Deterministic bidirectional communication and remote entanglement generation between superconducting qubits

We propose and experimentally demonstrate an efficient scheme for bidirectional and deterministic photonic communication between two remote superconducting modules. The two chips, each consists of a transmon, are connected through a one-meter long coaxial cable that is coupled to a dedicated “communication” resonator on each chip. The two communication resonators hybridize with a mode of the cable to form a dark “communication mode” that is highly immune to decay in the coaxial cable. We overcome the various restrictions of quantum communication channels established by other recent approaches in deterministic communication for superconducting qubits. Our approach enables bidirectional communication, and eliminates the high insertion loss and large volume footprint of circulators. We modulate the transmon frequency via a parametric drive to generate sideband interactions between the transmon and the communication mode. We demonstrate bidirectional single-photon transfer with a success probability exceeding 60%, and generate an entangled Bell pair with a fidelity of 79.3 ± 0.3%. Quantum information can be passed between qubit devices by using nonlinear interactions to control transmission through a connecting cable. The construction of quantum networks and larger-scale quantum computers requires interconnections that can coherently transfer quantum information over long distances and between separate computing modules. Recent experiments have used controlled emission and absorption of microwave photons to produce one-way transmission between superconducting circuits. Nelson Leung and Yao Lu from the University of Chicago, with collaborators in the USA, have demonstrated two-way communication through a one-metre long coaxial cable. Manipulating the nonlinearities of their superconducting circuits via external controls allows the coupling between the qubits and the cable, and hence the inter-module transmission, to be turned on and off as necessary. This approach avoids some of the drawbacks of other quantum communication solutions with one-way transmission.


INTRODUCTION
A practical quantum computer requires a large number of qubits working in cooperation, 1 a challenging task for any quantum hardware platform. For superconducting qubits, there is an ongoing effort to integrate increasing numbers of qubits on a single chip. [2][3][4][5][6][7][8] A promising approach to scaling up superconducting quantum computing hardware is to adopt a modular architecture [9][10][11] in which modules are connected together via communication channels to form a quantum network. This reduces the number of qubits required on a single chip, and allows greater flexibility in reconfiguring and extending the resulting information processing system. In such an architecture, each module is capable of performing universal operations on multiple-bits, and neighboring modules are connected through photonic channels, allowing communication and entanglement generation between remote modules.
Remote entanglement between superconducting qubits has been realized probabilistically. [12][13][14] A deterministic quantum communication channel is advantageous over a probabilistic one because it lowers the threshold requirement for fault-tolerant quantum computation and can achieve higher entanglement rates. 15 Realizing deterministic photonic communication requires releasing a single photon from one qubit and catching it with the remote qubit. In the long-distance limit, the photon emission and absorption are from a continuum density of states. In this limit, static coupling limits the maximum transfer fidelity to only 54%. 16,17 This limit is exceeded by dynamically tailoring the emission and absorption profiles. [18][19][20][21] These capabilities are presently being used to perform photonic communication between superconducting qubits connected by a transmission line within a cryostat. 14,[22][23][24] In these experiments, the use of a circulator enables the finite-length transmission line to be modeled as a long line with a continuum density of states, at the cost of added transmission loss.
Here, we establish bidirectional photonic communication between two superconducting qubits through a multimodal communication channel. Rather than inserting a circulator, the multimode nature of the finite length transmission line is made manifest and exploited. 25 For intra-cryostat communication, the required connection coaxial cable length of 1 m or less results in a free spectral range on the order of hundreds of MHz. In this setting, the resonances of the coaxial cable form hybridized normal modes with on-chip communication resonators, and photons are transferred coherently through the discrete modes of the channel in contrast to emission/absorption through a continuum. We use parametric flux modulation of the qubit frequency to generate resonant sideband interactions between the qubit and the communication channel. [26][27][28][29][30] This approach avoids the loss due to the circulator that significantly limits the communication fidelity, and enables bidirectional quantum communication. Moreover, avoiding large volume footprint circulators facilitates the establishment of communication channels between multiple devices within a cryostat.

Bidirectional communication
To demonstrate photonic communication between the two chips, we send a single photon from one chip to the other. First, we excite the sender qubit, then we switch on sideband interactions simultaneously on both qubits, targeting the communication channel. This effectively creates a Λ-system between single photon states of the qubits and the communication mode. We send a photon in the reverse direction using the same sideband sequence but instead exciting the other qubit, thus demonstrating bidirectional photon transfer. Figure 1 shows the transmon population plotted as a function of the sideband pulse length. The master equation simulation results (solid lines) are shown along with the experimental data (dots). We are able to obtain photon transfer with a success rate of {P |ge〉 , P |eg〉 } ≈ 61%. We use simultaneous square pulses for the time-envelopes of the sideband interactions. From the simulations detailed in the appendix, we found that square pulses gave superior performance for our current circuit parameters. Note that the achieved transfer fidelity exceeds 54%, the maximum fidelity for absorbing a naturally shaped emission into a continuum. 16,17 This demonstrates a qualitative difference in transferring via a multimode cable compared to that of releasing and catching flying photonic qubits through a continuum.
The transfer fidelity is limited by qubit dephasing and photon decay in the communication mode. Qubit 1 has a higher dephasing rate T Ã 2 % 700 ns À Á than qubit 2 T Ã 2 % 1:4 μs À Á . The dephasing rate of qubit 1 is comparable to the sideband coupling rate, with the result that this qubit is not able to fully release its excitation during the transfer process. Conversely, for transfer in the other direction qubit 1 is not able to receive all of the excitations. This transfer infidelity can be largely mitigated by using a fixed-frequency qubit less susceptible to the flux noise, with its coupling strength to the communication mode parametrically controlled via a tunable coupler circuit. [29][30][31][32][33] The remaining loss of transfer fidelity comes from the loss in the communication mode. From our numerical simulations detailed in the appendix, we estimate that the overall photon loss in both the qubits and the communication mode contribute to an infidelity of 24%, while the dephasing error of the two qubits accounts for an infidelity of 15%. The sideband coupling rate of the transmon is limited by the range over which its frequency can be parametrically tuned, resulting in a maximum effective sideband coupling to the communication resonator of ≈2 MHz. With improved qubit coherence time, our simulation shows that more sophisticated transfer protocols such as STIRAP 34,35 can be employed to boost transfer efficiency.
Bell state entanglement We now entangle two qubits by creating a Bell state between the transmons on the respective chips. 36 We can create such a state by first applying the ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi iSWAP p gate between the excited qubit 1 and the communication mode, which generates the Bell state g1 j i þ e0 j i ð Þ = ffiffi ffi 2 p between them. We implement the ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi iSWAP p by applying a sideband modulation pulse to qubit 1 to perform a π/2 rotation. Subsequently, we transfer the state of the communication mode to qubit 2 through the iSWAP gate by applying a sideband modulation pulse to the latter to perform a π rotation. Ideally this sequence prepares the Bell shared between the two remote qubits. To minimize decoherence the sender and receiver pulses can be applied simultaneously, so long as the lengths and amplitudes of the pulses are adjusted appropriately. Choosing qubit 1 as the sender and using square pulses, we found-both in our simulation and in the experiment-that maximal fidelity was obtained by setting both pulses at the same coupling rate and the length of the receiver pulse to be slightly longer than twice that of the sender, demonstrated by the pulse sequence diagram in Fig. 2b. The resulting Bell state has a fidelity of Ψ þ h jρ exp Ψ þ j i= 79.3 ± 0.3%. We obtained the density matrix ρ exp using quantum state tomography with an over-complete set of measurements complemented with the maximum likelihood method, 37 and we corrected the measurement error by constructing a confusion matrix (detailed in the appendix). It can be inferred from the data that the fidelity is almost equally limited by photon decay in the cable and the qubit dephasing errors. We also note that the Bell state fidelity is significantly higher than the success probability we achieved for photon transfer. This is because the superposition state that is transferred has less participation in the communication mode which is the primary source of loss. The dephasing losses on both qubits are also reduced due to either shorter interaction time or lower average excitations.

Photonic communication by parametric interaction
Each chip consists of a single flux-tunable transmon and two additional resonators. 38 The first of these resonators is used for readout, and the second is coupled to the coaxial cable to enable the inter-module communication. The transmon can resonantly couple to the communication resonator through parametric flux modulation to realize inter-module photonic communications. Figure 3  The inset at top right shows the pulse sequence used to implement excitation transfer. The labels c1, c2 denote the charge drives on qubits 1 and 2, respectively, and f1, f2 the respective flux drives. We first apply a π pulse to excite one of the qubits, then simultaneously switch on the sideband flux pulse to drive the transfer process. Using the same sideband sequence, but instead applying the π pulse to the other qubit, we can send a single photon in the opposite direction. The transfer fidelity is limited by qubit dephasing and photon decay in the communication mode. Described in the following, the transfer process in different directions have slightly different loss mechanisms. a Excitation transfer from qubit 1 to qubit 2. Notice that in this transfer process the sender qubit is not able to fully receive its excitation (population of |eg〉 does not reach zero). As confirmed by the master equation simulation, this is due to the dephasing of qubit 1. The remaining errors arise from communication cavity loss and dephasing of qubit 2, which is less than that of qubit 1. b Excitation transfer from qubit 2 to qubit 1. In this process, while qubit 2 releases most of its excitations (population of |ge〉 comes close to zero), the dephasing of qubit 1 prevents it from capturing all the excitations in the communication mode, resulting in a slightly higher final population in |gg〉. The resulting fidelities for the transfer in the two directions are similar: {P |ge〉 , P |eg〉 } ≈ 61%, confirming the results from our numerical simulation eigenmode frequencies spanning from 5.8 to 7.7 GHz. These quantum memory modes can be utilized for universal quantum computation on each module locally but are not used in this work. We induce resonant interactions between the transmon and the communication channel by modulating the transmon frequency via its flux bias. The modulation creates sidebands of the transmon excited state, detuned from the original resonance by the frequency of the applied flux tone. When one of these sidebands is resonant with a mode, the system experiences stimulated vacuum Rabi oscillations. 39 This process is similar to resonant vacuum Rabi oscillations, 40 but occur at a rate that is controlled by the modulation amplitude. 27,28 To illustrate the application of parametric control, we employ the following experimental sequence. First, the transmon is excited via its charge bias. Subsequently, we modulate the flux bias to create sidebands of the transmon excited state at the modulation frequency. This is repeated for different flux pulse durations and frequencies, with the population of the transmon excited state measured at the end of each sequence. When the frequency matches the detuning between the transmon and a given eigenmode, we observe fullcontrast stimulated vacuum Rabi oscillations. Figure 4 shows that the transmon can selectively interact with each of the eigenmodes of the communication channel by choosing the appropriate modulation frequency. The photon transfer process between two remote qubits is initiated by switching on the sideband interactions targeting the communication resonator on each chip. As the bare frequencies of the transmon and the communication resonator are far detuned (Δ ≈ 3 GHz, g ≈ 50 MHz), the sideband coupling scheme for photonic communication achieves a high on/off ratio. The following section explain the details of the multimode nature of our communication channel.

Multimode communication channel
The two communication resonators are designed to have identical frequencies. They are chosen to be coplanar waveguide resonators with a large center pin and gap width to make the frequency insensitive to fabrication variations. 41 These resonators are coupled via the one-meter long coaxial cable, where the cable can be thought of as a multimode resonator with a free spectral range of around 100 MHz. The coupling strength between the cable and the communication resonators is g l ≈ 7 MHz. The cable mode that we use for communication has a frequency that is within g l of the frequencies of the communication modes. Since the free spectral range of the coaxial cable is an order of magnitude larger than g l , we consider the cable as a single mode nearly resonant with the communication resonators. The cable and the communication resonators thus together produce three hybridized normal modes which are depicted in Fig. 4. The near-degeneracy of the coaxial cable with the two communication resonators give rise to this almost equally-spaced threemode structure, which can be seen from the three stimulated vacuum Rabi chevrons in Fig. 4b. The center normal mode used for communication ideally has no participation in the cable mode, and as a result, its loss rate is limited by the internal quality factors of the communication resonators and small Purcell losses from neighboring cable modes. The transfer fidelity is ultimately limited by the single pass loss of the cable (see Appendix). In comparison to the neighboring modes, the center normal mode couples more strongly to both qubits due to higher wavefunction participation at the communication resonators. Thus, this communication mode has both the advantages of high-quality factor and high coupling rate. For any practical device, the center normal mode does have a nonzero participation in the lossy coaxial cable due to a frequency mismatch between the two on-chip communication resonators. For our device, we found this frequency mismatch to be 5 MHz (detailed in the appendix), resulting in a less than 7% of cable mode participation in the communication mode.
The coherence time of the communication mode can be characterized using protocols analogous to those for the transmon. For instance, for mode T 1 one would first excite the transmon, then apply an iSWAP to transfer the excitation to the mode. After some variable time, another iSWAP is applied to transfer the excitation back to the transmon and we measure the transmon state subsequently. 39 We find T 1 = 550 ns and T Ã 2 = 1 μs, corresponding to a quality factor of about 27,000. This quality factor is b Expectation values of two-qubit Pauli operators. We create a Bell state between two remote qubits, one on each module. This is achieved by first applying the ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi iSWAP p gate between the excited qubit 1 and the communication mode, which is implemented by a sideband modulation pulse to qubit 1 to perform a π/2 rotation. A similar pulse, this time a π rotation, applied to the second qubit performs an iSWAP that transfers the entanglement from the communimation mode to the second qubit. As shown in the inset, we implement the two pulses simultaneously to reduce decoherence. We obtain the resulting Bell state with fidelity Ψ þ h jρ exp Ψ þ j i= 79.3 ± 0.3% reasonably high, considering that it involves losses from the long lossy cable, wirebonds, solder of the SMA connector, and the copper leads of the sample holder. The two neighboring normal modes have much lower coherence times due to the higher participation of the lossy cable mode. From fitting to Fig. 4b we estimate the T 1 for the bare communication resonator and the bare cable mode to be~5 us and~40 ns, respectively (see Appendix). The dark state communication mode explored in this work can thus be utilized to improve transfer efficiency. In the following, we demonstrate the use of the dark state communication mode for bidirectional photon transfer and Bell-state generation.
We have realized photonic communication between two remote modules, a first step in realizing a modular network. The sideband modulation of the transmon qubit in each module can be applied to implement local operations on the multimode resonators and to perform photon transfer between the two modules. The multimode characteristic of the communication channel (a coaxial cable) is enabled by the absence of a circulator. This mode structure results in normal modes that are superpositions of a mode of the inter-module communication cable and the on-chip resonators. One of the these normal modes is "dark" to the coaxial cable mode, thus avoiding much of the cable loss and allowing for high fidelity photon transfer. We characterized our system by performing single photon transfer with 61% fidelity and Bell-state preparation with 79.3% fidelity. These fidelities can be increased by improving the qubit coherence time and the strength of the coupling to the communication channel. This work sets the stage for future exploration in high fidelity and scalable quantum communications. Fidelity can be further improved by implementing more sophisticated photon transfer protocols (e.g., STIRAP). Another exciting direction is scaling to multiple communication channels on the same cable, and exploring the crossover between distributed modes and the continuum as the cable gets longer. Finally, one can employ heralding, 42,43 and exploit the ability to perform high fidelity local gates in conjunction with photonic communication to distill entanglement [44][45][46] or perform forward error correction. 47

DATA AVAILABILITY
Data available on request from authors.