Logical measurement-based quantum computation in circuit-QED

We propose a new scheme of measurement-based quantum computation (MBQC) using an error-correcting code against photon-loss in circuit quantum electrodynamics. We describe a specific protocol of logical single-qubit gates given by sequential cavity measurements for logical MBQC and a generalised Schrödinger cat state is used for a continuous-variable (CV) logical qubit captured in a microwave cavity. To apply an error-correcting scheme on the logical qubit, we utilise a d-dimensional quantum system called a qudit. It is assumed that a three CV-qudit entangled state is initially prepared in three jointed cavities and the microwave qudit states are individually controlled, operated, and measured through a readout resonator coupled with an ancillary superconducting qubit. We then examine a practical approach of how to create the CV-qudit cluster state via a cross-Kerr interaction induced by intermediary superconducting qubits between neighbouring cavities under the Jaynes-Cummings Hamiltonian. This approach could be scalable for building 2D logical cluster states and therefore will pave a new pathway of logical MBQC in superconducting circuits toward fault-tolerant quantum computing.

and = ± e X {1, } with Pauli operators X, Z. Thus, it is interpreted as the single-qubit gate HR z (θ) is performed on |+〉 by the measurement of qubit A with the chosen angle θ on |2CS〉 AB . It is therefore of essence to demonstrate efficiently building such a useful entangled resource state and performing single-qubit measurements on the resource state for practical MBQC.
The MBQC in continuous variables (CVs) has been firstly well developed in quantum optics since such CV cluster states are achievable using traveling squeezed states through optical parametric amplifiers [4][5][6][7] . For example, the recent development of creating one-dimensional (1D) and 2D CV cluster states has been demonstrated in quantum optics using quantum memory and in time/frequency domain [8][9][10][11] . In these methods, a phase-space translation operator is in general used for single-qubit gates while a two-qubit controlled-Z gate is implemented in a sequence of beam-splitters 12,13 . Toward fault-tolerant CV MBQC using this approach, a scheme of high squeezing photons (20.5 dB) has been required to reach the error tolerance threshold with 10 −6 through concatenated codes 14 , and is very challenging with the state-of-the-art experiments in quantum optics. Recently, an alternative method of creating four-qubit CV cluster states has been suggested in a circuit quantum electrodynamics (circuit-QED 15 ) system 16 .
One of the advantages of using CVs is that the optical cluster states are built in a deterministic manner and can in principle store information in infinite dimension [17][18][19][20] while alternative optical methods of creating discrete-variable

Results
Circuit-QED architecture for entangled cavity states. The platform of superconducting circuits has been rapidly developed for QI processing over two decades 38 . The artificial qubits are intrinsically scalable and manufacturable in the forms of different qubit types with precise control of desired parameters [39][40][41][42][43] . In experiment, one utilises only superconducting qubits (mainly transmon qubits 39,41 ) for QI unit while it has also been successfully shown that a coupled system of superconducting qubits and 3D cavities offers excellent capability of creating quantum cavity states through the nonilnearity of an intermediary superconducting qubit, e.g., deterministic generation of Schrödinger cat states and entangling CV states inside the cavities 44,45 .
As shown in the left figure of Fig. 1, we consider a circuit-QED architecture for creating entangled microwave states and the neighboring cavities are connected with each other via a middle transmon qubit M i (i = 1, 2) enabling to entangle cavity states. This approach shows a unique advantage that a massive 1D CV-qudit cluster state can be built in one step as the key resource state for MBQC. Since two cavities are simple harmonic oscillators, a superconducting qubit inserted in between two cavities brings induced Kerr effects on the joint cavity modes. For an ideal case, it is assumed that two neighboring cavities are only coupled by a cross-Kerr interaction, which is induced by the intermediary superconducting qubit.
In a real circuit-QED setup, this architecture might cause unwanted nonlinear effects over the cavities (e.g., self-Kerr distortion effects and non-identical cross-Kerr effects). In general, the cavity self-Kerr effect makes the amount of distortion in the cavity state and could prevent building ideal CV-qudit entangled states and to measure the cavity qubit accurately at an appropriate time. For example, let us consider the Jaynes-Cummings (JC) Hamiltonian for two cavities with an intermediary transmon is given by Figure 1. Schematics of logical MBQC in a circuit-QED architecture. (Left) Three cavities (A, B, C) have the intersected superconducting qubits M 1 and M 2 used for inducing the Kerr interactions between cavities. When a 3-qudit logical cluster state is built in the cavities by cross-Ker interaction (K ij ), logical MBQC is performed by a sequential measurement of each cavity. The colours of transmons energy states represent the anharmonicity of the energy levels in a transmon. (Right) the tunability of Kerr effects between the neighbouring cavities provided with the help of tunable on-site superconducting qubits and an extra (tunable) intermediary qubit in the same architecture (the details are shown in 29,64 ). For example, the self-Kerr effects can be only reduced by shifting energy levels in on-site qubits at point (a) and the simultaneous entangling gates are performed by cross-Kerr K ij between (a,b). From (b) to (c), the cavities are uncoupled and the sequential measurements of each cavity are performed for MBQC.
The generalised Pauli operators for the qudits are defined by ). Note that the normalisation coefficients α M i are approximately equal to 1/2 for α ≥ 2, which implies the validity of orthogonality in qudit |k 4 〉 for QI unit (k = 1, 2, 3, 4). In other words, if the average photon number is large enough to distinguish between coherent states, the qudits can be treated as logical qubits against photon-loss errors (details in Section 2.5) 27 .
How to create ideal three CV-qudit cluster states. We here show an mathematical description of how to build 1D CV-qudit cluster states with an ideal cross-Kerr interaction 49,50 . The cross-Kerr interaction shows a natural way to entangle two coherent states (see the details in Section 4.1). For a three-cavity case, an initial state |ψ int 〉 ABC is prepared in three cavities and a time-evolved state at time t is given by The cross-Kerr Hamiltonian is ideally given in With the assumption K AB = K BC for simplicity, the three CV-qudit state at a quarter of the revival time is written in It could be crucial to match the strength values of two cross-Kerr interactions between neighbouring cavities (K AB = K BC ) to create the target state in Eq. (7). Otherwise, the cavity state becomes maximally entangled in A and B at a certain time but it does not in B and C. In ref. 29 , a slight modification of the circuit-QED architecture has been investigated with additional superconducting qubits to control self-and cross-Kerr interactions independently. This modified architecture might thus be beneficial for building a multi-partite entangled state in many cavities at once toward practical MBQC.
Three single-qudit gates in cavity states. For logical MBQC, three specific single CV-qudit operations are required in each cavity such as (1) coherent-state projection P Coh , (2) parity measurement P Par , and (3) SNAP phase gates. Note that all the gates have already been demonstrated in a qubit-cavity architecture experimentally. In a dispersive regime of the JC Hamiltonian, which is defined by much smaller coupling strength than the difference between cavity and qubit frequencies, it is feasible to perform the projection measurement on Fock states in a cavity-transmon coupled system (see the details in Section 4.2).
To describe the operations, we define an arbitrary CV-qudit state |Ψ 4 〉 A given in cavity A by First, the projection set of a coherent-state is given by 4 and is viable in a microwave cavity coupled with a superconducting qubit and a readout resonator 24 for | 〉〈 |   0 0 4 4 and the definitions and details are presented in Section 4.2. Second, a QND parity measurement of cavity states has been successfully demonstrated with the assistance of an ancillary superconducting qubit in ref. 51 . The cavity state is projected on the even-or odd-photon subspace such as 4 and its parity is imprinted in the state of an ancillary readout qubit. For example, the state |Ψ 4 〉 A is collapsed by the parity measurement into with the outcome of the qubit state in |e〉 or 4 with |g〉. Therefore, the cavity state is projected in either the even-or odd-photon subspace through the parity measurement performed by the readout qubit. (see details in Section 4.3).
Finally, the SNAP gate is essential for performing photon-phase operations for CV-qudits and originally designed for the correction of phase distortion induced by self-Kerr effects 32 . The injection of a group of microwaves into a cavity induces a sum of the phase-rotation gates on each photon-Fock state |m〉 given by In our scheme, four groups of microwaves are applied due to d = 4 to obtain the same phase rotations on each |k 4 〉 (k = 0, 1, 2, 3) and the grouped phase gate is acheived on each |k 4 〉 independently. For example, if we apply the SNAP operation with four-group phase gates, e.g., In particular, we utilise two specific SNAP gates for logical phase gates. The first is a parity-conditional phase The other gate is given by , which is applied to only selected Fock states with where Schrödinger cat states are given with The two types of logical qubits span only either even-or odd-photon states and a photon-loss error can be monitored and corrected by the real-time parity measurement on the final state 51 .
For example, let us assume that a logical qubit is encoded in , which implies that the information of an arbitrary single qubit can be written in even photon subspace as a logical state. By real-time parity measurements, the cavity state is monitored through a superconducting qubit coupled with a readout resonator. Before cavity photon-loss, the parity measurement always results in the even state |Ψ 〉 e L . If the parity changes from even to odd, the updated logical state is equivalent to . Thus, the parity change tells us that the quantum information is preserved against photon-loss but the relative phase is altered.
Logical single-qubit gates in a three-qudit cluster state. The essence of MBQC is to create a designed multipartite entangled state initially and to apply sequential measurements on individual qubits will operate one-and two-qubit gates for universal quantum computing 2,3 . We now propose a specific protocol to perform a modified MBQC protocol from a three CV-qudit entangled state |3CS 4 〉 ABC given in Eq. (7)  www.nature.com/scientificreports www.nature.com/scientificreports/ MBQC from a three-qubit cluster state is described in Section 4.5. The CV-qudit measurement schemes are all experimentally viable for logical MBQC using the photon-loss error-correcting code 27,28 .
The first step is to determine the photon parity in the cavity state of the final outcome using the parity measurement on B from Eq. (7). Although any alternative implementation of building |ψ ideal (τ r /4)〉 ABC is applicable for our initial CV-qudit states (e.g., a scheme in ref. 44 ), we simply assume that |ψ ideal (τ r /4)〉 ABC is initially prepared by a cross-Kerr interaction among the cavities. Then, after the decoupling of all the Kerr-interactions (see Fig. 1), the middle cavity state is projected by the parity measurement such as ψ τ | 〉 P ( /4)

B par ideal r
A BC and is given in the even or odd parity state on B such as Note that this is the only initialisation operation on B to choose the parity of the outcome state, and we do not touch the cavity state in B afterwards. Without loss of generality, we will assume that the state is subjected in , however, the odd parity case is identical except the definition of logical qubits given in We now consider the cavity operations in A and C with two parameters (θ 1 and θ 2 ), which make desired single-qubit gates on B. Because of the technical limitations of cavity measurements in real experiment, it is not feasible to directly perform a single-cavity measurement in |±θ〉 〈±θ|. However, we theoretically suggest an alternative measurement scheme consisting of a logical single-qubit phase operation and a cavity measurement along the logical Z-axis because this alternative is equivalent to the measurement in |±θ〉〈±θ| ∝ R Z (−θ)|±〉 〈±|(R Z (−θ)) † . To implement the logical phase gate, SNAP gates are used for encoding the desired operations on logical qubits. More precisely, two SNAP gates, on C, are applied for mimicking a single-qubit phase gate with θ 1 . Note that Ŝ p1 is a parity-conditional phase gate as shown in Section 4.4 and the phase information is embeded in the three CV-qudit state Because the SNAP gates of θ 1 are QND operations, the total cavity state is not collapsed into a single cavity state yet.
In the next step, phase θ 2 is imprinted by θ S ( ) Although we showed a preferred sequence of SNAP gates performed by Ŝ p1 on A and C first and Ŝ p2 on C second, one can choose an alternative sequence depending on each cavity (e.g., Ŝ p1 on A first and ˆŜ S p p 2 1 on C second).
Finally, we are able to gain the designed logical state in B from |Out 3 (θ 1 , θ 2 )〉 ABC given by two cavity measurements (parity and coherent-state measurements) on A and C. When we perform the parity measurement on A, the resultant state is equal to . The state for the even parity is given in Then, if we project the qubit C by the coherent state-measurement α , , } as shown in Section 4.2, the successful detection gives the logical qubit in   where  z and  are a logical rotation gate on z-axis and a logical Hadamard gate defined by logical qubits in |0 L 〉 and |1 L 〉. Note that a repeat-until-success method can be used for approximated orthogonal projection of the cavity states on the measurement set of {|α〉〈α|, |iα〉〈iα|, |−α〉〈−α|, |−iα〉〈−iα|} for large α ≥ 2. The details of logical gates with respect to each measurement outcomes are presented in Table 1, which shows one-to-one correspondence with the orginal MBQC operations with three qubits in Table 2. Note that logical Pauli operators ≡Ẑ X ( ) Implementation of a two CV-qudit state in the JC Hamiltonian. We here mainly examine how to build two-qudit entangled states in the model of the JC generalised Hamiltonian (Ĥ ABM JC ), which describes the nonlinear effects given from the contribution of the intermediary transmon qubit (upto the third level). From the JC Hamiltonian in Eq. (1) with two coherent states, the total state in the two cavities with the qubit evolves in time and the state of two cavities are given by

Outcome
Logical gate Outcome Logical gate ).
Outcome state in C Single-qubit operations   35)), one may obtain the fidelity between the two states, however, this value might not represent the characteristics of the time-evolved state ρ t ( ) AB JC 0 because the distortion of the cavity state from the self-Kerr effects suppress the fidelity very low. In the spirit of MBQC, one of the simple verifications of the measured states is to compare between the projected cavity states of ρ AB JC and of |ψ ieal 〉 AB . In Fig. 2(a), the state in A is given by ρ tr t ( ( )) B AB JC 0 , in which we expect to obtain the mixture of four coherent states at t 0 = 39.45 μs. From (b) to (f), we plot the Wigner functions of the cavity state in mode A (B) at t 0 given by the projection of the certain states in mode B (A) such as for k = 0, 1, 2, 3. In the bottom of Fig. 2, we show that the maximum fidelity is approximately 0.978 at t 0 ≈ 40 μs in (b) and some levels of self-Kerr distortions occur during the time evolution from (c) to (f). We neglect decoherence processes in the cavities since the state-of-the-art lifetime of a 3D cavity is above 1.2 ms and the decoherence is expected to be not dominant until the period t 0 ≈ 40 μs. Apparently, this period of creating multi-partite microwave entangled state could not grow up much with increasing the number of cavities.

conclusion and Remarks
In summary, we introduce a new-type of CV logical MBQC in three microwave cavities coupled with superconducting qubits in a circuit-QED system. After the CV-qudits are defined, three specific circuit-QED gates are introduced to realise logical gate operations for the protocol of logical MBQC. We deliver the method of a logical single-qubit gate in photon-loss correcting codes from the three CV-qudit entangled state. Finally, the implementation of the two CV-qudit state and measured cavity states are numerically investigated under the JC Hamiltonian in a two-cavity system coupled with a superconducting qubit. The results show that the entangled CV-qudit states can be efficiently built with high fidelity via the cross-Kerr effect induced by the intermediary superconducting qubit between cavities.
Although the main goal of this paper is demonstrating the feasibility of the scheme in superconducting circuits, the improvement of the fidelity is necessary for wider range of quantum computing applications. For example, it has been known that a high-fidelity operation with a few percentage errors is only acceptable with specific quantum codes for full fault-tolerant quantum computation 52 . Some applications for noisy intermediate-scale quantum computation, e.g., digital quantum simulation, are open with the level of fidelity because it requires only to calculate the expectation values of quantum operators. In addition, there is an important technique called the error-mitigation scheme, which could provide ideal expectation values (without errors) by using an extrapolation method 53 . Since we only optimized the evolution time for two-qubit gate fidelity, all parameters (e.g., cavity frequencies and interaction strengths) can be tuned to achieve higher fidelity than we demonstrated (0.977). These higher fidelity issues and a full analysis of the associated parameter optimizations will be addressed in our future work.

Methods
How to build two CV-qudit states. When an initial state |ψ int 〉 AB is prepared in cavities A and B, the timeevolved state at time t is given by , and the evolved state is in general written in an entangled (inseparable) state between two modes at t ≠ τ r . For t = τ r /d, it is given by For example, for t = τ r /2 with |ψ int 〉 AB = |α〉 A |α〉 B , the state evolves such as This state is known as an entangled coherent state 49,50 , which is also of excellence for quantum metrology and other QI processing methods [54][55][56] and has been recently demonstrated in a deterministic method in circuit-QED 57 and probabilistically in quantum optics 58,59 . In fact, the entangled coherent state can be used as a simplest resource state for MBQC with no error-correction because CV quantum teleportation, which is the building block for MBQC, has been demonstrated in quantum optics [60][61][62][63] and investigated in circuit-QED 64 . The similar method of implementing the states has been suggested with the assumption of the cross-Kerr interaction in a circuit-QED system 65 .
For d = 4, the desired evolution time is the half period of |ψ ideal (τ r /2)〉 AB . The evolved state at t = τ r /4 is written by where η e i n is an undesired operation in π R m ( , ) A J y due to self-Kerr interaction but does not influence our result because we only use the outcome state |e〉 in a heralded way 45 . Then, when the outcome is measured in |e〉 J , the cavity state is also projected in |M〉 A and the operator of this Fock-state projection on the m-th photon is given by π = ⊗| 〉 〈 | .P m e e R m ( ) (1 ) ( , ) www.nature.com/scientificreports www.nature.com/scientificreports/ A coherent-state projection can be also performed by adding displacement operation = α α α − − ⁎ † D e a a on cavity states 45 . the coherent-state projection on |α〉 is given in As shown in Table 2, this protocol is equivalent to two single-qubit rotations and two sequential projective measurements on A and B. Thus, this procedure of MBQC is equivalent to the operation of two single-qubit gates with phases θ 1 and θ 2 .