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Quantum information processing with bosonic qubits in circuit QED
Atharv Joshi, Kyungjoo Noh, Yvonne Y. Gao
TL;DR
Conventional QEC can require many interconnected two-level systems, creating engineering challenges and potentially degrading device performance. This review examines bosonic codes in cQED, highlighting progress toward effective QEC and information processing with logical elements encoded in superconducting cavities.
Problem
Conventional QEC schemes can require tens of interconnected two-level systems, ancillary qubits, and measurement elements, making robust hardware difficult to construct.
Method
The article reviews theory and implementations of bosonic codes in cQED, including cavity encodings, logical operations, and multi-mode couplings.
Results
The review reports progress demonstrating effective QEC and information processing with bosonic logical elements in cQED.
Takeaways & Limitations
Bosonic codes in cQED show potential as hardware-efficient building blocks for robust universal quantum computing.
Takeaways & Limitations
Fault-tolerant operation remains constrained by ancilla decoherence, two-photon loss, Kerr effects, and photon-loss-induced logical errors.
Abstract
from arXiv · showhide
The unique features of quantum theory offer a powerful new paradigm for information processing. Translating these mathematical abstractions into useful algorithms and applications requires quantum systems with significant complexity and sufficiently low error rates. Such quantum systems must be made from robust hardware that can coherently store, process, and extract the encoded information, as well as possess effective quantum error correction (QEC) protocols to detect and correct errors. Circuit quantum electrodynamics (cQED) provides a promising hardware platform for implementing robust quantum devices. In particular, bosonic encodings in cQED that use multi-photon states of superconducting cavities to encode information have shown success in realizing hardware-efficient QEC. Here, we review recent developments in the theory and implementation of quantum error correction with bosonic codes and report the progress made towards realizing fault-tolerant quantum information processing with cQED devices.
I. INTRODUCTION
Quantum error correction protects logical information by encoding it in symmetry-structured physical systems, but conventional multiqubit implementations face substantial hardware complexity and have not consistently surpassed break-even. This review examines bosonic codes in cQED as a hardware-efficient route toward robust quantum information processing.
- Motivation: Quantum errors can accumulate and scramble information, motivating encodings that extract error syndromes without disturbing the logical qubit.QEC maps physical elements onto a logical bit with symmetry properties supporting error detection and correction.
- Motivation: The break-even point is reached when a logical qubit outlives the best single physical element, making it a prerequisite for fault-tolerant gates.The added elements and operations must introduce less degradation than the protection they provide.
- Limitations of conventional QEC: Conventional multiqubit QEC requires many interconnected qubits, ancillas, and measurement elements, which create engineering challenges and additional uncorrectable errors such as cross-talk.Proof-of-principle demonstrations based on two-level systems had not deterministically extended logical-qubit performance beyond the best physical qubit.
- Bosonic approach: Bosonic cQED encodings store logical information in multi-photon states of a single superconducting cavity controlled and measured through an anharmonic ancilla.The cavity's large Hilbert space enables compact encoding, while the ancilla supplies nonlinearity for control and measurement.
- Review scope and progress: Bosonic-code implementations have demonstrated break-even QEC, robust operations, and fault-tolerant error-syndrome measurement, motivating their review as building blocks for robust quantum computing.The article surveys code designs, cQED hardware, single- and two-mode control, error correction, fault tolerance, and scaling perspectives.
- Bosonic approach: The kitten code and 4-qubit code can offer comparable single-error protection, but the kitten implementation uses one bosonic mode and one ancilla instead of four data qubits, three ancillae, and additional cavity modes.The comparison illustrates the hardware-overhead advantage of bosonic encoding in cQED.
III. PERFORMANCE OF BOSONIC CODES FOR LOSS AND DEPHASING ERRORS
Bosonic modes in cQED are primarily affected by photon loss, while ancilla coupling can induce dephasing and non-Markovian behavior. Different bosonic-code families therefore provide distinct protection profiles against loss and dephasing.
- Error mechanisms: Photon loss is the dominant error channel in superconducting cavities, with its rate determined by the cavity's internal quality factor; intrinsic cavity dephasing is usually insignificant.The loss and dephasing rates are represented by κ and κφ, respectively.
- Error mechanisms: Ancilla absorption or emission can induce logical dephasing through dispersive coupling, with the rate depending on Γ↑, Γ↓, and χ.An ancilla transition rotates the logical qubit by an angle approximately χt while the ancilla remains in the resulting state.
- Error mechanisms: The Lindblad description models photon loss and dephasing using jump operators a and n = a†a, respectively.The dissipative evolution uses κ and κφ as the corresponding rates.
- Implementation limitations: Ancilla-induced dephasing is generally non-Markovian in typical experimental regimes, limiting the performance of some current bosonic-QEC implementations.This limitation arises even when the cavity itself has insignificant intrinsic dephasing.
- Code families: Rotation-symmetric codes are robust against both photon loss and dephasing, whereas GKP codes are highly robust against photon loss but susceptible to dephasing.Two-component cat codes instead suit dephasing-dominated systems when stabilized by engineered two-photon dissipation.
A. Rotation-symmetric codes: Binomial and cat codes
Rotation-symmetric bosonic codes use discrete phase-space rotations to structure photon-number encodings, with binomial and cat codes offering different protections against photon loss and dephasing.
- Rotation-symmetric codes remain invariant under discrete phase-space rotations and are stabilized by photon-number super-parity operators.
- Binomial codes: The modified binomial code protects against single-photon loss and single dephasing by satisfying the Knill-Laflamme condition for {I, a, a†a}.Its logical states have three photons on average, compared with two for the kitten code.
- Binomial codes: Using 120° rotational symmetry, a generalized binomial code detects single- and two-photon loss events and single dephasing errors.Its photon numbers are integer multiples of 3, and its coefficients enforce the relevant Knill-Laflamme condition.
- Cat codes: Four-component cat codes use coherent-state superpositions, while larger cat codes can improve protection against photon loss and dephasing.Six-component cat codes can detect up to two-photon loss events, but exploiting dephasing protection requires engineered multi-photon dissipation.
B. Translation-symmetric codes: GKP codes
GKP codes encode qubits through translation symmetry in oscillator phase space, enabling correction of small displacement errors while exposing important preparation and dephasing limitations.
- Square-lattice GKP codes encode a logical qubit using a harmonic oscillator stabilized by two commuting displacement operators.This permits simultaneous modular measurements of the position and momentum quadratures.
- Limitations: Ideal GKP states are infeasible, so practical implementations use finitely squeezed states with Gaussian envelopes; cQED realizations have reached 5.5–9.5 dB squeezing.
- Error correction: Square-lattice GKP codes correct displacement errors smaller than √π/2 by measuring quadratures modulo √π and applying the corresponding correction.
- Error correction: For large, highly squeezed GKP states, standard decoding fails under photon loss, whereas amplification followed by modular quadrature correction can recover excellent performance.
- Limitations: Highly squeezed GKP codes lack analogous techniques for dephasing, because small random rotations can produce large shift errors.
- Operations: GKP logical Pauli and Clifford operations can use displacement or Gaussian operations, leaving codeword preparation as the required non-Gaussian operation for universality.
C. Biased-noise bosonic qubits: Two-component cat codes
Two-component cat codes engineer strongly biased noise, suppressing one error type so higher-level QEC can focus on the dominant residual errors.
- Biased-noise bosonic qubits are engineered so one error type occurs much more frequently than others, simplifying the next QEC layer.
- Code mechanism: Two-component cat codes use coherent states |±α⟩ and can correct dephasing when |α| is sufficiently large.
- Code mechanism: Engineered two-photon dissipation can exponentially suppress logical bit-flip errors caused by dephasing in stabilized 2-cat codes.
- Limitations: Single-photon loss flips parity and creates an uncorrectable phase-flip error, while sufficiently strong two-photon dissipation suppresses loss-induced bit flips exponentially in α2.
- Limitations: Ancilla thermal excitation can rotate the cavity state and completely dephase the encoded qubit, limiting the logical lifetime of stabilized 2-cat codes.
- Gate implementation: Universal gates on 2-cat codes must preserve the engineered noise asymmetry, and a bias-preserving universal gate set has been proposed.
D. Comparison of various bosonic codes for loss and dephasing errors
Bosonic code suitability depends on the dominant physical noise: rotation-symmetric codes address loss and dephasing, GKP codes favor loss, and 2-cat codes favor dephasing or noise-biased architectures.
- Rotation-symmetric codes such as 4-cat and binomial codes are robust against both photon loss and dephasing errors.
- Translation-symmetric GKP codes can be highly robust against photon loss but remain susceptible to dephasing, making them suited to loss-dominated systems.
- Two-component cat codes correct dephasing well with engineered two-photon dissipation but cannot correct photon loss, while their noise bias can still simplify higher-level QEC.
- Limits: If photon-loss and dephasing probabilities become too high, channel quantum capacity vanishes and reliable logical encoding becomes impossible even with optimal QEC.
- Additional error channels: Photon gain occurs less frequently than photon loss in typical cQED, and codes designed for loss generally also provide robustness against gain.
IV. THE CQED HARDWARE FOR BOSONIC CODES
Bosonic cQED implementations require coherent harmonic modes for encoding and nonlinear ancillae for controlling and characterizing encoded information.
- Bosonic codes use coherent harmonic modes to encode information and nonlinear ancillae to control and characterize it.
A. Components of the cQED architecture
The cQED architecture combines linear oscillator cavities with tunable anharmonic modes, whose coupling enables photon exchange, readout, memory, and mediated interactions.
- cQED devices couple a linear oscillator mode with an anharmonic mode implemented using Josephson-junction-based few-level systems.
- Superconducting microwave cavities provide common oscillator architectures, including CPW, 3D rectangular, 3D cylindrical co-axial, and micromachined designs.
- Resonant coupling enables direct energy exchange, while dispersive coupling produces state-dependent frequency shifts between the cavity and artificial atom.
- Strongly coupled cavities support efficient artificial-atom readout, whereas weakly environment-coupled cavities serve as coherent quantum memories.
B. Coherence of superconducting cavities
Superconducting cavity architectures trade footprint, fabrication simplicity, and coherence, with 3D co-axial cavities leading current bosonic-QEC implementations and 2.5D designs offering compact high-coherence alternatives.
- B. Coherence of superconducting cavities: 2D cavities have small footprints and straightforward fabrication but typically achieve Qint of ∼10^5−10^6 because of dielectric, surface, and interface losses.
- B. Coherence of superconducting cavities: 3D cavities achieve Qint of ∼10^7−10^8 by storing energy in vacuum, with 3D co-axial cavities reaching 1.1 × 10^8.
- B. Coherence of superconducting cavities: 2.5D cavities combine compact geometry with vacuum energy storage, and optimized indium bump bonding has produced internal quality factors exceeding 300 million.
- B. Coherence of superconducting cavities: Ancilla integration is necessary for bosonic logical qubits but degrades cavity Qint because best ancilla coherence times of ∼50−100 µs are 10−20 times shorter than state-of-the-art cavities.
- B. Coherence of superconducting cavities: 3D co-axial cavities currently lead the architectures used to realize bosonic qubits.
- B. Coherence of superconducting cavities: Improved cQED hardware has supported bosonic logical-qubit demonstrations and operations involving both single and two bosonic modes.
A. Operations on single bosonic modes
Single-mode bosonic processing combines displacement, dispersive controlled phases, SNAP gates, parity mapping, tomography, and optimal control to prepare, manipulate, and characterize cavity states.
- A. Operations on single bosonic modes: Standalone cavity control begins with displacement, but displacement alone generates coherent states from vacuum and cannot selectively address photon-number states.
- A. Operations on single bosonic modes: Controlled phase shifts use dispersive coupling to apply ancilla-state-dependent phases and map cavity photon-number parity onto the ancilla.
- A. Operations on single bosonic modes: Parity-mapping time decreases with larger χ, but stronger χ also increases inherited Kerr nonlinearity that can distort encoded information.
- A. Operations on single bosonic modes: Wigner tomography uses displacement, ancilla rotation, and controlled phase shifts to reconstruct density matrices and characterize cavity processes.
- A. Operations on single bosonic modes: SNAP gates selectively apply phases to number states, and optimized displacements and phases can create arbitrary cavity states through destructive interference.
- A. Operations on single bosonic modes: SNAP operations require O(n^2) gates for an n-photon operation, motivating optimal-control methods as efficient general-purpose alternatives.
- A. Operations on single bosonic modes: Parity mapping, repeat SNAP operations, and concurrent cavity-ancilla optimized pulses support parity measurement, arbitrary-state creation, and universal single-mode control.
B. Operations on multiple bosonic modes
Operations between bosonic modes must preserve individual-qubit coherence while providing entanglement for universal control. Engineered frequency-conversion and eSWAP interactions address this challenge, although some demonstrated gates remain encoding-specific.
- 98% gate fidelity was achieved for a CNOT between two bosonic qubits using an ancilla-mediated sideband transition and conditional phase gate.The operation was customized for a selective set of code words and does not readily generalize to other bosonic encodings.
- Four-wave mixing in a Josephson junction provides code-independent bilinear coupling for programmable interferometry between otherwise isolated cavity modes.The coupling supports continuous-variable tasks including boson sampling, molecular vibrational simulation, and distributed quantum sensing.
- For GKP encodings, engineered bilinear coupling provides a deterministic entangling operation because linear or bilinear operations suffice for universal control.
- The eSWAP operation provides deterministic, code-independent entanglement by implementing a programmable weighted superposition of identity and SWAP operations.It has been demonstrated across Fock, coherent, and binomial encoding schemes in two bosonic modes.
- A single transmon coupled to a multimode superconducting cavity has enabled universal control and operations on tens of bosonic qubits.
C. Implementations of QEC
Bosonic QEC implementations use active measurement and feedback, autonomous engineered dissipation, or passive noise bias to protect encoded information. Experiments have reached or approached break-even, while fault-tolerant operation remains limited by error propagation and imperfect correction of multiple error channels.
- Break-even performance: Break-even has been approached or achieved in three cQED bosonic-code studies without post-selection.The break-even point compares the lifetime of the logical qubit with that of the longest-lived uncorrected |0, 1⟩ Fock-state encoding.
- Active QEC: Active QEC measures error syndromes and applies real-time corrections; four-component cat-code recovery improved lifetime beyond both the uncorrected state and |0, 1⟩ Fock encoding.For rotationally symmetric codes, photon loss flips parity, allowing parity measurements to detect the error syndrome.
- Autonomous QEC: Autonomous QEC uses tailored dissipation or auxiliary coupling, including two-photon dissipation that stabilizes cat states and suppresses bit-flip errors.Other demonstrations engineered photon addition or jump operations to correct photon loss without repeatedly probing the logical qubit.
- Fault-tolerance challenges: Current autonomous demonstrations generally address only one error type, while fault-tolerant operations remain constrained by ancilla decoherence, two-photon loss, and Kerr effects.Measurement cadence trades more accurate syndrome tracking against ancilla relaxation, two-photon loss, and self-Kerr phase accumulation.
- Passive QEC: Passive QEC exploits exponentially increasing phase-to-bit-flip bias with coherent-state size α in 2-cat codes, providing intrinsic protection without syndrome probing.The protection arises from deliberately asymmetric noise channels rather than active correction operations.
- Fault-tolerant computation: GKP-based fault-tolerant proposals require squeezing above 11dB for circuit-based computation or 8dB with post-selection in measurement-based schemes.The 8dB threshold comes with higher resource overhead because the measurement-based schemes use post-selection.
VII. OUTLOOK AND PERSPECTIVES
The outlook identifies logical-operation break-even as the next milestone for bosonic QEC, alongside scalable hardware integration. Bosonic codes are presented as compatible with hardware-agnostic implementations and modular architectures that can reduce cross-talk and local-failure risks.
- Logical-operation performance: The next milestone is break-even for single- and two-mode logical operations, requiring gate fidelities above those of the best uncorrected physical element.Break-even in both QEC and logical operations is described as a foundation for fault-tolerant gates and algorithms.
- Hardware scalability: Bosonic-code techniques developed in 3D cQED are hardware-architecture agnostic and can be adapted to planar or 2.5D devices as materials and fabrication improve.The outlook explicitly connects this adaptability to continued performance improvements in alternative device designs.
- Modular architectures: Modular bosonic architectures encode, protect, and optimize logical elements individually before connecting them through on-demand communication channels.The stated benefits include reduced cross-talk, robustness against local failure modes, and enhanced reconfiguration.