⇦Chinese Physics Letters, 2023, Vol. 40, No. 11, Article code 110303Viewpoint Continuous-Variable Quantum Computation in Circuit QED Xiaozhou Pan1†, Pengtao Song1†, and Yvonne Y. Gao1,2* Affiliations 1Centre for Quantum Technologies, National University of Singapore, Singapore 2Department of Physics, National University of Singapore, Singapore Received 25 August 2023; accepted manuscript online 8 October 2023; published online 13 November 2023 †These authors contributed equally to this work.
*Corresponding author. Email: yvonne.gao@nus.edu.sg Citation Text: Pan X, Song P, and Gao Y Y 2023 Chin. Phys. Lett. 40 110303    Abstract
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 DOI:10.1088/0256-307X/40/11/110303 © 2023 Chinese Physics Society Article Text Quantum computation (QC) offers the promise of enhanced computational capabilities and drastic accelerations in solving complex tasks such as quantum chemistry[1,2] and prime factorization.[3] Various physical implementations for practical QC are being pursued across the academic and industrial research initiatives. Exemplary ones include trapped ions,[4] nuclear spin,[5] quantum dot,[6] photonics,[7] etc. Among these options, the circuit quantum electrodynamics (cQED) platform has gained significant traction over the past two decades,[8] thanks to its robustness and versatility. An overview of QC with cQED platform is shown in Fig. 1. The key building blocks of cQED are harmonic oscillators, realized using superconducting resonators, and anharmonic elements based on Josephson junctions.[9] The engineered interactions between these two components provide the core capabilities of information encoding, storage, control, and measurement in cQED devices. The anharmonic elements, such as transmon,[10,11] flux qubit,[12-14] and fluxonium,[15] are natural candidates for discrete-variable (DV) physical qubits. In such devices, the harmonic oscillators are typically employed for efficient readout[16] or as quantum buses to mediate fast multi-qubit gates.[17-20] With increasingly sophisticated fabrication, control, and measurement techniques, cQED devices consisting of tens or hundreds of DV physical qubits have achieved many crucial milestones.[21-23] However, to realize a practical fault-tolerant QC,[24] the challenges of quantum error correction (QEC) must be addressed. A single physical qubit is susceptible to environmental noise, which inevitably leads to the rapid collapse of the encoded information.[25] Therefore, we must devise strategies to implement logical qubits in which the quantum information can be preserved even in the presence of noise. To this end, many DV-based logical encodings and error correction schemes have been proposed and implemented on small-scales.[26,27] In parallel, instead of encoding information on the multiple physical qubits of two-level system, i.e., DV encoding, an alternative continuous-variable (CV) approach that encodes information on the cavity utilizing infinite dimension of Hilbert space is also being pursued.[28,29] Bosonic QEC utilizes harmonic oscillators to encode logical qubits, and ancillary anharmonic elements to provide the error tracking and correction capabilities. In this Viewpoint, we will provide a brief overview of the main ingredients needed to achieve robust QC using such CV-based encodings, and highlight the remarkable progress that have been demonstrated towards this goal. Bosonic CV encodings in cQED that take advantage of multi-photon states of superconducting cavities have successfully realized the effective encoding of logical quantum information, robust control, as well as efficient error tracking and correction. Several bosonic codes for QEC have been demonstrated in recent years as shown in Fig. 2(a), including cat codes,[30-32] binomial codes,[33] Gottesman–Kitaev–Preskill (GKP) codes,[34-36] and others.[37-39] These codes have different error-sensing capabilities and are selected for QEC based on the specific local error channels. Cat codes involve superpositions of coherent states. The first QEC experiment beyond break-even point has been demonstrated using a four-component cat state, where the lifetime of logical qubit is approximately 1.1 times that of the best physical qubit.[40] Additionally, a two-component cat state together with manifold stabilization has been shown as a type of biased-noise code with high robustness against dephasing but vulnerability against photon loss.[41] Binomial codes exploit superpositions of Fock states weighted with binomial coefficients and can exactly correct the errors up to a specific degree.[42] A repetitive QEC implemented using binomial mode has been shown with a logical qubit lifetime 2.8 times longer than that of its uncorrected counterpart.[43] Most recently, the surpass of the break-even point with binomial codes has been demonstrated, resulting in approximately 16$\%$ lifetime enhancement.[44] GKP codes, consisting of superpositions of highly squeezed coherent states, are protected against small shifts and perform well in both dephasing and energy relaxation. An implementation of GKP code has been generated using a feedback protocol.[45] Subsequently, a real-time error correction technique is employed to realize a fully stabilized GKP code beyond the break-even point. The lifetime is more than doubled compared to the best passive qubit encoding.[46] Furthermore, an autonomous QEC using truncated 4-component cat (T4C) code has been demonstrated by adopting quantum dissipation engineering with neither high-fidelity readout nor fast digital feedback.[47] The utilization of those bosonic codes for QEC represents a remarkable step forward in the field of quantum information processing, which brings us closer to realizing practical and scalable quantum technologies.
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Fig. 1. Overview of key ingredients for universal QC with cQED. As is seen, cQED hardwares provide crucial elements such as high-performance physical qubits and cavities, high-fidelity gates and high-fidelity QND readout. With these elements being continuously improved, it is now feasible to develop tailored quantum processors built upon discrete physical qubits. The broader goal is to achieve fault-tolerant QC, which necessitates the implementation of QEC. Initial step involves encoding logical qubits and using QEC to bolster the performances of these logical qubits. Subsequently, operations are carried out on single or multiple logical qubits, followed by the execution of quantum algorithms to realize fault-tolerant QC. The encoding of logical qubits can be accomplished through two distinct methods: DV encoding (surface code etc.) and CV encoding (cat code, squeezed cat code, binomial code, GKP code, etc.). The latter has successfully demonstrated the ability to prolong the logical lifetimes of the encoded information.
Quantum control is another essential component of QC using bosonic cQED systems. Presently, researchers have demonstrated various control gates on CV quantum states encoded in cavities as illustrated in Fig. 2(b). Universal control plays a key role for manipulating quantum systems in a way that enables the execution of any quantum algorithm. One approach for universal control is through analytical method, which applies the cavity and qubit drives sequentially as a series of unitary gates. In strong dispersive coupling regime, a gate between a qubit and a cavity (qcMAP) which maps the qubit state onto a superposition of two quasiorthogonal coherent states with opposite phases[48,49] can be used for the generation of arbitrary superposition of coherent states while selective number-dependent arbitrary phase (SNAP) gate[50-53] enables the generation of Fock states. In weak dispersive coupling regime, which takes the advantage of low cavity self-Kerr, echo conditional displacement (ECD) gate[45,54-56] and conditional not displacement (CNOD) gate[57] have realized fast universal control. In multi-cavity systems, a controlled-NOT (CNOT) gate utilizing a parametrically driven sideband interaction entangles two encoded multiphoton qubits[58] and further enables teleported control.[59] Furthermore, bilinear coupling can be engineered to realize an exponential-SWAP (eSWAP) gate that entangles two bosonic modes regardless of the logical codes, such as realizing entanglement between Fock- and coherent-state encodings.[60,61] The eSWAP can be also reduced to a SWAP with different choices of parameters, facilitating state exchange.[62] Another alternative approach of universal control involves numerical optimization algorithms and simultaneously drives the cavities and qubits.[63-67] This numerical optimization technique can also be used to enhance the fidelity of analytical gates through pulse optimization.[68] Further efforts are also underway to make the control of CV bosonic states more robust against certain loss mechanism. For example, bias-noise gates based on Hamiltonian engineering (HE)[37,69-73] have been proposed and realized for 2-component cat codes. In addition, error-transparent (ET) gates that are non-sensitive to a certain noise counterpart have also been developed.[74]
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Fig. 2. (a) Progress in QEC $\&$ quantum control in bosonic cQED systems. Gain of logical lifetime is obtained from the ratio between logical qubit lifetime and single photon lifetime in the cavity. Blue dash line shows the break-even point. (b) Fidelity of different gates on bosonic modes demonstrated experimentally.
The advancements of QEC and quantum control in bosonic cQED systems are a powerful demonstration of the potential of realizing scalable and fault-tolerant QC using the CV-based strategy. Looking ahead, there are several avenues for further development. One aspect is hardware improvement. Firstly, improving the performance of physical elements, for instance extending the intrinsic coherence properties of ancillary qubits, is crucial. Through the dispersive coupling, decoherence channels of these ancillary elements can significantly degrade the encoding, operation, and measurement of the CV logical states in the cavities. To address this issue, there are many ongoing efforts focusing on enhancing the performance of these ancillary qubits though advanced materials[75,76] and the design of new architectures.[77] Secondly, scalability remains a challenge in CV regime. To overcome this, modularization and miniaturization of devices can be employed, where each bosonic mode encoded into a single cavity can be treated as a unit, individually protected through error correction. By connecting these units via on-demand channels,[78-80] large-scale quantum devices can be constructed with flexibility. With the ability to connect the individual CV devices endowed with first layer QEC, fault-tolerant device can be constructed by incorporating second-layer, concatenated QEC strategies.[81,82] This is another active and rapidly progressing area of research. As we move towards more complex bosonic cQED systems, we also need to more efficiently characterize them. This requires us to explore new concepts such as shadow tomography[83] and machine learning.[84] Finally, while a few algorithms have been implemented in DV regime,[22,85] there is still a need for more quantum algorithms applicable to practical problems. Overall, the success of all these endeavors requires extensive collaboration between experts of different domains, and bosonic cQED systems are an ideal testbed for these new ideas and techniques to be investigated and perfected. 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