E h C t Q m q Z E J w F 1 9 e J k 2 v 6 p 5 V v d v z S u 0 q j 6 M I R 3 A M p + D C B d T g B u r Q A A o S n u E V 3 q x H 6 8 V 6 t z 7 m r Q U r n z m E P 7 A + f w C A w Z H e < / l a t e x i t > |1 1 1 2 i < l a t e x i t s h a 1 _ b a s e 6 4 = " s L Q F H E 1 E 2 i E A P r O Z 9 x i w M r c L f k w = " > A A A B 9 X i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I v g q W y q o M e i F 4 8 V 7 A e 0 6 5 J N Z 9 v Q b H Z J s k p Z + z + 8 e F D E q / / F m / / G t N 2 D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g o r q 2 v r G 8 X N 0 t b 2 z u 5 e e f + g p e N U M W i y W M S q E 1 A N g k t o G m 4 E d B I F N A o E t I P R 9 d R v P 4 D S P J Z 3 Z p y A F 9 G B 5 C F n 1 F j p / o n 4 h P i 1 n q J y I M A v V 9 y q O w N e J i Q n F Z S j 4 Z e / e v 2 Y p R F I w w T V u k v c x H g Z V Y Y z A Z N S L 9 W Q U D a i A + h a K m k E 2 s t m V 0 / w i V X 6 O I y V L W n w T P 0 9 k d F I 6 3 E U 2 M 6 I m q F e 9 K b i f 1 4 3 N e G l l 3 G Z p A Y k m y 8 K U 4 F N j K c R 4 D 5 X w I w Y W 0 K Z 4 v Z W z I Z U U W Z s U C U b A l l 8 e Z m 0 a l V y V q 3 d n l f q V 3 k c R X S E j t E p I u g C 1 d E N a q A m Y k i h Z / S K 3 p x H 5 8 V 5 d z 7 m r Q U n n z l E f + B 8 / g C A v p H e < / l a t e x i t > |0 1 2 2 i < l a t e x i t s h a 1 _ b a s e 6 4 = " F r z X g k a G M X z K 2 4 e O k K J C 4 r j m L S M = " > A A A B 9 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 B I v g q S R R 0 G P R i 8 c K 9 g P a G D b b S b t 0 s w m 7 G 6 X U / g 8 v H h T x 6 n / x 5 r 9 x 2 + a g r Q 8 G H u / N M D M v T D l T 2 n G + r c L K 6 t r 6 R n G z t L W 9 s 7 t X 3 j 9 o q i S T F B s 0 4 Y l s h 0 Q h Z w I b m m m O 7 V Q i i U O O r X B 4 P f V b D y g V S 8 S d H q X o x 6 Q v W M Q o 0 U a 6 f 3 I C 1 w u 8 r i S i z z E o V 5 y q M 4 O 9 T N y c V C B H P S h / d X s J z W I U m n K i V M d 1 U u 2 P i d S M c p y U u p n C l N A h 6 W P H U E F i V P 5 4 d v X E P j F K z 4 4 S a U p o e 6 b + n h i T W K l R H J r O m O i B W v S m 4 n 9 e J 9 P R p T 9 m I s 0 0 C j p f F G X c 1 o k 9 j c D u M Y l U 8 5 E h h E p m b r X p g E h C t Q m q Z E J w F 1 9 e J k 2 v 6 p 5 V v d v z S u 0 q j 6 M I R 3 A M p + D C B d T g B u r Q A A o S n u E V 3 q x H 6 8 V 6 t z 7 m r Q U r n z m E P 7 A + f w C A u 5 H e < / l a t e x i t > |Ri < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 O F L 9 Y K M R Q c x a C E j M T 2 Z P V P u B I M = " > A A A B 8 H i c b V D L T g J B E O z F F + I L 9 e h l I j H x R H b R R I 9 E L x 7 R y M P A h s w O v T B h d n Y z M 2 t C k K / w 4 k F j v P o 5 3 v w b B 9 i D g p V 0 U q n q T n d X k A i u j e t + O 7 m V 1 b X 1 j f x m Y W t 7 Z 3 e v u H / Q 0 H G q G N Z Z L G L V C q h G w S X W D T c C W 4 l C G g U C m 8 H w e u o 3 H 1 F p H s t 7 M 0 r Q j 2 h f 8 p A z a q z 0 8 H T X U V T 2 B X a L J b f s z k C W i Z e R E m S o d Y t f n V 7 M 0 g i l Y Y J q 3 f b c x P h j q g x n A i e F T q o x o W x I + 9 i 2 V N I I t T + e H T w h J 1 b p k T B W t q Q h M / X 3 x J h G W o + i w H Z G 1 A z 0 o j c V / / P a q Q k v / T G X S W p Q s v m i M B X E x G T 6 P e l x h c y I k S W U K W 5 v J W x A F W X G Z l S w I X i L L y + T R q X s n Z U r t + e l 6 l U W R x 6 O 4 B h O w Y M L q M I N 1 K A O D C J 4 h l d 4 c 5 T z 4 r w 7 H / P W n J P N H M I f O J 8 / 6 u 2 Q e w = = < / l a t e x i t > |Li < l a t e x i t s h a 1 _ b a s e 6 4 = " E 9 c 9 j 9 A Y Z v A 0 S 3 w 9 w h 0 0 1 x B / D N g = " > A A A B 8 H i c b V A 9 S w N B E J 2 L X z F + R S 1 t F o N g F e 6 i o G X Q x s I i g v m Q 5 A h 7 m 7 l k y d 7 e s b s n h J h f Y W O h i K 0 / x 8 5 / 4 y a 5 Q h M f D D z e m 2 F m X p A I r o 3 r f j u 5 l d W 1 9 Y 3 8 Z m F r e 2 d 3 r 7 h / 0 N B x q h j W W S x i 1 Q q o R s E l 1 g 0 3 A l u J Q h o F A p v B 8 H r q N x 9 R a R 7 L e z N K 0 I 9 o X / K Q M 2 q s 9 P B 0 2 1 F U 9 g V 2 i y W 3 7 M 5 A l o m X k R J k q H W L X 5 1 e z N I I p W G C a t 3 2 3 M T 4 Y 6 o M Z w I n h U 6 q M a F s S P v Y t l T S C L U / n h 0 8 I S d W 6 Z E w V r a k I T P 1 9 8 S Y R l q P o s B 2 R t Q M 9 K I 3 F f / z 2 q k J L / 0 x l 0 l q U L L 5 o j A V x M R k + j 3 p c Y X M i J E l l C l u b y V s Q B V l x m Z U s C F 4 i y 8 v k 0 a l 7 J 2 V K 3 f n p e p V F k c e j u A Y T s G D C 6 j C D d S g D g w i e I Z X e H O U 8 + K 8 O x / z 1 p y T z R z C H z i f P + G r k H U = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " V O n 2 Q I Y G l s q T p 5 w C k D + j X b I O 4 H s = " > A A A B 7 X i c d V B N S w M x E M 3 W r 1 q / q h 6 9 B I v g a d n U a t t b 0 Y s 3 K 9 h a a J e S T b N t b L J Z k q x Q l v 4 H L x 4 U 8 e r / 8 e a / M d t W U N E H A 4 / 3 Z p i Z F 8 S c a e N 5 H 0 5 u a X l l d S 2 / X t j Y 3 N r e K e 7 u t b V M F K E t I r l U n Q B r y l l E W 4 Y Z T j u x o l g E n N 4 G 4 4 v M v 7 2 n S j M Z 3 Z h J T H 2 B h x E L G c H G S u 3 e l a B D 3 C + W P L d a K 6 N q H X o u 8 u q o X s k I K p + i M 4 h c b 4 Y S W K D Z L 7 7 3 B p I k g k a G c K x 1 F 3 m x 8 V O s D C O c T g u 9 R N M Y k z E e 0 q 6 l E R Z U + + n s 2 i k 8 s s o A h l L Z i g y c q d 8 n U i y 0 n o j A d g p s R v q 3 l 4 l / e d 3 E h D U / Z V G c G B q R + a I w 4 d B I m L 0 O B 0 x R Y v j E E k w U s 7 d C M s I K E 2 M D K t g Q v j 6 F / 5 N 2 2 U U n b v m 6 U m q c L + L I g w N w C I 4 B A l X Q A J e g C V q A g D v w A J 7 A s y O d R + f F e Z 2 3 5 p z F z D 7 4 A e f t E 8 w Y j 0 k = < / l a t e x i t > |0 1 0 2 i < l a t e x i t s h a 1 _ b a s e 6 4 = " D V + P / 8 / Z w P o K j C f P 9 k Z 4 n D O 2 X F I = " > A A A B 9 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 B I v g q S R V 0 G P R i 8 c K 9 g P a G D b b S b t 0 s w m 7 G 6 X E / g 8 v H h T x 6 n / x 5 r 9 x 2 + a g r Q 8 G H u / N M D M v S D h T 2 n G + r c L K 6 t r 6 R n G z t L W 9 s 7 t X 3 j 9 o q T i V F J s 0 5 r H s + F q 0 5 J 5 s 5 h j 9 w P n 8 A i f m N T g = = < / l a t e x i t > (b) < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 t V g < l a t e x i t s h a 1 _ b a s e 6 4 = " l l x architecture is passive and tunable in situ, and, as we will show, the required experimental parameters and imperfections are achievable with current technology.
Our results presented below are organized as follows.In Sec.II we describe and analyse the design of giant unidirectional emitters (GUEs) as composite artificial atoms with an effective V-level structure, with each transition absorbing and emitting photons along a single direction in a waveguide, and present in Sec.III a possible implementation with superconducting transmon qubits.Next, we study in Sec.IV the cascaded driven-dissipative dynamics arising when several such unidirectional emitters are driven via the waveguide.Finally, in Sec.V we describe how these emitters can act as unidirectional photonic interfaces for additional long-lived stationary qubits, which enables applications for quantum networking such as quantum state transfer between distant stationary qubits, and the generation and manipulation of stabilizer codes for quantum error correction.

II. MODEL OF UNIDIRECTIONAL QUANTUM EMITTERS
Our model for designing unidirectional quantum emitters is represented in Fig. 1(a), and consists of two inter-acting artificial atoms as anharmonic oscillators coupled at two distant points to a waveguide.The dynamics of these two atoms, within the rotating wave approximation, is described by the Hamiltonian (with = 1) Here ω k is the transition frequency of each atom k, U k denotes their anharmonicity, and âk is their annihilation operator, which satisfies [â k , â † l ] = δ k,l .The second line in Eq. ( 1) describes the interaction between atoms, with linear exchange interaction rate J, and non-linear cross-Kerr frequency χ, which can be implemented with two superconducting transmon qubits coupled via a SQUID (see Fig.
Here L1 = √ γ 1 (â 1 + r 2 â2 ) and L2 = √ γ 2 (â 2 + r 1 â1 ) are the coupling operators associated to each coupling point, with coupling rates γ k (which we assume constant over the relevant bandwidth) and small cross-coupling coefficients r k (see implementation below), d is the distance of separation between the two coupling points along the waveguide, and v g is the group velocity of photons in the waveguide.Within a markovian approximation (i.e., assuming , the dynamics of the field can be integrated and treated as a reservoir for the atoms, and we obtain for the Heisenberg equation of motion for an arbitrary atomic operator Ô(t) the quantum Langevin equation (see details in Supplementary Section A) expressed in a rotating frame with respect to a central frequency ω 0 , and in an interaction picture with respect to the waveguide Hamiltonian Ĥph .Here the effective Hamiltonian reads ∆ k = ω 0 − ω k , where the last term emerges from a coherent exchange of photons propagating in the waveguide between the two coupling points, with φ = ω 0 d/v g the phase acquired by a photon in the propagation.On the other hand, the collective coupling operators in Eq. ( 3) represent the collective couplings of the atoms to right-and left-propagating photons due to interference of photon emission and absorption in the reservoir, and are defined respectively as LR (t) = e iφ L1 (t) + L2 (t) and LL (t) = L1 (t) + e iφ L2 (t).Finally, bin d (t) represents the input fields of the waveguide propagating along direction d, and is related to the output fields via [25] bout with [ bin/out The emergence of unidirectional coupling between propagating photons and the composite two-atom system, from Eqs. ( 3) and ( 5), occurs under the following two conditions.
(II) Second, the excitations associated to these two modes âR and âL must be eigenstates of the effective Hamiltonian Ĥeff .For states with a single atomic excitation, i.e., |R = â † R |G and |L = âL |G with |G = |0 1 0 2 the ground state of both atoms, this is achieved by taking symmetric detunings ∆ 1 = ∆ 2 ≡ ∆+2rγ sin(φ opt ) and J = J opt , with the optimal hopping rate given by J opt = −γ(1 + r 2 ) sin(φ opt ).The two excited states |R and |L are then eigenstate of Ĥeff with eigenenergies −∆.The non-linear cross-Kerr interaction with frequency χ, on the other hand, is introduced in the model in order to prevent the excitation of the doubly-excited state |1 1 1 2 when driving the system via the input fields, as we will consider below.
When these two conditions are fulfilled, the composite emitter will absorb and reemit propagating photons along the same direction.In order to assess this directionality in a more general case, we assume the emitter is prepared in state |R at time t = 0 with the waveguide in the vacuum state, and solve the dynamics of the system, which yields the emission of a photon in the waveguide, with the emitter returning to its ground state |G .The temporal shapes of the wavepacket amplitudes of the emitted photon propagating to the right/left are then obtained using a Wigner-Weisskopf ansatz (see details in Supplementary Section A) as where L[•](s) denotes the Laplace transform, and the evolution of the atomic excitation amplitudes is governed by the operator We then define the directionality of photon emission as . This directionality of emitted photons is represented in Fig. 1(e,f).Fig. 1(e) shows that very good directionalities can be achieved even with relatively large imprecisions on J and φ around their optimal values, e.g.due to fabrication imperfections.
Here we obtain β dir > 99% for |J − J opt | γ/10 and |φ − φ opt | π/10.This robustness to imperfections is also observable in Fig. 1(f), where we show the average directionality β dir obtained with random static deviations of r k and γ k .We obtain β dir > 99% as long as the fluctuation in the coupling parameters are below δγ 0.1γ and δr 0.05.

III. IMPLEMENTATION WITH SUPERCONDUCTING CIRCUITS
Our model can be implemented with the circuit represented in Fig. 1(c), which consists of two superconducting transmon qubits (k = 1, 2) with flux-tunable Josephson energies E k J and charging energies E k C = e 2 /(2C eff k ) [26], where e is the elementary charge and C eff k are the effective transmon capacitances (see details in Supplementary Section B).The interaction between transmons is mediated by a SQUID, acting as a non-linear element with flux-tunable Josephson energy E J and with capacitance C. We note that such tunable non-linear couplings mediated by Josephson junctions were demonstrated in recent experiments [27][28][29], and find applications for quantum simulation [30][31][32] and quantum information processing [33].
Following standard quantization procedures, the Hamiltonian for the circuit can be expressed as in Eq. ( 1) (see details in Supplementary Section B).In particular, analytical insight on the resulting system parameters can be gained in the regime of weakly coupled transmons, with In this limit, an estimation of the various parameters of the model can be made in terms of the circuit parameters, with the atomic transition frequencies taking the expression The interaction between atoms contains a linear hopping term J = J C − J I , with a capacitive (J C ) and an inductive (J I ) contribution reading while the cross-Kerr interaction term reads We note that the three Josephson energies in Fig. 1(c) can be independently controlled via flux biases, allowing for an independent in situ fine-tuning of the detunings ∆ k and the hopping rate J.The couplings to the waveguide on the other hand are given by , with c k the coupling capacitances and Z 0 the transmission line impedance [34,35].The capacitance C introduces as well small crosscoupling coefficients r k = C/C eff k , resulting in photon emission from each artificial atom via both coupling points.

IV. DRIVEN-DISSIPATIVE DYNAMICS OF CASCADED QUANTUM NETWORKS
Although the properties of unidirectional emission of our GUE studied above preserve Lorentz reciprocity, i.e., they are invariant under the exchange of left-and right-propagating modes, driving the system through the waveguide allows one to effectively achieve non-reciprocal interactions between artificial atoms.A paradigmatic example of such a situation is represented in Fig. 1(d), where several GUEs are coherently driven via rightpropagating modes, thus driving the âR transition as represented in Fig. 1(b).Photons emitted by each emitter will then also propagate to the right, leading to an effective cascaded quantum dynamics, where each GUE drives the other ones downstream, without any back-action [36][37][38].
This scenario has been studied in recent years in a different context, in a field known in the literature as "chiral quantum optics" [19], which originated from experiments with quantum emitters in the optical domain, such as atoms [39][40][41][42] or quantum dots [43][44][45][46], coupled to photonic 1D nanostructures.The strong confinement of light in these structures gives rise to a socalled "spin-momentum locking" effect [47], allowing for unidirectional couplings between photons and emitters which, in an analogous way to our GUE, does not by itself break Lorentz reciprocity.Besides, building on non-local couplings of quantum emitters to 1D reservoirs, chiral quantum optical systems could also be realized in AMO platforms with broken reciprocity [48][49][50].While photon losses inherent to optical platforms form experimental challenges, the near-ideal mode matching of artificial atoms coupled to 1D transmission lines presents new opportunities to realize this paradigm, in the microwave domain [51,52].Interestingly, it has been predicted that, for several quantum emitters, the ensuing cascaded dynamics in the presence of a coherent drive results in the dissipative preparation of quantum dimers, with quantum emitters pairing up in a dark, entangled state [53][54][55], as we will show below.
In order to study the dynamics of an ensemble of N GUEs (labeled n = 1, . . ., N ) interacting via a common waveguide, we employ the SLH input-output formalism [56][57][58].The SLH framework provides a methodical approach for modeling such composite quantum systems interacting via the exchange of propagating photons, where we assume that non-Markovian effects, due e.g. to the finite propagation time of photons exchanged by the emitters [59], can be neglected.As detailed in the Supplementary Section D, the dynamics of the network of N GUEs can then be obtained from the input-output properties of each individual GUE, by recursively applying composition rules of the SLH formalism in a "bottom-up" fashion.The evolution of an arbitrary atomic operator Ô(t) in the rotating frame then obeys a quantum Langevin equation as expressed in Eq. ( 3), with a redefinition of the effective Hamiltonian and of the coupling operators.Denoting the various parameters and operators associated with each GUE with a corresponding superscript n, we obtain for the effective Hamiltonian Ĥeff with the photon propagation phase φ = ω 0 l/v g where l is the distance between two neighbouring composite emitters along the waveguide.We note that the two new terms in Eq. ( 9) correspond to excitation exchange interactions between different GUEs, mediated respectively by right-and left-propagating photons.For the coupling operators on the other hand, we obtain LR = n e i φ(N −n) Ln R and LL = n e i φ(n−1) Ln L , which represent interference in the atom-field coupling between the emitters.
The presence of a coherent drive via the rightpropagating waveguide modes, with amplitude α(t) [and corresponding Rabi frequency Ω(t) = √ γ r α(t)], can be accounted for by assuming the initial state of the waveguide |α R satisfies bin , with ρ the atomic density matrix, the temporal evolution from Eq. ( 3) then yields the master equation where D[â]ρ = âρâ † − 1 2 {â † â, ρ}.Eq. ( 10) allows to access the evolution and steady-state values of observables with a finite drive amplitude α.In order to account for additional imperfections, we also add in Eq. ( 10) dephasing terms 2γ ϕ n,k D[(â n k ) † ân k ] and non-radiative decay terms γ nr n,k D[â n k ].In Fig. 2(a,b) we represent the ratio of left-and rightpropagating emitted photons obtained in the steadystate of the dynamics for N = 1, with J and φ set to their optimal values, and a constant real Rabi frequency Ω (i.e., the drive frequency is ω 0 ).Fig. 2(a) shows that, since directionality arises in our setup as interference of emission of the two atoms, the dephasing rate γ ϕ spoils the interference and induces some emission to the left with an intensity scaling linearly for low Rabi frequency Ω.As Ω increases with respect to the effective anharmonicities χ and U k , the intensity of left-propagating photons increases, as states with more than a single excitation get populated.This population increase can also be observed as the dashed red curves in Fig. 2(b), and we thus require Ω χ in order to retain a two-level dynamics.We also note that when χ = U 1 = U 2 in Fig. 2(b), the emission to the left vanishes even when states with several excitations are populated, as for these parameters states with several excitations (â † R ) n R (â † L ) n L |G become eigenstates of Ĥeff for all n R/L ≥ 0, thus preserving the property of unidirectional emission.Note that in + F q 0 5 J 5 s 5 h j 9 w P n 8 A i f m N T g = = < / l a t e x i t > (b) < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 t V g   the regime of weakly coupled transmons (C C k and E J E k J ) considered above, the value of U is limited by the fact that, from Eq. ( 8) and Achieving larger values thus requires going beyond the weak coupling regime.This is discussed in the Supplementary Section B, where we also study the validity of the analytical expressions for the effective model in Eqs.(7) and (8).Typical achievable values for χ range from 0 to ∼ 2π × 50 MHz with U k = 2π × 300 MHz.
In the ideal case where the parameters satisfy the properties of unidirectional coupling and the anharmonicities χ and U k are large enough with respect to the Rabi frequency Ω of the drive, the state of the emitters will thus remain within the two-level manifold n {|G n , |R n }.Denoting here σn + = e i φn |R n G|, the dynamics of Eq. ( 10) then reduces to a cascaded master equation [36,37] where ρeff denotes the density matrix of the system expressed in the reduced 2 N -dimensional manifold, and where the effective non-Hermitian Hamiltonian reads, as-suming Ω real, The dynamics generated by Eq. ( 11) induces an effective non-reciprocal interaction between the qubits: as seen from the expression of Eq. ( 12), an excitation in each qubit m can be coherently transferred only to qubits n > m located to its right.While the reduced density matrix of any single GUE is in general mixed, for even N the state of the whole system dissipates towards a pure steady-state , where, as represented in Fig. 1(d), neighbouring qubits pair up as dimers in a two-qubit entangled state [53][54][55] with Remarkably, once the system has reached this dark state |D , all photons emitted by qubit 2n − 1 are coherently absorbed by qubit 2n, such that each dimer effectively decouples from the waveguide radiation field.
The dynamics obtained for a pair of N = 2 GUEs is represented in Fig. 2(c,d).In Fig. 2(c) we observe the purification process described above where, in the steadystate, the system dissipates towards the pure state |D , as represented in the red curves.Strikingly, although the atoms are excited (see green curve), the amount of scattered photons, represented in blue, vanishes in the steady-state, i.e., the system becomes dark and decouples from the waveguide.We note that in the transient dynamics, i.e., before reaching the steady-state, photons are scattered unidirectionally by the emitters, which leads to a decrease of the purity Tr(ρ 2 ).Moreover, the purity of the reduced density matrix ρ(n) for each GUE n remains low in the steady-state (see black curves), as they become entangled.The steady-state overlap D| ρ |D is represented in Fig. 2(d), which shows a requirement for a large χ with respect to the drive intensity |Ω| 2 /γ r .The effect of imperfections due to dephasing and finite excitation lifetimes is represented in the inset, which shows that the steady-state overlap with the dark state becomes unity in the limit χ → ∞ and γ nr = γ φ = 0.

V. QUANTUM INFORMATION ROUTING FOR QUANTUM NETWORKING AND COMPUTING
Our approach enables the realization of large scale quantum processing units, where quantum information is processed in local nodes, and routed using unidirectional emitters.The setup we have in mind is represented in Fig. 3(a), where we represent a possible such architecture, with a set of stationary atomic qubits acting as (a) < l a t e x i t s h a 1 _ b a s e 6 4 = " s O j Z U o N K 3 3 N c 8 n q m G B 8 y g q 4 L + < l a t e x i t s h a 1 _ b a s e 6 4 = " s O j Z U o N K 3 3 N c 8 n q m G B 8 y g q 4 L + + F q 0 5 J 5 s 5 h j 9 w P n 8 A i f m N T g = = < / l a t e x i t > (b) < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 t < l a t e x i t s h a 1 _ b a s e 6 4 = " s O j Z U o N K 3 3 N c 8 n q m G B 8 y g q 4 L + + F q 0 5 J 5 s 5 h j 9 w P n 8 A i f m N T g = = < / l a t e x i t > (b) < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 t < l a t e x i t s h a 1 _ b a s e 6 4 = " s O j Z U o N K 3 3 N c 8 n q m G B 8 y g q 4 L + + F q 0 5 J 5 s 5 h j 9 w P n 8 A i f m N T g = = < / l a t e x i t > (b) < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 t 3. Architecture for quantum information routing.(a) An array of qubits (n = 1, . . ., N ) is coupled to one of two transmission lines, labeled "up" and "down", via GUEs.A propagating photon scatters sequentially on the qubits, while linear optical elements performing unitary transformations Un couple the transmission lines.A projective measurement of the qubits is performed upon detecting the photon at the output.(b) Model for the qubit-GUE interaction in each node n with cross-Kerr frequencies V n 1 and V n 2 , and (c) corresponding superconducting implementation adapted from Fig. 1(c).(d) Quantum circuit realized using the setup in (a), where the double circles represent controlled-Z gates between the qubits and the photon as virtual "flying qubit" with states |up ph and |down ph , corresponding to the photon propagating in transmission line "up" and "down", respectively.quantum register, and GUEs acting as an interface between a waveguide and the stationary qubits.The idea is to mediate effective long-range multi-qubit interactions by using (i) sequences of scattering events induced by unidirectional couplings between a single photon as "flying qubit" and each stationary qubit, (ii) local single-qubit operations, and (iii) linear optics represented by unitary operations U n acting on two waveguides, including in particular 50/50 beam-splitter operations.The applicability of this architecture is illustrated below for quantum state transfer between distant stationary qubits, as well as the generation and manipulation of stabilizer codes.
The scattering events are designed as follows [see Fig. 3(b)].Denoting the parameters and operators associated with node n = 1, . . .N with an index n, each GUE is initially prepared in its ground state |G n , and returns to this state after the photon scattering.The coupling between each stationary qubit (with states {|0 q,n , |1 q,n }) and its GUE consists of a purely non-linear cross-Kerr interaction, which can be described the Hamiltonian ĤV = n Ĥn V , where (see details in Supplementary Section C) ideally with identical frequencies V n 1 = V n 2 ≡ V .The effect of this interaction is then to shift the frequency of the excited states of the GUEs by V , conditional on qubit atom n being in state |1 q,n , without breaking the properties of unidirectional coupling discussed above.A possible implementation of this interaction term with superconducting circuits, adapted from Fig. 1(c), is represented in Fig. 3(c), where the qubit atom is coupled via two SQUIDs to the GUE atoms.We note that (i) the anharmonicity of the GUEs is inconsequential for the applications considered in this section as we consider the scattering of single photons, hence for simplicity the coupling between the artificial atoms of the GUEs are taken purely capacitive, and (ii) the presence of capacitances in the coupling SQUIDs between the stationary qubit and the GUE induces a small direct coupling between the qubit and the waveguide modes, which could deteriorate the qubit lifetime; however, this coupling can be cancelled by subradiance due to interference in the photon emission from both coupling points, by taking the qubit transition frequency ω q such that ω q d/v g is an odd multiple of π (see details in Supplementary Section C).
The scattering of a photon on a single node n, represented in Fig. 3(b), is described within the input-output formalism by a single-photon scattering operator where |vac, G n denotes the vacuum state of the waveguide, with the GUE in its ground state |G n , and the input and output field operators in the frequency domain are defined via bin/out The single-photon scattering operator represents the action of the temporal evolution operator on qubit n, conditional on having an input photon with detuning δ p (with respect to ω 0 ), propagating in direction d [either right (R) or left (L)] be scattered in direction d with detuning ν p .We consider a right-propagating input photon with frequency distribution given by some function f (δ p ) with qubit atom n in some state |ψ q,n , and write the state of the system before the scattering as |in = dδ p f (δ p )[ bin R (δ p )] † |vac, G n |ψ q,n .The state after the scattering can then be expressed from Eq. ( 15) as |out = The single-photon scattering operator in Eq. ( 15) can be obtained by using the quantum Langevin equation ( 3) and the input-output relation (5) (see details in Supplementary Section D).In particular, under the conditions for unidirectional coupling of the GUEs to the waveguide as discussed above, we find Ŝn L,R (ν p , δ p ) = 0 and Ŝn R,R (ν p , δ p ) = δ(ν p −δ p )σ n (δ p ), with the Dirac δ-function representing the conservation of the photon frequency in the scattering, and where with the phase shift t(δ p ) = (2iδ p + γ r )/(2iδ p − γ r ).The operator σn (δ p ) realizes a generic phase gate on qubit n.Assuming the photon has a sharp frequency distribution f (δ p ) around δ p = 0 relative to γ r , by taking V = γ r this phase gate can be parametrized by the value of the tunable detuning ∆ n from GUE n.When ∆ n = −γ r /2, the two terms in Eq. ( 16) acquire an opposite π/2 phase, and the phase gate becomes the Pauli operator σn z = |0 q,n 0| − |1 q,n 1|, up to an irrelevant global phase which can be absorbed in a redefinition of the phase of the output field operator bout R (δ p ).When ∆ n γ r on the other hand, these two terms become identical, and the phase gate reduces to the identity operator 1.
This effective unidirectional photon -qubit interaction finds immediate applications for the detection of individual itinerant microwave photons, which is a current technological challenge [60][61][62][63][64][65][66].This can be realized here with a Ramsey sequence, by preparing the atomic qubit in state |+ q,n , with |± ≡ (± |0 + |1 )/ √ 2. With ∆ n = −γ r /2, a resonant photon will be scattered unidirectionally by the GUE, while qubit atom n will be left in state |− q,n .The photon can then be detected by measuring the qubit state after applying a Ramsey π/2-pulse, which realizes a quantum non-demolition measurement of the itinerant photon, in analogy to the cavity-QED experiments in Refs.[60][61][62][63][64].The resonance frequency ω 0 of this detector can be tuned, while the detection bandwidth is given by γ r (see details in Supplementary Section E).
In order to describe the more generic setup in Fig. 3(a), which now includes two waveguides as well as N nodes, we make use of the SLH input-output formalism as discussed above (see details in Supplementary Section D).We write the input and output field operators in the frequency domain as bin/out d,j (δ), which now contains an additional index j ∈ {up, down} labelling the two waveguides.The single-photon scattering operator for the whole system where |vac, G = |vac N n=1 |G n , then contains two additional indices representing the input line i and the output line j of the scattered photon.The derivation and general expression of this operator are provided in the Supplementary Section D.
In the ideal case where each GUE scatters photons unidirectionally, the scattering operator factorizes as Ŝj,i L,R (ν, δ) = 0 and we obtain The dashed red frame represents the action of the scattered photon, with the corresponding quantum circuit realizing a controlled-Z gate between the two qubits.Upon detection of the photon at the output and reading out the final state of qubit 1, the initial superposition state |ψ q,1 is transferred to |ψ q,N .channels, as shown in Fig. 3(a).They can be represented as 2-dimensional unitary matrices acting on a vectorial space which we denote as {|up ph , |down ph }, where the basis vectors |up/down ph , correspond to the transmission line (either "up" or "down") in which the photon propagates.On this vectorial space the objects Ŝn (δ) are diagonal matrices of qubit operators, which represent the photon scattering on each node.They are defined as Ŝn (δ p ) |down ph = |down ph and Ŝn (δ p ) |up ph = |up ph σn (δ p ) as expressed in Eq. ( 16).
The operator Ŝn (δ p ) thus realizes a frequencydependent controlled-phase gate between the propagating photon as a "flying" control qubit with states |down ph and |up ph , and qubit atom n.For the applications discussed in the following the parameter ∆ n will always be chosen such that the effective interaction in Ŝn (δ p = 0), between a resonant photon and qubit atom n, is either trivial (with ∆ n γ r ), or realizes a controlled-Z gate |down ph down| + |up ph up| σn z (with ∆ n = −γ r /2) as represented in Fig. 3(d).
The entanglement structure of the scattering operator Ŝj,i R,R (ν p , δ p ) in Eq. ( 18) is that of a matrix product operator [67] with bond dimension 2, which is a consequence of quantum information being carried in the network by a propagating photonic qubit.The photon scattering will thus generate entanglement in the qubit array, which can be used e.g. to prepare it in a matrix product state [67] such as a GHZ state or 1D cluster state [68] (see details in Supplementary Section F).We note that this bond dimension, i.e., the amount of entanglement generatable by scattering a photon in the system, can in principle be increased by expanding the dimensionality of the photonic Hilbert space, e.g. by adding more waveguides.
As a first illustration of the working principles of this passive architecture, we consider one of the most basic protocol requiring quantum information routing, namely quantum state transfer between two stationary qubits.
Here, the goal is to transfer a superposition state from one qubit atom, e.g. with n = 1, to another (possibly distant) one, e.g. with n = N , as represented in Fig. 4(a).This is achieved by engineering the effective photonqubit interaction in such a way that the scattering operator in Eq. ( 18) realizes an effective controlled-Z gate between the distant qubits, thereby enabling universal quantum computation in our architecture.The corresponding protocol circuit is represented in Fig. 4(b), which shows how the initial state of qubit 1 |ψ q,1 = c 0 |0 q,1 +c 1 |1 q,1 (with ) is transferred as |ψ q,N upon detection of the photon at the output, while quantum information is erased from qubit 1.Here σz gates are applied conditional on the measurement of the photonic qubit in state |up ph , and of qubit 1 in state |1 q,1 .The Hadamard gates are defined for the atomic qubits as Ĥ = |+ q,n 0|+|− q,n 1|, and are similarly defined for the photonic qubit by replacing |0/1 q,n with |down/up ph .
Assuming perfect control over the other parameters of the system, the average fidelity for the quantum state transfer protocol, as defined in the Supplementary Section G, will depend on the photon frequency distribution f (δ p ) as 3 .This sets a bound to the bandwidth ∆ω of f (δ p ) as ∆ω γ r , and thus to the duration T of the protocol as T ≥ 1/∆ω (see below).Standard strategies for heralded quantum communication [69] can be translated to our protocol in Fig. 4(a), by adding ancillary stationary qubits to each node as quantum state "backups", thus enabling quantum communication with high fidelity, even with photon losses due for instance to amplitude attenuation in the waveguides or imperfect photon detection (see Supplementary Section G).We note that, as discussed above, the photon detection can also be realized using additional nodes as detectors.
As a second application of our architecture for quantum networking, we now show that the setup of Fig. 3(a) allows to perform entangling operations on many stationary qubits, and can be used to passively probe and measure many-body operators, such as stabilizers of stabilizer codes for quantum error correction [24].A standard approach for measuring such stabilizer operators consists in entangling the qubits with an ancilla using two-body interactions; the stabilizers can then be accessed by measuring the ancilla [70][71][72].Building on a previous protocol for measuring the parity of a pair of quantum dots as unidirectional emitters [73], the measurement of stabilizers is achieved here using an interferometric setup with photonic qubits as ancillas, where the only non-trivial operations on the photons are U 0 = U N = Ĥ, and one obtains for the scattering operator of Eq. ( 18) (19) We recall that, with the parameters discussed above, for each stationary qubit n we chose the parameters of the system such that the operator σn (δ p ) is either the identity operator 1 or the Pauli operator σn z when δ p = 0. Defining an arbitrary subset I of the qubit array, the operator in Eq. ( 19) can thus be applied to entangle the state of the output photonic qubit (given by the index j) with the parity PI = n∈I σn z of the interacting qubits, which can then be measured by detecting the photon.More generally, allowing local unitary operations to be performed on the stationary qubits before and after the scattering enables the measurement of any operator of the form n∈I σn , where σn is an arbitrary rotation of σn z on the Bloch sphere.Examples of such operators are the stabilizers of cluster states, which are universal resources for quantum computation [74], and of stabilizer codes, where logical qubits are redundantly encoded in many physical qubits and protected by topology [24].
Despite tremendous recent experimental progress towards the realization of stabilizer codes in superconducting platforms [75][76][77][78][79][80], scaling up the code distance (i.e., the number of physical qubits) beyond a few qubits remains a great challenge.As we show in the following, our architecture offers a naturally scalable approach to passively probe stabilizers, and thus generate and manipulate stabilizer codes.As an example of stabilizer code, we consider the toric code [81], where qubits are located (a) < l a t e x i t s h a 1 _ b a s e 6 4 = " s O j Z U o N K 3 3 N c 8 n q m G B 8 y g q 4 L + + F q 0 5 J 5 s 5 h j 9 w P n 8 A i f m N T g = = < / l a t e x i t > (b) < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 t V g + A h P e i a M n Y 8 Z H 9 9 m / J Z r < / l a t e x i t > J/J opt < l a t e x i t s h a 1 _ b a s e 6 4 = " z H 2 p 4 Q i d w A s s h q U 2 / U e M 7 e W M w f w = " > A A A B 9 X i c b V D L S s N A F J 3 4 r P V V d e l m s A i u a m r F x 6 7 o R r q q Y B / Q x j K Z T t q h k 0 m Y u V F L 6 H + 4 c a G I W / / F n X / j J A 2 i 1 g M X D u f c y 7 3 3 u K H g G m z 7 0 5 q b X 1 h c W s 6

T T p x e P c H 7 R u l j L 1 C m J O B U / T k R E 1 / r s e + a T p / A U P / 1 E v E / r x O B d + b E X I Y R M E m n i 7 x I Y A h w E g H u c 8 U o i L E h h C p u b s V 0 S B S h Y I L K p y G c
on edges of a lattice with periodic boundary conditions.A minimal instance with N = 8 qubits is represented in Fig. 5(a).The toric code has two types of stabilizers: for each plaquette p and each vertex v of the lattice we associate the stabilizers Âp = n∈p σn z and Bv = n∈v σn x , with σn x = |0 q,n 1| + |1 q,n 0|.The logical subspace for encoding quantum information then consists of the four states which are eigenstates of all these stabilizers, with eigenvalue +1.A protocol for preparing the system in one of these four states consists in initializing all qubits in state n |+ q,n .The plaquette operators Âp are then sequentially measured, and the system can be brought to the desired state by applying single-qubit σn x gates afterwards, conditioned on the measurement outcomes (see Supplementary Section H).
In Fig. 5(b,c) we represent the quantum circuit and the setup realizing the measurement of the operator Âp shown in Fig. 5(a).Similar protocols, realized by scattering single photons, can be devised for (i) transferring a superposition state from a single additional stationary qubit to a logical quantum superposition state of the stabilizer code, as well as the reverse process, and (ii) realizing arbitrary logical qubit gates on the code subspace, as well as exponentiated string operators for quantum simulation of anyonic [71] and fermionic models [82] (see details in Supplementary Section H).
In order to quantify the efficiency of our scheme, we consider the task of performing a measurement of the parity operator PI on n G = |I| qubits, with the qubits initially prepared in state |Ψ + = n |+ q,n .Ideally, detecting the photon at the output of waveguide "up" or "down" projects this state to state The average fidelity of this process, defined in the Supplementary Section H, takes here the expression which we represent in Fig. 6.In Fig. 6(a,b) we show this fidelity in situations where the photon scattering is not perfectly unidirectional, with the explicit expression of the scattering operator Ŝj,down d ,R (ν p , δ p ) from Eq. ( 17) provided in the Supplementary Section D. In these cases where the dynamics is not purely cascaded, the fidelity also depends on the propagation phase φ, in contrast to Eq. ( 19).We observe robust fidelities of F Z (δ p ) 99% for small fluctuations of V n 1,2 below ∼ 2% and J below ∼ 5% around their optimal values.Fig. 6(c,d) represents situations where the photon scatters unidirectionally on each node, and shows that the infidelity 1−F Z (δ p ) scales quadratically with the deviation of V around γ r , with the number of interacting qubits n G , and with the detuning of the photon δ p .
As an estimation of experimentally achievable performances, we consider V = γ r = 2π × 50 MHz.From Fig. 6(d), the gate infidelity intrinsic to our protocol remains below 1% as long as the photon detuning is below |δ p | 0.1γ r /n G .This sets a bound to the duration T of a stabilizer measurement, as the bandwidth ∆ω of the photon frequency distribution f (δ p ) must satisfy T ∆ω ≥ 1.For instance, assuming the photon wavepacket has a truncated gaussian temporal distribution, we obtain an average fidelity F Z above 99% with T = 400 ns for n G = 4 (see Supplementary Section H).All 6 independent stabilizers of the toric code with N = 8 qubits can then be measured sequentially in a total time 2.4 µs.We note that measurements of several stabilizers involving non-overlapping subsets of qubits can be performed in parallel using frequency-multiplexing techniques, as the frequency of their respective GUEs can be tuned to be resonant with probe fields with different frequencies.This allows to scale up stabilizer codes without increasing the total measurement time.

VI. CONCLUSION
To conclude, we presented the design of a unidirectional artificial atom, and demonstrated its application as an on-chip interface between itinerant photons and stationary qubits.This design can be integrated in a modular architecture of photonic quantum networks, where controllable multi-qubit operations are realized by passively scattering itinerant photons, which we illustrated with the realization of quantum state transfer protocols with high fidelity, as well as the measurement of manybody stabilizer operators, pertinent for topological quantum error correction.
In contrast to standard strategies for routing quantum information between nodes of a quantum network, our approach does not make use of circulators.In fact, rather than breaking Lorentz reciprocity for the electromagnetic field (i.e., the invariance under the exchange of source and detector) to control and route an itinerant quantum signal, here the propagation of the quantum signal is set by the itinerant photons injected in the network.This allows to achieve an effective non-reciprocal interaction between stationary qubits with a rather simple design, and an architecture resilient to noise and perturbations.
Note added.We recently became aware of related unpublished work by N. Gheeraert, S. Kono and Y. Nakamura.
where ∆ k = ω 0 − ω k .In an interaction picture with respect to the photonic Hamiltonian Ĥph , the interaction between the artificial atoms and the waveguide reads and the total Hamiltonian is given by Ĥtot (t) = Ĥa + Ĥint (t).Denoting for the initial time t 0 , the Heisenberg equations of motion for the field operators then yield bR (ω, t) = bR (ω, dt e i(ω−ω0)t L1 + e iωd/vg L2 .
Injecting these expression in the Heisenberg equation for an arbitrary atomic operator Ô(t), which read (d/dt) Ô(t) = −i[ Ô(t), Ĥtot (t)], we obtain the quantum Langevin equation where we defined the collective coupling operators as LR = e iφ L1 + L2 and LL = L1 + e iφ L2 with φ = ω 0 d/v g , and the input fields as bin . In deriving Eq. ( 24), we used integrals of the form dωe iωt = 2πδ(t), and we made use of a markovian approximation where any retardation effects, due e.g. to the finite time-delays in the propagation of photons between the quantum emitters, is set to 0 + in the final expression.We then obtain the expression of Eq. ( 3) by rearranging terms and by defining the effective Hamiltonian Ĥeff = Ĥa + sin(φ with arbitrary t 1 > t, we have [ bout , and the input-output field relations read [25] bout The quantum Langevin equation ( 24) can be interpreted according to Ito quantum stochastic calculus, and integrated.In the particular case where the waveguide is initially in its vacuum state |vac , we have bin d (t) |vac = 0. Moving back to the Schrödinger picture, we then express the average value of the arbitrary atomic operator Ô(t) in Eq. ( 24) as Ô(t) = Tr[ Ô ρ(t)], where ρ(t) is the density matrix for the atoms, and we obtain from Eq. ( 24) the master equation where D[â]ρ = âρâ † − 1 2 {â † â, ρ}.In order to obtain Eq. ( 6), we consider the situation where the atoms are prepared in state |R at time t = 0.Over time, the system will spontaneously emit a photon in the waveguide, with the atoms returning to their ground state |G .We then make a Wigner-Weisskopf ansatz for the density matrix of the GUE as where , which provides from Eq. ( 28) Denoting the Laplace transform of |Ψ(t) as | Ψ(s) ≡ L[|Ψ(t) ](s), Eq. ( 30) can be solve as | Ψ(s) = F −1 (s) |Ψ(t 0 ) , with F (s) defined in Eq. ( 6) as From Eq. ( 27), this provides for the wavepacket of the emitted photon f R/L (t) = G| LR/L L −1 [ F −1 (s) |R ](t), and we define the emission directionality as

B. SUPERCONDUCTING CIRCUIT IMPLEMENTATION OF UNIDIRECTIONAL EMITTERS
The circuit implementing the GUE is represented in Fig. 1(c), and consists of two transmons interacting via a SQUID and coupled at two points to an open transmission line.Following standard quantization procedures [34,35,83], we decompose the transmission line, with inductance and capacitance per unit length l 0 and c 0 , into segments of finite lengths ∆x, and write the Lagrangian of the system as L = 1 2 φT C φ − V , where ϕ = (ϕ 1 , ϕ 2 , ϕ TL,1 , ϕ TL,2 , ϕ TL,3 , . ..)T contains the superconducting phase variables associated to the transmons (ϕ 1 and ϕ 2 ), and to each segment of the transmission line (ϕ TL,i ), indexed from left to right.Denoting the indices for the segments coupled to each transmon as i 1 and i 2 , the capacitance matrix reads C = Ca − Ca,TL − CT a,TL

CTL
, with CTL . The potential energy, on the other hand, reads with ϕ 0 = /2e (e is the elementary charge).
Defining the conjugate variables Q = ∂L ∂ φ = C φ, we obtain the Hamiltonian of the full system which can be decomposed into H tot = H a + H ph + H int , with an atomic term H a , a term for the transmission line H ph , and an interaction term H int .For the artificial atoms we obtain We then promote the phase and charge variables to operators satisfying [ φk , Ql ] = iδ k,l , and express the Hamiltonian in terms of bosonic annihilation operators â1 and â2 , with Here The atomic Hamiltonian Ĥa then takes the expression of Eq. ( 1) by expanding the cosine functions in Eq. ( 35) up to fourth order, in the limit | φk | ϕ 0 , which is achieved in the transmon regime E k C E k J , and discarding counter-rotating terms in a rotating wave approximation valid for C C k and E J E k J .To estimate the value of the parameters in Eq. ( 1), we keep only the leading order terms, and find for the transition frequencies The linear interaction terms have a capacitive component J C and an inductive component J I as expressed in Eq. (7).We note that the conditions of C C k and E J E k J are required here in order to be able to neglect counter-rotating terms such as (J C + J I )(â † 1 â † 2 + â1 â2 ).Similar considerations apply for the non-linear cross-Kerr interaction χ as expressed in Eq. (8).
For the transmission line Hamiltonian on the other hand, in the limit ∆x → 0 the only non-vanishing terms are 2 /2l 0 + q(x) 2 /2c 0 , where ϕ(x) is the phase variable at position x in the waveguide, and q(x) the charge density.We then express these fields in second quantization in terms of the bosonic operators bR (ω) and bL (ω) as where v g = 1/ √ l 0 c 0 is the group velocity of photons in the transmission line and Z 0 = l 0 /c 0 the transmission line characteristic impedance, and we obtain Ĥph Finally, for the interaction term we obtain where x 1 and x 2 denote the position of the two coupling points along the waveguide.We then obtain the expression in Eq. ( 2) by setting x 1 = 0 and x 2 = d, redefining the phase of the right-propagating modes as bR (ω) → bR (ω)e −iωd/vg and approximating the couplings to be constant over the relevant bandwidth.Assuming for simplicity J /E 2 C , the coupling rates express as ) and the cross-coupling coefficients as r k = C/C eff k .We now study numerically the validity of the model of Ĥa in Eq. ( 1), for our implementation with superconducting circuits.We identify the model parameters (namely J, χ k , U and ω k ) from the full atomic Hamiltonian expressed in Eq. (35), with the phase and charge operators operators in Eq. ( 36).

(a)
< l a t e x i t s h a 1 _ b a s e 6 4 = " s O j Z U o N K 3 3 N c 8 n q m G B 8 y g q 4 L + + F q 0 5 J 5 s 5 h j 9 w P n 8 A i f m N T g = = < / l a t e x i t > (b) < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 t Moreover, the effect of counter-rotating terms in the full Hamiltonian, which are the terms that do not preserve the number of excitations N exc = â † 1 â1 + â † 2 â2 , are accounted for by treating them as perturbation, and applying standard second-order perturbation theory.That is, we decompose the full Hamiltonian into Ĥa = For experimentally realistic parameters, the resulting cross-Kerr interaction χ and anharmonicities U k are shown in Fig. 7, where the Josephson energies E k J and E J are optimized such that J = J opt and ω k = ω 0 .Fig. 7(a) shows a linear scaling of χ for weak cross-coupling coefficients r k , while the anharmonicity U k decreases, and displays an optimal value for χ which is achieved with a small but non-negligible r k .This optimal value is represented as a function of the photon frequency ω 0 and the ratio E

C. IMPLEMENTATION OF UNIDIRECTIONAL QUBIT -PHOTON INTERFACE
Here we discuss the superconducting circuit implementation of the GUE as unidirectional photonic interface for an additional transmon qubit, as represented in Fig. 8.Following the quantization procedure as described in Sec.B for the GUE, the full system, including the transmission line, is described by a Hamiltonian Ĥtot = Ĥph + Ĥa + Ĥint , with the transmission line Hamiltonian reading In the regime of weakly coupled transmons, where (c < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 q K e G 5 T r Z m < l a t e x i t s h a 1 _ b a s e 6 4 = " + I q w X 6 9 3 6 m I 0 u W X l m H / 7 A + v w B e y i T f A = = < / l a t e x i t > FIG. 8. Superconducting circuit implementation of unidirectional photonic interface.A transmon qubit, represented in green, with superconducting phase variable ϕq, is coupled with a purely non-linear cross-Kerr interaction mediated by two SQUIDs to a GUE, in yellow, which acts as unidirectional photonic interface.
the Hamiltonian for the artificial atoms, including both the GUE and the additional qubit, reduces to In analogy to the quantization of the GUE variables yielding Eq. ( 1), here we also quantized the variables for the qubit charge (Q q ) and phase (ϕ q ) as φq where E q C = e 2 /2C eff q , with C eff q ≈ C q + k C c,k the effective qubit capacitance, and we assumed E q C E q J .In Eq. ( 41), the qubit frequency reads ω q ≈ 8E q C E q J , while the qubit anharmonicity is given by U q ≈ E q C .For the qubit -GUE interaction terms, we find a linear exchange interaction term with a capacitive (J C,k ) and an inductive (J I,k ) contribution, where For the non-linear cross-Kerr terms, we have on the other hand which is constrained by the rotating wave approximation [which requires J I,k (ω 0 , ω q )] as The interaction between qubit and GUE reduces to the expression of Eq. ( 14), by setting J C,k = J I,k .We note that (i) the Josephson energies of the coupling SQUIDs E J,k can be independently controlled via flux biases, allowing to fine-tune V k such that V 1 = V 2 = γ r , as required in the main text, (ii) the frequency of the qubit (ω q ) and of the GUEs (∼ ω 0 ) can be far detuned (by several GHz), allowing to relax the condition J C,k ≈ J I,k , which does not need to be met exactly in order to cancel deleterious excitation exchanges between the GUE and the qubit, and (iii) as all the applications discussed here are achieved by scattering single photons with the GUE in its ground state |G , the cross-Kerr interaction term within the GUE, arising from the coupling SQUID in Fig. 1(c), is irrelevant here, and has been removed for simplicity.
Finally, for the interaction between artificial atoms and the transmission line, we obtain, in analogy to Eq. ( 2) where the first component corresponds to the interaction of propagating photons with the GUE, while the second is an additional spurious interaction term coupling directly the qubit atom to the transmission line, with the rates Even though we typically have γ q,k γ k as C k C q , the presence of these additional couplings could lower the lifetime of the qubit atom by spontaneous photon emission in the transmission line.This can however be remedied by properly choosing the qubit frequency ω q .Indeed, let us consider the situation where the qubit is prepared in state |1 q , with the GUE in its ground state |G and the transmission line in the vacuum state |vac .The dynamics of this system can be solved by means of a Wigner-Weisskopf ansatz, where the state of the system takes the expression Inserting in these equations the formal solutions for the dynamics of the field variables, obtained from (d/dt) |ψ we obtain ċq (t) = − γ q,1 + γ q,2 + 2 √ γ q,1 γ q,2 e iωqd/vg c q (t) − √ γγ q,1 + √ γγ q,2 e iω0d/vg − i(J C,1 − J I,1 ) e i(ωq−ω0)t c 1 (t) One can then readily solve these differential equations.In particular, working in a regime with the qubit frequency far detuned with respect to the GUE, i.e., |ω q − ω 0 | ≫ √ γγ q,k and |ω q − ω 0 | |J C,k − J I,k |, the residual linear exchange rates (J C,k − J I,k ) between the qubit and the GUE in Eq. ( 41) adds a contribution to the linear coupling between qubit and waveguide as This yields for the qubit amplitude c q (t) = e −i∆qt e −γqt , where the qubit atom frequency is shifted by ∆ q = 2 γ eff q,1 γ eff q,2 sin(ω q d/v g ), and the qubit undergoes a spontaneous photon emission with rate γ q = k γ eff q,k + 2 γ eff q,1 γ eff q,2 cos(ω q d/v g ).In particular, this decay rate vanishes if γ eff q,1 ≈ γ eff q,2 and if ω q is taken such that ω q d/v g is an odd multiple of π.This is a manifestation of subradiance due to the destructive interference of photons emitted by the qubit atom via the two coupling points.

D. SLH FORMALISM FOR MODELING INPUT-OUTPUT PHOTONIC QUANTUM NETWORKS
Here we provide details on the SLH formalism employed for modeling input-output photonic quantum networks, with several GUEs coupled to a waveguide.We start by giving a brief introductory overview of the formalism in Sec.D 1, including the composition rules for modeling composite photonic systems in a "bottom-up" approach.For more details, we refer the reader to the review in Ref. [58].In Sec.D 2 we apply the formalism to the network represented in Fig. 1(d), with an ensemble of N GUEs coupled to a common waveguide as photonic bath.In Sec.D 3, we derive the expression of the single-photon scattering operator, first for a single qubit coupled to a GUE, as represented in Fig. 3(b), and then for the more generic setup represented in Fig. 3(a).

Definitions and properties
In the SLH formalism, each element of an open input-output photonic quantum network with N c input and output photonic channels is represented by a triplet G = S, L, Ĥ .Here S is an N c × N c scattering matrix describing the coupling between photonic quantum channels, L is an N c × 1 vector of coupling operators representing the interaction between the system and the photonic channels, and Ĥ is the Hamiltonian of the system.For instance, in the situation represented in Fig. 1(d) where an ensemble of N GUEs interacts via a common waveguide, each individual GUE couples to N c = 2 photonic channels, corresponding to the right-and left-propagating modes of the waveguide.Denoting the various parameters and operators associated with each composite emitter with a corresponding superscript n, the SLH triplet for each GUE n is given by 1, Ln R Ln L T , Ĥn eff .On the other hand, the propagation of photons between nodes is described by another triplet e i φ, 0, 0 , where φ = ω 0 l/v g with l the distance between two neighbouring GUEs along the waveguide.
To describe larger composite quantum systems, triplets can be combined in a bottom-up approach using different composition rules.In the following we will make use of two composition rules: the series product and the concatenation product.The series product, represented in Fig. 9(a), allows to combine cascaded quantum systems, where the output channel of a first system G 1 becomes the input channel of a second one G 2 , and is denoted G 2 G 1 .The composition rule is The concatenation product on the other hand, represented in Fig. 9(b), combines different photonic channels in parallel, and is denoted G 2 G 1 .The composition rule is For a generic (possibly large) quantum system with triplet Similarly, the collective coupling of the emitters to left-propagating waveguide modes G L is obtained by recursively applying the series product as represented in Fig. 9(c).Note that the ordering of the triplets is however reversed, and we have The system is then finally described by the concatenation of the right-and left-propagating mode contributions as with Ĥeff = n Ĥn eff + ĤR + ĤL .The dynamics of the system then follows from Eq. ( 55), and expresses as in Eq. ( 3), up to a redefinition of the phase of the input field operators as bin R/L (t) → e −i φ(N −1)bin R/L (t).

Single-photon scattering operator
In the following we derive the expression of the single-photon scattering operator for the setup of Fig. 3(a).We start by defining the single-photon scattering operator, and derive its expression for the situation of a single node (i.e., a single qubit atom interacting with a GUE) coupled to a waveguide, as represented in Fig. 3(b).We then extend the situation to the more generic setup of Fig. 3(a).

Definitions
Within the SLH formalism, we recall that a (possibly composite) open quantum system with N c photonic input and output channels, described by the SLH triplet in Eq. ( 54), follows the dynamics in Eqs.(55) and (56).In this framework, the single-photon scattering problem consists in solving for the single-photon scattering operator Ŝj,i (ν p , δ p ), where Ŝj,i (ν p , δ p ) = vac| bout j (ν p )[ bin i (δ p )] † |vac .
Here, Ŝ(ν p , δ p ) is a matrix of operators acting on the quantum system, and its elements Ŝj,i (ν p , δ p ) represent the back-action on the quantum system when a photon with detuning δ p (with respect to the central frequency ω 0 ) scatters on the system from input channel i, and leaves the system in output channel j with detuning ν p , with the the input/output Fourier transform operators defined as bin/out with [ bin/out j (ν p ), [ bin/out i (δ p )] † ] = δ j,i δ(ν p − δ p ).We stress that the single-photon scattering operator in Eq. ( 62) is a different quantity from the scattering matrix S in Eq. (54).From the input-output relation in Eq. (56) where the last term must be evaluated using the quantum Langevin equation (55).

E. APPLICATION FOR SINGLE-PHOTON DETECTORS
Here we discuss how the setup of a single qubit atom coupled to a GUE can be used to detect individual itinerant photons.Considering the setup as in Fig. 3(b), the scattering of a photon on qubit atom n, assuming a unidirectional coupling, is described by the operator in Eq. ( 77), which, with ∆ n = −γ r /2 and V = γ r , yields σn (δ p ) = 2i(δ p − γ r /2) + γ r 2i(δ p − γ r /2) − γ r |0 q,n 0| + 2i(δ p + γ r /2) + γ r 2i(δ p + γ r /2) − γ r |1 q,n 1| , which becomes the Pauli operator σn z = |0 q,n 0| − |1 q,n 1|, up to an irrelevant global phase, for resonant photons (i.e., with δ p = 0).Resonant photons can thus be detected by preparing qubit n in state |+ q,n , which will be flipped to the orthogonal state |− q,n upon photon scattering.The photon is then effectively detected by measuring qubit n, after applying a Ramsey π/2-pulse on the qubit.For off-resonant photons, the detection probability is given by which represents a detection bandwidth of ∼ γ r .
For photons with finite wavepacket bandwidth, the back-action of this detection will also alter the shape of the wavepacket.
After subsequently measuring the qubit, the frequency distribution of the wavepacket will be deformed, up to normalization constants, as upon successful detection with probability dδ p P det (δ p )|f (δ p )| 2 , and if the detection fails.Note that the fact that the phases of two factors in Eqs.(98) and (99) depend on δ p /γ r is due to the temporal deformation of the wavepacket in the dynamics of the photon absorption and reemission.On the other hand, the fact that their norms depend on δ p /γ r is a consequence of the frequency filtering due to the measurement back-action.

F. APPLICATION FOR PREPARATION OF MATRIX PRODUCT STATES
In this section we provide examples of generation of matrix product states with a single photon scattering, namely, GHZ and 1D cluster states.

G. QUANTUM STATE TRANSFER PROTOCOL
The average fidelity F QST for the quantum state transfer protocol is evaluated by applying the protocol on an initially maximally entangled state between qubit 1 and a virtual ancilla qubit (denoted a) [85] as |Ψ i = (|0 q,1 |0 a + |1 q,1 |1 a ) |+ q,N / √ 2. After performing the protocol as represented in Fig. 4, we obtain the state of the system of qubit N and ancilla as a density matrix ρf .Ideally, the state of qubit N and the ancilla should be pure and entangled as |Ψ ideal = (|0 q,N |0 a + |1 q,N |1 a )/ √ 2. The average fidelity is then defined as Here, the state of the system before the scattering of the photon, injected in the system from line "down", reads |in = dδ p f (δ p )[ bin R,down (δ p )] † |vac, G |Ψ i , with frequency distribution f (δ p ). Assuming unidirectional photon -GUE interactions, from the expression in Eq. ( 18) the state after the scattering reads |out = Including in the description the finite probability P d of losing the photon in the process, due for instance to amplitude attenuation in the waveguides or to a faulty photon detection, the overall transfer fidelity is (1 − P d )F QST .Standard strategies for quantum error correction can however be applied to correct for such photon losses.For example, following Ref.[69], we can add an ancillary backup stationary qubit b to node 1 and, before performing the state transfer protocol, entangle it with qubit 1 as In case the photon is not detected after the scattering, the initial superposition can then be retrieved by measuring qubit 1, as the photon scattering operator in Eq. ( 16) is diagonal in the computational basis of the qubits.From Eq. (107), for the measurement outcome |0 q,1 , the state of the backup qubit is projected to |1 b |0 b , while the outcome . This allows to prepare the system back to the entangled state (107), and repeat the procedure until the photon is successfully detected at the output, which requires on average 1/(1 − P d ) trials.At this stage, the state transfer protocol can resume normally, which transfers the entanglement with the backup qubit b from qubit 1 to qubit N , yielding The qubit superposition is then finally transferred to qubit N by measuring the backup qubit b and, depending on the outcome, performing a local σN x gate on qubit N .

H. PROTOCOLS FOR TORIC CODE GENERATION AND MANIPULATION 1. Toric code generation
The toric code is a stabilizer code where physical qubits are located on the edges of a 2D lattice with periodic boundary conditions [81].The code has two types of stabilizers: as represented in Fig. 5(a), for each plaquette p of the lattice we define an operator Âp = n∈p σn z , and, similarly, for each vertex v we define Bv = n∈v σn x .With an N l × N l lattice [e.g.N l = 2 in Fig. 5(a)], the number of physical qubits is N = 2N 2 l , while the number of independent (n = 0) located to the left of the other "topological quantum memory" qubits with n = 1, . . .N .There, the controlled-Ŝ gate is realized by (i) performing single qubit rotations on the quantum memory (denoted L) before and after the photon scattering, and (ii) engineering the photon scattering in Eq. ( 16) such that σn (δ p = 0) = σ n z if n ∈ I, and σn (δ p = 0) = 1 otherwise.The exponentiated gate is performed on the ancilla qubit which, after measurement, is transferred to the quantum memory.Assuming a logical qubit in the memory |Ψ L is encoded in a superposition of states |Φ 1 and |Φ 2 = Ẑ1 |Φ 1 , any logical single-qubit gate can be decomposed into a product e iϕ1 X1 e iϕ2 Ẑ1 e iϕ3 X1 , which is thus performed with our protocol in three steps.

Quantum state write-in and read-out
The protocol described above can also be used to write a qubit superposition state in the quantum memory, with the same setup.This "write-in" protocol, adapted from the quantum state transfer protocol of Fig. 4, is represented in Fig. 10(c), where the ancilla qubit is initialized in a superposition state |Ψ q,0 = c 0 |0 q,0 + c 1 |1 q,0 (with |c 0 | 2 + |c 1 | 2 = 1).After the protocol, the superposition state is transferred to the quantum memory as |Ψ L = c 0 |Φ 1 L + c 1 |Φ 2 L .The inverse protocol, consisting in reading-out the quantum memory by mapping the superposition state back to the ancilla, is represented in Fig. 10(d,e).This requires to invert the setup, and use left-propagating photons to carry the quantum information from the quantum memory to the ancilla.

U 1 < 2 <
l a t e x i t s h a 1 _ b a s e 6 4 = " u g 4 O t S r n h m 6 C K o W R Z 8 t r j X l N J v M = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h6 9 L B b B U 0 m s + H E r e v F Y 0 b S F N p T N d t M u 3 W z C 7 k Y o o T / B i w d F v P q L v P l v 3 K R B 1 P p g 4 P H e D D P z / J g z p W 3 7 0 y o t L a + s r p X X K x u b W 9 s 7 1 d 2 9 t o o S S a h L I h 7 J r o 8 V 5 U x Q V z P N a T e W F I c + p x 1 / c p 3 5 nQ c q F Y v E v Z 7 G 1 A v x S L C A E a y N d O c O n E G 1 Z t f t H G i R O A W p Q Y H W o P r R H 0 Y k C a n Q h G O l e o 4 d a y / F U j P C 6 a z S T x S N M Z n g E e 0 Z K n B I l Z f m p 8 7 Q k V G G K I i k K a F R r v 6 c S H G o 1 D T 0 T W e I 9 V j 9 9 T L x P 6 + X 6 O D C S 5 m I E 0 0 F m S 8 K E o 5 0 h L K / 0 Z B J S j S f G o K J Z O Z W R M Z Y Y q J N O p U 8 h M s M Z 9 8 v L 5 L 2 S d 1 p 1 B u 3 p 7 X m V R F H G Q 7 g E I 7 B g X N o w g 2 0 w A U C I 3 i E Z 3 i x u P V k v V p v8 9 a S V c z s w y 9 Y 7 1 / v F 4 2 0 < / l a t e x i t >U l a t e x i t s h a 1 _ b a s e 6 4 = " A 8 q A M O A x 2 n S 1 X V S K O Z 5 X T I 5 5 D 3 U = " >A A A B 6 n i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k r f t y K X j x W N G 2 h D W W z 3 b R L N 5 u w u x F K 6 E / w 4 k E R r / 4 i b / 4 b N 2 k Q t T 4 Y e L w 3 w 8 w 8 P + Z M a d v + t J a W V 1 b X 1 k s b 5 c 2 t 7 Z 3 d y t 5 + W 0 W J J N Q l E Y 9 k 1 8 e K c i a o q 5 n m t B t L i k O f 0 4 4 / u c 7 8 z g O V i k X i X k 9 j 6 o V 4 J F j A C N Z G u n M H 9 U G l a t f s H G i R O A W p Q o H W o P L R H 0 Y k C a n Q h G Ol e o 4 d a y / F U j P C 6 a z c T x S N M Z n g E e 0 Z K n B I l Z f m p 8 7 Q s V G G K I i k K a F R r v 6 c S H G o 1 D T 0 T W e I 9 V j 9 9 T L x P 6 + X 6 O D C S 5 m I E 0 0 F m S 8 K E o 5 0 h L K / 0 Z B J S j S f G o K J Z O Z W R M Z Y Y q J N O u U 8 h M s M Z 9 8 v L 5 J 2 v e Y 0 a o 3 b 0 2 r z q o i j B I d w B C f g w D k 0 4 Q Z a 4 A K B E T z C M 7 photon 8 H b h w O O d e 7 r 0 n i B l V 2 n E + r N z a + s b m V n 6 7 s L O 7 t 3 9 Q P D y 6 V V E i M e n g i E W y F y B F G B W k o 6 l m p B d L g n j A S D e Y X m Z + 9 5 5 I R S N x o 2 c x 8 T k a C x p S j L S R e o M x 4 h w N 5 b B Y c u x m r d p s g w 8 g C f w b N 1 Z j 9 a L 9 b p s z V m r m W P w A 9 b b J 7 g S k J I = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " 3 L aZ S f k u n Y 9 O E 9 0 N Z u 6 H J N J Z a c o = " > A A A B 7 3 i c b V D L S g M x F M 3 U V 6 2 v q k s 3 w S K 4 G m Y 6 f e 6 K b l x W s L b Q D i W T Z t r Q J D M m G a E M / Q k 3 L h R x 6 + + 4 8 2 / M t E V8 H b h w O O d e 7 r 0 n i B l V 2 n E + r N z a + s b m V n 6 7 s L O 7 t 3 9 Q P D y 6 V V E i M e n g i E W y F y B F G B W k o 6 l m p B d L g n j A S D e Y X m Z + 9 5 5 I R S N x o 2 c x 8 T k a C x p S j L S R e o M x 4 h w N 5 b B Y c u x m r d p s a V m P D C d H O m J + u 1 l 4 n 9 e P 9 F h w 0 + p i B N N B F 4 u C h M G d Q S z 5 + G I S o I 1 m x m C s K T m V o g n S C K s T U S F Z Q g Z a l 8 v / y W 3 Z d v 1 b O + 6 U m p d r O L I g x N w C s 6 B C + q g B a 5 A G 3 Q A B g w 8 g C f w b N 1 Z j 9 a L 9 b p s z V m r m W P w A 9 b b J 7 g S k J I = < / l a t e x i t > dir < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 d r / S V F V F + + x 5 l X 9 e F 9 n s r j T j B 0 = " > A A A B + X i c b V B N S 8 N A E N 3 4 W e t X 1 K O X Y B E 8 l a Q K e i x 6 8 V j B f k A T w m Y 7 b Z d u N m F 3 U i y h / 8 S L B 0 W 8 + k + 8 + W / c t j l o 6 4 O B x 3 s z z M y L U s E 1 u u 6 3 t b a + s b m 1 X d o p 7 + 7 t H x z a R 8 c t n W S K Q Z M l I l G d i G o Q X E I T O Q r o p A p o H A l o R 6 O 7 m d 8 e g 9 I 8 k Y 8 4 S S G I 6 U D y P m c U j R T a t h 8 B 0 t B H e M K 8 x 9 U 0 t C t u 1 Z 3 D W S V e Q S q k Q C O 0 v / x e w r I Y J D J B t e 5 6 b o p B T h V y J m B a 9 j M N K W U j O o C u o Z L G o I N 8 f v n U O T d K z + k n y p R E Z 6 7 + n s h p r P U k j k x n T H G o l 7 2 Z + J / X z b B / E + R c p h m C Z I t F / U w 4 m r z Y b U k U Z m u Y K p g R 3 P v I i a V T K 7 l m 5 c n t e q l 5 l d e T J A T k i J 8 Q l F 6 R K b k i N 1 A k j j + S Z v J I 3 6 8 l 6 s d 6 t j 9 l o z s p 2 9 s k f W J 8 / K o e Z l g = = < / l a t e x i t > (a)< l a t e x i t s h a 1 _ b a s e 6 4 = " s O j Z U o N K 3 3 N c 8 n q m G B 8 y g q 4 H N k c 6 L 8 + 5 8 L F p z T j Z z D H / g f P 4 A i 3 6 N T w = = < / l a t e x i t > (d) < l a t e x i t s h a 1 _ b a s e 6 4 = " N 6 4 w f C Y x v c T j 6 p h g 2 D Q U a j 6 P A d k b U D P W y N x P / 8 z q p C a / 9 C Z d J a l C y x a I w F c T E Z P Y 3 6 X O F z I i x J Z Q p b m 8 l b E g V Z c a m U 7 A h e M s v r 5 J m t e J d V K r 3 l 6 X a T R Z H H k 7 g F M r g w R X U 4 A 7 q 0 A A G A 3 i G V 3 h z h P P i v D s f i 9 a c k 8 0 c w x 8 4 n z + Q D Y 1 S < / l a t e x i t > (f ) < l a t e x i t s h a 1 _ b a s e 6 4 = " U W O l h z I 7 7 x V L b s W d g 6 w S L y M l y F D v F b + 6 / Z i l E U r D B N W 6 4 7 m J 8 S d U G c 4 E T g v d V G N C 2 Y g O s G O p p B F q f z I / d U r O r N I n Y a x s S U P m 6 u + J C Y 2 0 H k e B 7 Y y o G e p l b y b + 5 3 V S E 1 7 7 E y 6 T 1 K B + F q 0 5 J 5 s 5 h j 9 w P n 8 A j Q O N U A = = < / l a t e x i t > l < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 C 8 s f U J Y b r j C 4 3 r 1 L R B Z 8 O g M 3 f g = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m q o M e i F 4 8 t 2 F p o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g p r 6 x u b W 8 X t 0 s 7 u 3 v 5 B + f C o r e N U M W y x W M S q E 1 C N g k t s G W 4 E d h K F N A o E P g T j 2 5 n / 8 I R K 8 1 j e m 0 m C f k S H k o e c U W O l p u i X K 2 7 V n Y O s E i 8 n F c j R 6 J e / e o O Y p R F K w w T V u u u 5 i f E z q g x n A q e l X q o x o W x M h 9 i 1 V N I I t Z / N D 5 2 S M 6 s M S B g r W 9 K Q u f p 7 I q O R 1 p M o s J 0 R N S O 9 7 M 3 E / 7 x u a s J r P + M y S Q 1 K t l g U p o K Y m M y + J g O u k B k x s Y Q y x e 2 t h I 2 o o s z Y b E o 2 B G / 5 5 V X S r l W 9 i 2 q t e V m p 3 + R x F O E E T u E c P L i C O t x B A 1 r A A O E Z X u H N e X R e n H f n Y 9 F a c P K Z Y / g D 5 / M H 1 V e M 9 A = = < / l a t e x i t > d < l a t e x i t s h a 1 _ b a s e 6 4 = " F E T z 1 U e K h c j G u j B e 9 Z p K b G 4 P g J k = " > A A A B 8 n i c b V D L S g M x F L 3 j s 9 Z X 1 a W b Y B F c l Z k q 6 L L o x m U F + 4 D p U D K Z T B u a S Y Y k I 5 S h n + H G h S J u / R p 3 / o 2 Z d h b a e i B w O O c e c u 8 J U 8 6 0 c d 1 v Z 2 1 9 Y 3 N r u 7 J T 3 d 3 b P

2 <
2 p a k v w l k 9 e J d 1 m w 7 t s N B + u 6 q 3 b s o 4 K n M I Z X I A H 1 9 C C e 2 h D B w h I e I Z X e H O M 8 + K 8 O x + L 0 T W n z J z A H z i f P z E D k T A = < / l a t e x i t > ' l a t e x i t s h a 1 _ b a s e 6 4 = " C f I 1 7 Q G 5 f 5 E + I 6 k i E d b Y 4 7 p Y s g M

2 <
d t q 4 4 + c w R + A H n 7 R P o D J B 7 < / l a t e x i t > C l a t e x i t s h a 1 _ b a s e 6 4 = " F A 8 A t O a 2 Q C q L I r 0 T E C 4 a t I + C g k Q = " > A A A B 6 n i c d V D L S g M x F M 3 U V 6 2 v q u D G T b A I r o Z M a 9 V l a T c u W 7 Q P a I e S S T N t a C Y z J B m h D P 0 E N y 4 U c e v W v / A L 3 L n x W 0 w 7 F V T 0 w I X D O f d y 7 z 1 e x J n S C L 1 b m a X l l d W 1 7 H p u Y 3 N r e y e / u 9 d S Y S w J b Z K Q h 7 L j Y U U 5 E 7 S p m e a 0 E 0 m K A 4 / T t j e u z f z 2 D Z W K h e J a T y L q B n g o m M 8 I 1 k a 6 q

2 J
T b p 5 E w I X 5 / C / 0 m r a D s l u 9 g w a V R B i i w 4 B E f g B D j g H F T A J a i D J i B g C G 7 B P X i w u H V n P V p P a W v G W s z s g x + w n j 8 B 2 / G R O Q = = < / l a t e x i t > E < l a t e x i t s h a 1 _ b a s e 6 4 = " I 7 k s a P q w a y 9 h j B a Q W p G B Z r 5 P 65 E = " >A A A B 7 H i c d V B N S 8 N A E N 3 U r 1 q / q h 6 9 L C 1 C T y F p r f V Y F E E 8 V T B t o Y1 l s 9 2 0 S z e b s L s R S u h v 8 O J B K V 7 9 L 1 6 9 i f 4 Y t 0 0 F F X 0 w 8 H h v h p l 5 X s S o V J b 1 Z m S W l l d W 1 7 L r u Y 3 N r e 2 d / O 5 e U 4 a x w M T

1 <
H H s C j w Y 1 7 Y 2 o 8 p a 0 Z Y z G z D 3 7 A e P 4 E e M K S M g = = < / l a t e x i t > ' l a t e x i t s h a 1 _ b a s e 6 4 = " 9 q K e G 5 T r Z m L r E 6 W c a n Q x B 2 S B n Q w = " > A A A B 8 H i c d V D L S g N B E O z 1 G e M r 6 t H L Y B A 8 L b t 5 q M e g F 4 8 R z E O S J c x O Z p M h M 7 P L z G w g h H y F F w + K e P V z v P k 3 T p I V V L S g o a j q p r s r T D j T

1 <
W 0 I X 5 + i / 0 m z 5 P p l t 3 R b K d a u s j h y c A w n c A Y + X E A N b q A O D S A g 4 A G e 4 N l R z q P z 4 r w u W 1 e c b O Y I f s B 5 + w T m i J B 6 < / l a t e x i t > C l a t e x i t s h a 1 _ b a s e 6 4 = " B O S H X A u a K 8 z w 0 y 4 6 c u C r H l 3 v R w c = " > A A A B 6 n i c d V D L S g M x F M 3 U V 6 2 v q u D G T b A I r o Z M a 9 V l a T c u W 7 Q P a I e S S T N t a C Y z J B m h D P 0 E N y 4 U c e v W v / A L 3 L n x W 0 w 7 F V T 0 w I X D O f d y 7 z 1 e x J n S C L 1 b m a X l l d W 1 7 H p u Y 3 N r e y e / u 9 d S Y S w J b Z K Q h 7 L j Y U U 5 E 7 S p m e a 0 E 0 m K A 4 / T t j e u z f z 2 D Z W K h e J a T y L q B n g o m M 8 I 1 k a

1 J
5 / a b Z E 4 < / l a t e x i t > E < l a t e x i t s h a 1 _ b a s e 6 4 = " b V H G a u l t J D S D D e m v B G m S 0 U N j + 0 k = " > A A A B 7 H i c d V B N S 8 N A E N 3 U r 1 q / q h 6 9 L C 1 C T y F p r f V Y F E E 8 V T B t o Y 1 l s 9 2 0 S z e b s L s R S u h v 8 O J B K V 7 9 L 1 6 9 i f 4 Y t 0 0 F F X 0 w 8 H h v h p l 5 X s S o V J b 1 Z m S W l l d W 1 7 L r u Y 3 N r e 2 d / O 5 e U 4 a x w M T

2 <
6 4 X S x 3 v t Z d r o 5 V + 7 / R D H A e E K M y R l x 7 Y i 5 S Z I K I o Z m e S 6 s S Q R w i M 0 I B 1 N O Q q I d J P 5 s R N 4 q J U + 9 E O h i y s 4 V 7 9 P J C i Q c h x 4 u j N A a i h / e z P x L 6 8 T K / / E T S i P Y k U 4 T h f 5 M Y M q h L P P Y Z 8 K g h U b a 4 K w o P p W i I d I I K x 0 P j k d w t e n 8 H / S L J t 2 x S x f 6 T R O Q Y o s O A A F U A I 2 q I E 6 u A A N 4 A A M K L g D D + D R 4 M a 9 M T W e 0 t a M s Z j Z B z 9 g P H 8 C d z 6 S M Q = = < / l a t e x i t > c 0 l a t e x i t s h a 1 _ b a s e 6 4 = " V d O k 6 t 7 P h Z D n k

1 <
e 5 n 4 l 9 d N T V Q N p k w m q a G S L B Z F K Y c m h t n j c M A U J Y Z P L M F E M X s r J C O s M D E 2 n o I N 4 e t T + D 9 p + S 4 6 d / 3 b c q l + t Y w j D 4 7 A M T g D C F R A H d y A B m g C A k b g A T y B Z 0 c 4 j 8 6 L 8 7 p o z T n L m U P w A 8 7 b J 7 r S j g s = < / l a t e x i t > c 0 l a t e x i t s h a 1 _ b a s e 6 4 = " U s 8 C W N / m z P v X 6 w u 6 5 Y M 8 l a S Z g jE = " > A A A B 6 3 i c d V D L S g M x F M 3 U V 6 2 v q k s 3 w S K 6 G i Z j t e 2 u 6 M Z l B f u A d i i Z N N O G J p k h y Q i l 9 B f c u F D E r T / k z r 8 x 0 1 Z Q 0 Q M X D u f c y 7 3 3 h A l n 2 n j e h 5 N b W V 1 b 3 8 h v F r a 2 d 3 b 3 i v s H L R 2 n i t A m i X m s O i H W l D N J m 4 Y Z T j u J o l i E n L b D 8 X X m t + + p 0 i y W d 2 a S 0 E D g o W Q R I 9 h k E j n t o 3 6 x 5 L m V q o 8 q N e i 5 y K u h W j k j y L 9 A l x C 5 3 h w l s E S j X 3 z v D W K S C i o N 4 V j r L v I S E 0 y x M o x w O i v 0 U k 0 T T M Z 4 S L u W S i y o D q b z W 2 f w x C o D G M X K l j R w r n 6 f m G K h 9 U S E t l N g M 9 K / v U z 8 y + u m J q o G U y a T 1 F B J F o u i l E M T w + x x O G C K E s M n l m C i m L 0 V k h F W m Bg b T 8 G G 8 P U p / J + 0 f B e d u / 5 t u V S / W s a R B 0 f g G J w B B C q g D m 5 A A z Q B A S P w A J 7 A s y O c R + f F e V 2 0 5 p z l z C H 4 A e f t E 7 l O j g o = < / l a t e x i t > E J < l a t e x i t s h a 1 _ b a s e 6 4 = " k j C 1 r a C j y A 5 o h Y u v S K 8 F i E N O 2 B c = " > A A A B 9 H i c b V D L S g M x F L 3 j s 9 Z X 1 a W b Y B F c l Z k q 6 L I o g r i q Y B / Q D i W T Z t r Q T D I m m U I Z + h 1 u X C j i 1 o 9 x 5 9 + Y a W e h r Q c C h 3 P u 4 d 6 c I O Z M G 9 f 9 d l Z W 1 9 Y 3 N g t b x e 2 d 3 b 3 9 0s F h U 8 t E E d o g k k v V D r C m n A n a M M x w 2 o 4 V x V H A a S s Y 3 W R + a 0 y V Z l I 8 m k l M / Q g P B A s Z w c Z K f l d a M 8 u i 2 9 5 9 r 1 R 2 K + 4 M a J l 4 O S l D j n q v 9 N X t S 5 J E V B j C s d Y d z 4 2 N n 2 J l G O F 0 W u w m m s a Y j P C A d i w V O K L a T 2 d H T 9 G p V f o o l M o + Y d B M / Z 1 I c a T 1 J A r s Z I T N U C 9 6 m f i f 1 0 l M e O W n T M S J o Y L M F 4 U J R 0 a i r A H U Z 4 o S w y e W Y K K Y v R W R I V a Y G N t T0 Z b g L X 5 5 m T S r F e + 8 U n 2 4 K N e u 8 z o K c A w n c A Y e X E I N 7 q A O D S D w B M / w C m / O 2 H l x 3 p 2 P + e i K k 2 e O 4 A + c z x 9 U 1 Z H O < / l a t e x i t > C < l a t e x i t s h a 1 _ b a s e 6 4 = " K C + q A U 1 X g m 4 X + i 2 e s D A z N U S c 4 a 4 = " > A A A B 8 n i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s x U Q Z f F b l x W s A + Y D i W T Z t r Q T D I k G a E M / Q w 3 L h R x 6 9 e 4 8 2 / M t L P Q 1 g O B w z n 3 k H t P m H C m j e t + O 6 W N z a 3 t n f J u Z W / / 4 P C o e n z S 1 T J V h H a I 5 F L 1 Q 6 w p Z 4 J 2 D D O c 9 h N F c R x y 2 g u n r d z v P V G l m R S P Z p b Q I M Z j w S J G s L G S P 5 D W z 9 c b D d a 1 5 V 9 R R h j M 4 h 0 v w 4 A a a c A 9 t 6 A A B C c / w C m + O c V 6 c d + d j O V p y i s w p / I H z + Q P + 8 J E P < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 f D 9 K X P 4 C N K F A 7 y 8 c Y v i F W R 6 T I s = " > A A A B 7 X i c b V D L S g M x F M 3 4 r P V V d e k m W A R X w 0 w 7 f e 2 K b l x W s A 9 o h 5 J J M 2 1 s M h m S j F C G / o M b F 4 q 4 9 X / c + T d m 2 i K + D l w 4 n H M v 9 9 4 T x I w q 7 T g f 1 t r 6 x u b W d m 4 n v 7 u 3 f 3B Y O D r u K J F I T N p Y M C F 7 A V K E 0 Y i 0 N d W M 9 G J J E A 8 Y 6 Q b T q 8 z v 3 h O p q I h u 9 S w m P k f j i I Y U I 2 2 k z m C M O E f D Q t G x G 5 7 n 1 S v Q s d 1 y v V Z p Z K R U K l U b 0 L W d B Y p g h d a w 8 D 4 Y C Z x w E m n M k F J 9 1 4 m 1 n y K p K W Z k n h 8 k i s Q I T 9 G Y 9 A 2 N E C f K T x f X z u G 5 U U Y w F N J U p O F C / T 6 R I q 7 U j Ae m k y M 9 U b + 9 T P z P 6 y c 6 r P s p j e J E k w g v F 4 U J g 1 r A 7 H U 4 o p J g z W a G I C y p u R X i C Z I I a x N Q f h l C h u r X y 3 9 J p 2 T y s c s 3 X r F 5 u Y o j B 0 7 B G b g A L q i B J r g G L d A G G N y B B / A E n i 1 h P V o v 1 u u y d c 1 a z Z y A H 7 D e P g E g 0 I + l < / l a t e x i t > < l a t e x i t s h a _ b a s e = " f D K X

FIG. 1 .
FIG. 1. Unidirectional coupling of quantum emitters to a transmission line.(a) Model for realizing a giant unidirectional emitter (GUE) using non-linear coupling between two artificial atoms coupled to a waveguide.(b) Corresponding level structure obtained with specific parameters (see text).An effective two-level system with states |0102 and |R is obtained, which couples to right-propagating modes of the transmission line.(c) Superconducting circuit implementation, where two transmons (k = 1, 2) are coupled at two points to a meandering transmission line, and interact via a SQUID.(d) Driven-dissipative cascaded quantum network realized with several GUEs as effective two-level emitters unidirectionally coupled to a transmission line.The system dissipates towards a pure steady-state with emitters pairing up in an entangled state |D .(e) Directionality β dir of emitted photons, with ∆ k = 0 and r k = 0.2, γ k = γ.(f) Averaged directionality β dir for J = Jopt, φ = φopt and ∆ k = 0, obtained with uniformly distributed r1, r2, γ1 and γ2, with means r k = 0.2, γ k = γ and standard deviations r 2 k − r k 2 = δr, γ 2 k − γ k 2 = δγ.
1(c) and discussion below).The waveguide has a continuous spectrum of modes described over the relevant bandwidth by the bare Hamiltonian Ĥph = dωω[ b † R (ω) bR (ω) + b † L (ω) bL (ω)], where bd (ω) is the annihilation operator for photons with frequency ω propagating to the right (with d = R) or to the left (with d = L), and satisfies [ bd (ω), b † d (ω )] = δ(ω − ω )δ d,d .Finally, the coupling between the atoms and the waveguide yields, within the rotating wave ap-proximation, the Hamiltonian Ĥint 1 w s g d e S I v 5 N W 5 d 5 6 d N + d 9 H J 1 y s p k N 8 g v O x y f g y K z k < / l a t e x i t > (a) < l a t e x i t s h a 1 _ b a s e 6 4 = " s O j Z U o N K 3 3 N c 8 n q m G B 8 y g q 4 H N k c 6 L 8 + 5 8 L F p z T j Z z D H / g f P 4 A i 3 6 N T w = = < / l a t e x i t > (c) < l a t e x i t s h a 1 _ b a s e 6 4 = " u m a C 7 x G r 7 w M U W O l h z I 7 7 x V L b s W d g 6 w S L y M l y F D v F b + 6 / Z i l E U r D B N W 6 4 7 m J 8 S d U G c 4 E T g v d V G N C 2 Y g O s G O p p B F q f z I / d U r O r N I n Y a x s S U P m 6 u + J C Y 2 0 H k e B 7 Y y o G e p l b y b + 5 3 V S E 1 7 7 E y 6 T 1 K B k i 0 V h K o i J y e x v 0 u c K m R F j S y h T 3 N 5 K 2 J A q y o x N p 2 B D 8 J Z f X i X N a s W 7 q F T v L 0 u 1 m y y O P J z A K Z T B g y u o w R 3 U o Q E M B v A M r / D m C O f F e X c + F q 0 5 J 5 s 5 h j 9 w P n 8 A j Q O N U A = = < / l a t e x i t > (d) < l a t e x i t s h a 1 _ b a s e 6 4 = " N 6 4 w f C Y x v c T j 6 p h g 2 D Q U FIG. 2.Driven-dissipative dynamics.(a,b) Ratio of left-and right-propagating photon intensities, in the steadystate, emitted by the artificial atoms when coherently driven through the waveguide, withγ k = γ, ∆ k = 0, r k = 0.2, U k = 100γ, γnr = 0.01γ, J = Jopt, φ = φopt.(a)χ = 50γ.Dashed red: γϕ/γr.(b) γϕ = 0.01γ.Dashed red: probability of 10 −2 and 10 −3 (resp.top and bottom) of having two or more excitations in the atoms.Dashed grey: χ = U1 = U2.(c,d) Cascaded dynamics with N = 2 GUEs, with r k = 0.2, U k = 500γ, φ = 0, and γnr = γϕ.(c) Ω = γ, χ = 50γ, γϕ = 0.01γ.(d) Steady-state overlap D| ρ |D , with Ω ∈ [1, 10]γ (light to dark blue), and γϕ = 0.01γ.Inset: χ → ∞, red dashed curve ∝ Ω 2 γϕ/γ 3 r .

FIG. 4 .
FIG.4.Protocol for quantum state transfer.(a) Setup and (b) corresponding quantum circuit realizing quantum state transfer from qubit atoms 1 to N .Hadamard photonic gates Ĥ are realized as 50/50 beam-splitters.The dashed red frame represents the action of the scattered photon, with the corresponding quantum circuit realizing a controlled-Z gate between the two qubits.Upon detection of the photon at the output and reading out the final state of qubit 1, the initial superposition state |ψ q,1 is transferred to |ψ q,N .

FIG. 5 .
FIG. 5. Toric code generation and manipulation.(a) Abstract representation of a toric code, where qubits are located on the edges of a 2D lattice with periodic boundary conditions, with here N = 8 qubits.The two types of stabilizers Âp and Bv are represented.(b) Quantum circuit realizing a measurement of the stabilizer Âp represented in (a), and (c) corresponding interferometric setup, with the detunings ∆ n of the GUEs chosen such that only nodes 1, 5, 7 and 8 are resonant with the photon.
H N k c 6 L 8 + 5 8 L F p z T j Z z D H / g f P 4 A i 3 6 N T w = = < / l a t e x i t > (c) < l a t e x i t s h a 1 _ b a s e 6 4 = " u m a C 7 x G r 7 w Mc J Y d n Z R Q j I M m S L k c = " > A A A B 6 n i c b V D L S g N B E O y N r x h f U Y 9 e B o M Q L 2 E 3 C n o M e v E Y 0 T w g W c L s p D c Z M j u 7 z M w K I e Q T v H h Q x K t f 5 M 2 / c Z L s Q R M L G o q q b r q 7 g k R w b V z 3 2 8 m t r W 9 s b u W 3 C z u 7 e / s H x c O j p o 5 T x b D B Y h G r d k A 1 C i 6 x Y b g R 2 E 4 U 0 i g Q 2 A p G t z O / 9 Y R K 8 1 g + m n G C f k Q H k o e cU W O l h z I 7 7 x V L b s W d g 6 w S L y M l y F D v F b + 6 / Z i l E U r D B N W 6 4 7 m J 8 S d U G c 4 E T g v d V G N C 2 Y g O s G O p p B F q f z I / d U r O r N I n Y a x s S U P m 6 u + J C Y 2 0 H k e B 7 Y y o G e p l b y b + 5 3 V S E 1 7 7 E y 6 T 1 K B k i 0 V h K o i J y e x v 0 u c K m R F j S y h T 3 N 5 K 2 J A q y o x N p 2 B D 8 J Z f X i X N a s W 7 q F T v L 0 u 1 m y y O P J z A K Z T B g y u o w R 3 U o Q E M B v A M r / D m C O f F e X c + F q 0 5 J 5 s 5 h j 9 w P n 8 A j Q O N U A = = < / l a t e x i t > (d) < l a t e x i t s h a 1 _ b a s e 6 4 = " N 6 4 w f C Y x v c T j 6 p h g 2 D Q U w 3 p u 0 e t P X B w O O 9 G W b m B Q l n 2 r j u t 1 N Y W 9 / Y 3 C p u l 3 Z 2 9 / Y P y o d H b S 1 T R W i L S C 5 V N 8 C a c i Z o y z D D a T d R F M c B p 5 1 g f D v z O 0 9 U a S b F o 5 k k 1 I / x U L C I E W y s 9 F A N z w f l i l t z 5 0 C r x M t J B X I 0B + W v f i h J G l N h C M d a 9 z w 3 M X 6 G l W G E 0 2 m p n 2 q a Y D L G Q 9 q z V O C Y a j + b n z p F Z 1 Y J U S S V L W H Q X P 0 9 k e F Y 6 0 k c 2 M 4 Y m 5 F e 9 m b i f 1 4 v N d G 1 n z G R p I Y K s l g U p R w Z i W Z / o 5 A p S g y f W I K J Y v Z W R E Z Y Y W J s O i U b g r f 8 8 i p p 1 2 v e R a 1 + f 1 l p 3 O R x F O E E T q E KH l x B A + 6 g C S 0 g M I R n e I U 3 h z s v z r v z s W g t O P n M M f y B 8 / k D j o i N U Q = = < / l a t e x i t > F Z < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 z D F w E O N P s Q 2 U S / F 0 r b 6 7 o h I 0 g E = " > A A A C A H i c b Z D L S s N A F I Y n 9 V b r L e r C h Z v B I r g q i R U v u 6 I g L i v Y C 7 Y h T K a T d u h k E m Y m Q g n Z + C p u X C j i 1 s d w 5 9 s 4 S U N R 6 w 8 D H / 8 5 h z n n 9 y J G p b K s L 6 O 0 s L i 0 v F J e r a y t b 2 x u m d s 7 FIG. 6. Stabilizer measurements.Fidelity FZ (δp) for the measurement of the parity operator PI on nG qubits prepared in state |Ψ+ , with φ = φopt, γ1 = γ2, r k = 0 and ∆n = −γr/2.(a) V n 1 = V n 2 = γr, δp = 0, nG = 4. (b-d) J = Jopt, φ = 0. (b) δp = 0, nG = 4. (c) δp = 0, V n 1 = V n 2 = V .(d) V n 1 = V n 2 = γr.

∞
ne,n e =0 Ĥne,n e a , where the diagonal part Ĥne,ne a is the projection of Ĥa on the subspace with n e excitations, while the off-diagonal part Ĥne,n e =ne a couples subspaces with different excitation numbers n e and n e .For small perturbations with respect to the optical frequency ω 0 , the renomalized Hamiltonian, which now includes the second-order contribution of these off-diagonal terms, is then obtained as Ĥ(2) a ≈ Ĥa − ne,n e =ne Ĥne,n e a Ĥn e ,ne a ω 0 (n e − n e ) .

kJ
/E k C in Fig.7(b), where E k J = E k J + E J , which shows that χ decreases for increasing ratio E k J /E k C .A trade-off must thus be made between working in the transmon regime (E k J /E k C 1) in order for the artificial atoms to have small sensitivity to charge noise, and having large effective anharmonicities.For concreteness, a reasonable such trade-off can be taken as ω 0 = 2π × 8 GHz and E k J /E k C = 100, in which case χ ≈ 2π × 80 MHz and U k ≈ 2π × 240 MHz.

e 4 8
e c K e 0 4 H 1 Z u a X l l d S 2 / X t j Y 3 N r e K e 7 u t V S U S E K b J O K R 7 P h Y U c 4 E b W q m O e 3 E k u L Q 5 7 T t j y 9 m f v u e S s U i c a s n M f V C P B Q s Y A R r I z U v + 9 d 3 b r 9 Y c u y a 4 5 R P H

u b 2 Y 2 J 1 < 2 <
c f s N 4 + A U m b j l k = < / l a t e x i t > E < l a t e x i t s h a 1 _ b a s e 6 4 = " g 8 G 0 7 6 T b K d M 7 Y M t W W S T Y p t E m Q U 8 = " > A A A B 7 H i c d V B N S 8 N A E N 3 U r 1 q / q h 6 9 L B b B U 9 i k r X o s i i C e K p i 2 0 M a y 2 W 7 a p Z t N 2 N 0 I J f Q 3 e P G g iF d / k D f / j d s 2 g o o + G H i 8 N 8 P M v C D h T G m E P q z C 0 v L K 6 l p x v b S x u b W 9 U 9 7 d a 6 k 4 l Y R 6 J O a x 7 A R Y U c 4 E 9 T T T n H Y S S X E U c N o O x h c z v 3 1 P p W K x u N W T h P o R H g o W M o K 1 k b z L / v W d 2 y 9 X k F 1 H y D 1 B c E F q 9 Z z U q 9 C x 0 R w V k K P Z L 7 / 3 B j F J I y o 0 4 V i p r o M S 7 W d Y a k Y 4 n Z Z 6 q a I J J m M 8 p F 1 D B Y 6 o 8 r P 5 s V N 4 Z J Q B D G N p S m g 4 V 7 9 P Z D h S a h I F p j P C e q R + e z P x L 6 + b 6 v D M z 5 h I U k 0 F W S w K U w 5 1 D G e f w w G T l G g + M Q Q T y c y t k I y w x E S b f E o m h K 9 P 4 f + k 5 d p O 1 X Z v a p X G e R 5 H E R y A Q 3 A M H H A K G u A K N I E H C G D g A T y B Z 0 t Y j 9 a L 9 b p o L V j 5 z D 7 4 A e v t E 0 s f j l o = < / l a t e x i t > C l a t e x i t s h a 1 _ b a s e 6 4 = " w O q E S l e L f s + B B n k V m L Y P v / u N 8 m g = " > A A A B 6 n i c d V D J S g N B E K 1 x j X G L e v T S G A R P Q 0 8 W 9 R j M x W N E s 0 A y h J 5 O T 9 K k p 2 f o 7 h H C k E / w 4 k E R r 3 6 R N / / G z i K o 6 I O C x 3 t V V N U L E s G 1 w f j D W V l d W 9 / Y z G 3 l t 3 d 2 9 / Y L B 4 c t H a e K s i a N R a w 6 A d F M c M m a h h v B O o l i J A o E a w f j + s x v 3 z O l e S z v z C R h f k S G k o e c E m O l 2 3 r f 6 x e K 2 K 1 i X D r H a E E q 1 S W p l p H n 4 j m K s E S j X 3 j v D W K a R k w a K o j W X Q 8 n x s + I M p w K N s 3 3 U s 0 S Q s d k y L q W S h I x 7 W f z U 6 f o 1 C o D F M b K l j R o r n 6 f y E i k 9 S Q K b G d E z E j / 9 m b i X 1 4 3 N e G l n 3 G Z p I Z J u l g U p g K Z G M 3 + R g O u G D V i Y g m h i t t b E R 0 R R a i x 6 e R t C F + f o v 9 J q + R 6 Z b d 0 U y n W r p Z x 5 O A Y T u A M P L i A G l x D A 5 p A Y Q g P 8 A T P j n A e n R f n d d G 6 4 i x n j u A H n L d P + 8 W N m w = = < / l a t e x i t >C l a t e x i t s h a 1 _ b a s e 6 4 = " D e q q x O q k u V x l c z X I l I L P 3 6 u i l g g = " >A A A B 6 n i c d V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 C k n a q s d i L x 4 r 2 g 9 o Q 9 l s N + 3 S z S b s b o Q S + h O 8 e F D E q 7 / I m / / G b R t B R R 8 M P N 6 b Y W Z e k H C m t O N 8 W C u r a + s b m 4 W t 4 v b O 7 t 5 + 6 e C w r e J U E t o i M Y 9 l N 8 C K c i Z o S z P N a T e R F E c B p 5 1 g 0 p j 7 n X s q F Y v F n Z 4 m 1 I / w S L C Q E a y N d N s Y e I N S 2 b F r j u O d O 2 h J q r W c 1 C r I t Z 0 F y p C j O S i 9 9 4 c x S S M q N O F Y q Z 7 r J N r P s N S M c D o r 9 l N F E 0 w m e E R 7 h g o c U e V n i 1 N n 6 N Q o Q x T G 0 p T Q a K F + n 8 h w p N Q 0 C k x n h P V Y / f b m 4 l 9 e L 9 X h p Z 8 x k a S a C r J c F K Y c 6 R j N / 0 Z D J i n R f G o I J p K Z W x E Z Y 4 m J N u k U T Q h fn 6 L / S d u z 3 Y r t 3 V T L 9 a s 8 j g I c w w m c g Q s X U I d r a E I L C I z g A Z 7 g 2 e L W o / V i v S 5 b V 6 x 8 5 g h + w H r 7 B P 1 J j Z w = < / l a t e x i t > E q J < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 T C O a e B b g 9 I b 4 b 6 H r u + j T T 1 W y k 0 = " > A A A B 7 H i c d V D J S g N B E K 2 J W 4 x b 1 K O X x i B 4 G m Y m I c s t K I J 4 i m A W S G L o 6 X S S J j 0 9 Y 3 e P E I Z 8 g x c P i n j 1 g 7 z 5 N 3 Y W Q U U f F D z e q 6 K q n h 9 x p r T j f F i p l d W 1 9 Y 3 0 Z m Z r e 2 d 3 L 7 t / 0 F B h L A m t k 5 C H s u V j R T k T t K 6 Z 5 r Q V S Y o D n 9 O m P z 6 f + c 1 7 K h U L x Y 2 e R L Q b 4 K F g A 0 a w N l L 9 o n d 1 e 9 f L 5 h z b C D B 7 g C Z 4 t Y T 1 a L 9 b r o j V l L W c O 4 Q e s t 0 / g b I 6 + < / l a t e x i t > C q < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 M r X e AE n K d G K x n F y 0 m 6 6 4 Z B r + f Q = " > A A A B 6 n i c d V D L S g N B E O y N r x h f U Y 9 e B o P g a d l N Q h 6 3 Y C 4 e I x o T S J Y w O 5 l N h s z O r j O z Q l j y C V 4 8 K O L V L / L m 3 z h 5 C C p a 0 F B U d d P d 5 c e c K e 0 4 H 1 Z m b X 1 j c y u 7 n d v Z 3 d s / y B 8 e 3 a o o k Y S 2 S c Q j 2 f W x o p w J 2 t Z M c 9 q N J c W h z 2 n H n z T n f u e e S s U i c a O n M f V C P B I s Y A R r I 1 0 3 B 3 e D f M G x i 9 V y t e 4 i x 6 6 U 3 F r N M c Q t 1 4 u V E n J t Z 4 E C r N A a 5 N / 7 w 4 g k I R W a c K x U z 3 V i 7 a V Y a k Y 4 n e X 6 i a I x J h M 8 o j 1 D B Q 6 p 8 t L F q T N 0 Z p Q h C i J p S m i 0 U L 9 P p D h U a h r 6 p j P E e q x + e 3 P x L 6 + X 6 K D m p U z E i a a C L B c F C U c 6 Q v O / 0 Z B J S j S f G o K J Z O Z W R M Z Y Y q J N O j k T w t e n 6 H 9 y W 7 T d k l 2 8 K h c a F 6 s 4 s n A C p 3 A O L l S h A Z f Q g j Y Q G M E D P M G z x a 1 H 6 8 V 6 X b Z m r N X M M f y A 9 f Y J k q W O A A = = < / l a t e x i t > ' q< l a t e x i t s h a 1 _ b a s e 6 4 = " S E s B 3 7 X d S R S l x K s z 9 b G + K q T 5 l T Y = " > A A A B 8 H i c d V D J S g N B E K 1 x j X G L e v T S G A R P w 0 w S s t y C X j x G M I s k Q + j p 9 C R N e n r G 7 p 5 A G P I V X j w o 4 t X P 8 e b f 2 F k E F X 1 Q 8 H i v i q p 6 f s y Z 0 o 7 z Y a 2 t b 2 x u b W d 2 s r t 7 + w e H u a P j l o o S S W i T R D y S H R 8 r y p m g T c 0 0 p 5 1 Y U h z 6 n L b 9 8 d X c b 0

2 < 1 <
W 0 I X 5 + i / 0 m z 5 P p l t 3 R b K d a u s j h y c A w n c A Y + X E A N b q A O D S A g 4 A G e 4 N l R z q P z 4 r w u W 1 e c b O Y I f s B 5 + w T m i J B 6 < / l a t e x i t > ' l a t e x i t s h a 1 _ b a s e 6 4 = "C f I 1 7 Q G 5 f 5 E + I 6 k i E d b Y 4 7 p Y s g M = " > A A A B 8 H i c d V D L S g M x F M 3 4 r P V V d e k m W A R X Q 2 b a q s u i G 5 c V 7 E P a o W T S T B u a Z I Y k U y h D v 8 K N C 0 X c + j n u / B v T d g Q V P X D h c M 6 9 3 H t P m H C m D U I f z s r q 2 v r G Z m G r u L 2 z u 7 d f O j h s 6 T h V h D Z J z G P V C b G m n E n a N M x w 2 k k U x S L k t B 2 O r + d + e 0 K V Z r G 8 M 9 O E B g I P J Y s Y w c Z K 9 7 0 J V s m I 9 f 1 + q Y z c G k L + O Y J L U q 3 l p F a B n o s W K I M c j X 7 p v T e I S S q o N I R j r b s e S k y Q Y W U Y 4 X R W 7 K W a J p i M 8 Z B 2 L Z V Y U B 1 k i 4 N n 8 N Q q A x j F y p Y 0 c K F + n 8 i w 0 H o q Q t s p s B n p 3 9 5 c / M v r p i a 6 D D I m k 9 R Q S Z a L o p R D E 8 P 5 9 3 D A F C W G T y 3 B R D F 7 K y Q j r D A x N q O i D e H r U / g / a f m u V 3 H 9 2 2 q 5 f p X H U Q D H 4 A S c A Q 9 c g D q 4 A Q 3 Q B A Q I 8 A C e w L O j n E f n x X l d t q 4 4 + c w R + A H n 7 R P o D J B 7 < / l a t e x i t > c 0 l a t e x i t s h a 1 _ b a s e 6 4 = " u p 6 7 s C r u L H G g V M S g 9 I b L 3 1 J i 8 b 0 = " > A A A B 6 3 i c d V D L S g M x F M 3 U V 6 2 v q k s 3 w S K 6 G i Z j c e q u 6 M Z l B f u A d i i Z N N O G J p k h y Q i l 9 B f c u F D E r T / k z r 8 x 0 1 Z Q 0 Q M X D u f c y 7 3 3 R C l n 2 n j e h 1 N Y W V 1 b 3 y h u l r a 2 d 3 b 3 y v s H L Z 1 k i t A m S X i i O h H W l D N J m 4 Y Z T j u p o l h E n L a j 8 X X u t + + p 0 i y Rd 2 a S 0 l D g o W Q x I 9 j k E j n t o 3 6 5 4 r m B d x n 4 V e i 5 K E C B X 8 t J 9 S K o I Y h c b 4 4 K W K L R L 7 / 3 B g n J B J W G c K x 1 F 3 m p C a d Y G U Y 4 n Z V 6 m a Y p J m M 8 p F 1 L J R Z U h 9 P 5 r T N 4 Y p U B j B N l S x o 4 V 7 9 P T L H Q e i I i 2 y m w G e n f L e i φ(m−n) − h.c. .

FIG. 10 .
FIG.10.Operations on the toric code and exponentiated string operators with an additional qubit.(a) Setup with an additional qubit atom n = 0, located to the left of the other qubit atoms as topological quantum memory.(b,c) Protocols for (b) applying an exponentiated gate e iϕ Ŝ on the quantum memory (denoted L), and (c) for transferring a qubit state superposition from qubit 0 to the quantum memory, with the setup depicted in (a).(d) Inverted setup, with a photon injected from the right and detected at the left output.(e) Protocol for transferring back a state superposition from the quantum memory to qubit 0, with the setup depicted in (d).