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Bosonic quantum error correction codes in superconducting quantum circuits

W. Cai, Y. Ma, W. Wang, C. -L. Zou, L. Sun

arXiv:2010.08699v1quant-ph

TL;DR

Quantum information requires QEC because environmental noise threatens reliable scalable computation, while qubit-based approaches face substantial overhead and scaling challenges. The article reviews bosonic QEC codes and their control techniques, applications, challenges, and outlook in superconducting circuits. It reports demonstrated break-even performance, universal control and fault-tolerant developments, alongside application-specific demonstrations and proposed extensions.

  • Problem

    Qubit-based QEC requires large physical-resource overhead and scaling, motivating alternative encodings that can protect quantum information against noise.

  • Method

    The article reviews GKP, cat, and binomial bosonic codes, their universal control and channel-simulation techniques, applications, challenges, and future directions.

  • Results

    Bosonic QEC has demonstrated break-even performance, while universal control, fault-tolerant operations, quantum simulation error suppression, and sensing beyond the shot-noise limit have also been demonstrated or estimated.

  • Takeaways & Limitations

    Bosonic codes can protect information from temporal, propagation, and gate errors, supporting longer storage, longer propagation, and deeper circuits in near-term applications.

Abstract

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Quantum information is vulnerable to environmental noise and experimental imperfections, hindering the reliability of practical quantum information processors. Therefore, quantum error correction (QEC) that can protect quantum information against noise is vital for universal and scalable quantum computation. Among many different experimental platforms, superconducting quantum circuits and bosonic encodings in superconducting microwave modes are appealing for their unprecedented potential in QEC. During the last few years, bosonic QEC is demonstrated to reach the break-even point, i.e. the lifetime of a logical qubit is enhanced to exceed that of any individual components composing the experimental system. Beyond that, universal gate sets and fault-tolerant operations on the bosonic codes are also realized, pushing quantum information processing towards the QEC era. In this article, we review the recent progress of the bosonic codes, including the Gottesman-Kitaev-Preskill codes, cat codes, and binomial codes, and discuss the opportunities of bosonic codes in various quantum applications, ranging from fault-tolerant quantum computation to quantum metrology. We also summarize the challenges associated with the bosonic codes and provide an outlook for the potential research directions in the long terms.

I. INTRODUCTION

Quantum information is fragile to environmental coupling, while qubit-based QEC faces substantial resource and scaling challenges. This review introduces bosonic modes as an alternative architecture and surveys their applications, experimental progress, and future directions.

  • Quantum states are vulnerable to environmental noise, making QEC necessary for reliable universal quantum computation.
  • Qubit-based QEC remains difficult because of large physical-resource overhead and challenges in scaling the number of qubits.
  • Bosonic modes support applications in quantum computation, communication, simulation, and metrology, including demonstrated QEC and fault-tolerant operations.
  • The article reviews GKP, cat, and binomial codes, their control techniques, applications, challenges, and future research directions.
  • Bosonic architectures store quantum information in harmonic modes while coupled qubits provide nonlinearity for control and readout.

II. BASICS OF QEC

QEC protects encoded quantum information by adding redundancy in an expanded Hilbert space and diagnosing errors through non-destructive syndromes. Bosonic modes offer a single-system alternative with large Hilbert space, but redundancy increases error exposure and requires careful control.

  • QEC encodes quantum information in a code space within an expanded Hilbert space and diagnoses errors through syndromes rather than directly measuring codewords.
  • Multi-qubit codes distribute encoded information non-locally, but scaling requires many physical qubits and introduces more error channels and non-local gates.
  • Bosonic QEC uses one harmonic oscillator’s infinitely large Fock-state Hilbert space to add redundancy within a single degree of freedom.
  • In the bosonic architecture, a coupled nonlinear element such as a two-level qubit enables arbitrary control and readout of the oscillator.
  • For single-photon-loss errors, photon-number parity or generalized parity provides the error syndrome that must be monitored continuously.
  • Adding redundancy with n average photons makes the error rate n times larger, although control and QEC may compensate through extended coherence.

III. QEC BASED ON BOSONIC CODES

The review focuses on three single-mode bosonic codes—GKP, cat, and binomial codes—and summarizes their theoretical foundations and experimental achievements.

  • The review concentrates on GKP, cat, and binomial codes based on a single bosonic mode.Their experimental achievements are summarized in a figure and table.

A. Cat codes

Cat codes encode logical information in coherent-state superpositions and use additional phase-space structure to detect and correct photon-loss errors. Their experimental development spans break-even QEC, universal logical control, and two-qubit gates, while stabilization and non-orthogonality remain practical challenges.

  • Cat-code construction: Two-component cat qubits use coherent states with opposite phases, while multicomponent cat codes add phase components for greater error tolerance.The two-component encoding alone lacks sufficient redundancy for photon-loss correction; four-component codes add separate code and error spaces.
  • Cat-code construction: At sufficiently large α, cat-code basis states can satisfy QEC conditions, with photon-number parity distinguishing code and error spaces.Parity provides a quantum-nondemolition error syndrome in circuit QED.
  • Challenges and stabilization: Cat-code stabilization requires replenishing energy because no-jump evolution shrinks coherent states, while the resulting non-orthogonality error cannot be fully corrected unitarily.Proposed stabilization strategies use engineered four-photon dissipation or a specific Hamiltonian, but relatively strong four-photon drives remain difficult experimentally.
  • Error correction: n/2 −1: n-component cat codes can correct photon-loss errors up to order n/2 −1, although larger average photon number is required for orthogonality.Pair-cat codes extend photon-loss protection across two modes under the stated single-mode-error condition.
  • Experimental progress: The four-component cat code was the first bosonic code to surpass break-even, and cat encodings subsequently demonstrated universal single-qubit control and a two-qubit controlled-phase gate.Its error-tracking property can eliminate the need for immediate recovery after each detected error.
  • Cat qubits: Stabilized cat qubits exhibit biased noise: bit-flip errors are exponentially suppressed with photon number, whereas phase-flip errors increase linearly.This bias supports bias-preserving gates and fault-tolerant syndrome detection, but cat qubits themselves are not protected against photon loss.

B. Binomial codes

Binomial codes use finite superpositions of Fock states to tailor protection against specified loss, gain, and dephasing errors. Their hardware-efficient encodings support syndrome-based QEC and increasingly universal logical operations, but no-error backaction remains a distortion requiring correction or mitigation.

  • Code construction: Binomial codes use superpositions of truncated Fock states weighted by binomial coefficients to correct polynomial photon-loss, photon-gain, and dephasing errors.The code parameters determine the correctable error orders, Fock-state spacing, and maximum occupied level.
  • Code construction: The generalized parity, obtained from photon number modulo S + 1, uniquely distinguishes the error spaces while the logical basis states remain orthogonal and equally energetic.Truncated occupation makes unitary operations on binomial codes potentially more convenient.
  • Error correction: ⟨n⟩ = 2: the lowest-order binomial code corrects single-photon loss with ε = {Î, â} using fewer oscillator levels than a comparable four-qubit code.The cited comparison describes five occupied oscillator levels and one error syndrome versus a 16-dimensional expanded Hilbert space and three syndromes for the four-qubit code.
  • Error correction: Higher-order binomial codes use larger Fock-space dimensions to correct broader error sets, and spacing three can jointly distinguish photon gain from two-photon loss.At large average photon number, binomial and cat codes asymptotically approach one another because their photon-number distributions become approximately normal.
  • Challenges: No-error evolution still causes non-unitary backaction in binomial codes, distorting code states even when no photon loss is detected.A two-mode construction with the same spacing and total excitation distributed across modes can mitigate this problem.
  • Experimental progress: The lowest-order binomial code achieved repetitive real-time-feedback QEC with a lifetime nearly at break-even, alongside a high-fidelity universal logical gate set and demonstrated two-qubit gates.Reported two-qubit operations include geometric controlled-phase, teleported CNOT, and CNOT gates involving binomial logical qubits.

C. GKP codes

GKP codes encode quantum information in oscillator phase-space grids, correcting small shift errors through stabilizer measurements. Their strong loss performance comes with challenges from nonphysical ideal states, rare large errors, and Kerr sensitivity.

  • GKP codes encode qudits in harmonic oscillators and define logical qubits using superpositions of infinitely squeezed position states spaced by 2√π.
  • Ideal GKP codewords are unphysical, while realistic states use finite squeezing and Gaussian envelopes; large errors remain insufficiently protected.
  • GKP codes often outperform cat, binomial, and numerically optimized codes under photon loss, but their broad photon-number distribution increases Kerr distortion.
  • Small shift errors satisfying |δq| < √π/2 and |δp| < √π/2 can be diagnosed by stabilizer measurements and corrected with minimal displacements.
  • Gaussian operations implement fault-tolerant Clifford gates because they preserve the locality of small phase-space deviations.
  • Non-Clifford gates and codeword preparation require difficult non-Gaussian operations or resources, such as magic-state ancillas.
  • Square and hexagonal GKP error correction, encoding, readout, and control have been experimentally demonstrated across trapped-ion and superconducting-cavity platforms.

IV. UNIVERSAL QUANTUM CONTROL OF BOSONIC CODES

Superconducting circuit QED provides universal control of bosonic modes through ancilla-assisted operations, SNAP and displacement gates, optimized pulses, and adaptive channel simulation. These tools support bosonic-code operations but become harder to optimize and implement as system size grows.

  • A transmon ancilla supplies the nonlinearity needed for universal control of a cavity mode and the combined circuit-QED system.
  • SNAP gates apply photon-number-selective phases, while displacement operations induce hopping between adjacent Fock states; together they provide universal cavity control.
  • GRAPE optimizes discretized control pulses by iteratively updating parameters using gradients of a performance function for a target unitary.
  • SNAP and GRAPE approaches extend to multiple modes and logical-qubit gates, but larger Hilbert spaces create major numerical-optimization and experimental challenges.
  • Arbitrary quantum channels can be simulated with repeated ancilla use, reset, measurement, and feedforward control, extending universal control to open systems.
  • Experiments have demonstrated single-qubit channel simulation, maximally mixed-state preparation, and autonomous QEC without feedback electronics.
  • Alternative universal-control routes use intrinsic Pockels and Kerr nonlinearities to simulate arbitrary Hamiltonians through Trotterization.

V. APPLICATIONS OF BOSONIC CODES

Bosonic codes are positioned for applications in quantum information processing and for reducing noise effects in quantum simulation and metrology. Demonstrated fault-tolerant operations illustrate this broader application potential.

  • A single bosonic mode supports an infinitely large Hilbert space, enabling applications in quantum computation, communication, simulation, and metrology.
  • Figure 7 summarizes fault-tolerant bosonic-code operations including error detection, path-independent gates, and error-transparent gates.
  • Bosonic codes can reduce noise and system imperfections in estimating selected outputs through QEC or approximate QEC on near-term platforms.

A. Fault-tolerant quantum computation

Fault-tolerant bosonic computation must prevent ancilla and data errors from propagating during every computational step. Experiments demonstrate protected error detection and error-transparent gates, while a bosonic-code threshold remains unresolved.

  • A. Fault-tolerant quantum computation: Fault-tolerant computation requires state preparation, error detection, gates, and measurements to prevent errors from propagating and accumulating.
  • A. Fault-tolerant quantum computation: Ancilla damping during cat- and binomial-code error detection can induce uncorrectable random phase shifts in encoded information.
  • A. Fault-tolerant quantum computation: Redundant ancilla energy levels yielded a fivefold suppression of ancilla errors in demonstrated fault-tolerant error detection.
  • A. Fault-tolerant quantum computation: Biased-noise ancillas can limit encoded-state damage because ancilla σ_z errors commute with the dispersive interaction and affect detection results instead.
  • A. Fault-tolerant quantum computation: Path-independent gates require encoded evolution to remain deterministic despite ancilla errors up to a specified order.
  • A. Fault-tolerant quantum computation: Error-transparent gates preserve compatible evolution in code and error spaces, allowing photon-loss errors to be corrected after the gate.
  • A. Fault-tolerant quantum computation: Bosonic codes still lack a demonstrated fault-tolerant threshold; proposed directions include higher-order single-mode encodings and multimode nonlocal encodings.

B. Quantum communications with bosonic codes

Bosonic codes support quantum networking through state transfer, entanglement, teleportation, and repeater components, but room-temperature thermal noise remains a practical constraint for superconducting implementations.

  • Quantum networks require efficient state transfer and long-lived memories, while photons support high-rate communication between distinct nodes.
  • Bosonic-network demonstrations include controlled photonic release, teleported CNOT gates, quantum buses, and on-demand state transfer or entanglement.
  • Bosonic-code nodes can convert stored states into traveling wavepackets and use parity information to track photon loss during entanglement generation.
  • Quantum repeaters and teleportation could deliver information over arbitrarily long distances with high fidelity using shared entanglement, local operations, and classical communication.
  • Room-temperature thermal noise mainly limits practical superconducting bosonic networks, motivating microwave-to-optical transducers.

C. Quantum simulations with bosonic codes

Bosonic modes offer hardware-efficient platforms for simulating vibrational, topological, and other intrinsically bosonic systems. Error correction and mitigation could extend useful circuit depth, although prior simulators had not directly used QEC codes.

  • Bosonic modes directly represent physical models including molecular vibrations, boson sampling, the quantum Rabi model, and Bose-Hubbard systems.
  • Superconducting bosonic processors have simulated molecular vibronic spectra and extracted Franck–Condon factors, including for water photoionization.
  • A digital bosonic simulator implemented a split-step quantum walk and measured a topological invariant through interference between components of a cavity Schrödinger cat state.
  • Prior bosonic simulators had not directly used QEC codes, but bosonic encoding provides hardware efficiency and supports deeper digital simulations.
  • With 5% imperfect logical-gate error and 99% error-detection efficiency, an estimate gives 1% operation error and increases circuit depth from 5 to 20 at about 36% success probability.

D. Quantum metrology with bosonic codes

Bosonic codes combine nonclassical state preparation with oscillator redundancy to improve quantum metrology. They can suppress shot noise and extend probing-state coherence through error correction.

  • Bosonic modes are relevant to sensing magnetic fields, acceleration, rotation, displacement, and distance because these systems can be described or approximated as bosonic modes.
  • Superconducting systems and hybridization with spins or mechanical resonators provide a platform for deterministic state engineering, processing, and detection in quantum metrology.
  • Coherent-state parameter estimation is limited by shot noise proportional to 1/N and coherence time proportional to 1/Tc, whereas the Heisenberg limit scales as 1/N.
  • Maximum-variance quantum states and interferometers using squeezed, number, or Schrödinger cat states can suppress shot noise; single-mode schemes have also been implemented experimentally.
  • Bosonic Hilbert-space redundancy enables QEC code subspaces that protect probing states from environmental noise and extend their coherence time.

VI. DISCUSSIONS AND OUTLOOK

The outlook emphasizes extending bosonic systems, demonstrating near-term advantages, and addressing fabrication, control, scaling, and fault-tolerance challenges. Bosonic codes are projected for applications across quantum information platforms and tasks.

  • Overview: Bosonic modes offer hardware efficiency, large Hilbert spaces, and long-distance transfer, but many challenges remain despite preliminary experimental evidence.
  • Quantum metrology: Quantum metrology experiments reported results surpassing the standard limit by about 9.1 dB at N = 12.
  • Short-term outlook: Near-term bosonic QEC can protect information from temporal, propagation-loss, and gate errors, supporting longer storage, transmission distance, and circuit depth.
  • System extensions: Future work includes multiple oscillators, scalable resonator arrays, and low-noise microwave-to-optical transducers for long-distance communication.
  • System extensions: Hybrid phononic architectures could enable multimode mechanical memory and mechanical-oscillator force or inertial sensing.
  • Long-term challenges: Long-term universal computation requires clearer bosonic-code fault-tolerance thresholds, higher-energy or higher-dimensional codes, and extensions to multiple modes.
  • Long-term challenges: Limited gate fidelity obstructs high-order bosonic codes because larger Hilbert spaces require more complicated control pulse sequences.
  • Outlook: The review argues that superconducting bosonic demonstrations can extend to optical, mechanical, spin-wave, spin-phonon, and optical-cavity-QED platforms.
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