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Towards Scalable Bosonic Quantum Error Correction

Barbara M. Terhal, Jonathan Conrad, Christophe Vuillot

arXiv:2002.11008v3quant-ph

TL;DR

Bosonic qubit architectures face challenges in preserving oscillator harmonicity, validating engineered-Hamiltonian approximations, and managing error processes. The paper reviews cat and GKP encodings, presents finite-squeezing GKP decoding results, discusses circuit-QED controlled displacements, and considers surface-code architectures using GKP and regular qubits.

  • Problem

    Bosonic qubit architectures must preserve oscillator harmonicity during coupling while maintaining accurate engineered dynamics and useful error-correction properties.

  • Method

    The paper reviews bosonic qubit encodings, analyzes repeated GKP error-correction decoding with finitely squeezed ancillas, and discusses circuit-QED gates and surface-code architectures.

  • Results

    The paper identifies differences between finite-squeezing GKP decoding dynamics and probability-level stochastic shift-error models, and describes controlled-displacement constructions from dispersive interactions.

  • Takeaways & Limitations

    GKP and other bosonic encodings provide candidate building blocks for superconducting surface-code architectures, with data and ancilla qubits potentially chosen differently.

  • Takeaways & Limitations

    The practicality of proposed noise-biased gates and the noise threshold for an all-regular-qubit-ancilla architecture remain open questions.

Abstract

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We review some of the recent efforts in devising and engineering bosonic qubits for superconducting devices, with emphasis on the Gottesman-Kitaev-Preskill (GKP) qubit. We present some new results on decoding repeated GKP error correction using finitely-squeezed GKP ancilla qubits, exhibiting differences with previously studied stochastic error models. We discuss circuit-QED ways to realize CZ gates between GKP qubits and we discuss different scenario's for using GKP and regular qubits as building blocks in a scalable superconducting surface code architecture.

1. Introduction

The paper reviews bosonic quantum error correction with emphasis on scalable GKP schemes in superconducting circuit-QED. It frames scalability around oscillator engineering, suitable noise regimes, and integrating bosonic and regular qubits into surface-code architectures.

  • Bosonic quantum error correction encodes a qubit in a two-dimensional oscillator subspace and combines error correction with universal computation techniques.
  • High-Q microwave cavities provide a natural photon-loss-dominated platform, with frequencies of 3−10 GHz and single-photon lifetimes up to 1−10 ms.
  • No finite code corrects all errors, so bosonic correction aims to provide logical qubits as building blocks for further coding schemes.
  • Surface-code scalability depends on engineering choices about data and ancilla qubits, while the review focuses on superconducting circuit-QED rather than comprehensively covering all bosonic codes.
  • Scalable bosonic architectures must preserve oscillator harmonicity during coupling while keeping engineered-Hamiltonian approximations accurate in a useful photon-number regime.
  • Superconducting implementations exchange the roles of harmonic and anharmonic oscillators: bosonic modes store information, while anharmonic elements prepare states and generate effective nonlinearities.

2. Bosonic Qubits and Their Components

Bosonic qubits trade hardware-engineering complexity against error-correction advantages, with cat and GKP encodings offering distinct noise and control properties. The GKP construction provides lattice-based protection and approximate-state tools, but finite energy, gate inaccuracies, and noise-model assumptions remain important constraints.

  • Noise-Biased Cat Qubit: Photon loss and Kerr nonlinearity produce uncorrectable effects in cat codes, including distortion, two-photon logical bit flips, and dephasing.These limitations constrain the protection offered by repeated photon-parity measurements.
  • Noise-Biased Cat Qubit: Cat qubits can achieve exponentially suppressed bit-flip rates while phase-flip rates increase linearly with |α|^2.The paper describes a large phase-space energy barrier and reports experimental observation of both trends.
  • The GKP Qubit: The GKP qubit encodes information through commuting phase-space translations, with logical operators given by half-translations that anticommute.Ideal code states have support on integer multiples of √π in both quadratures.
  • The GKP Qubit: A hexagonal GKP lattice has a larger correctable Wigner-Seitz cell than the square lattice under stochastic Gaussian displacement noise.The larger cell corresponds to a greater correctable probability volume in that noise model.
  • Approximate GKP States: Approximate GKP states can be asymmetric in q and p, and their photon-number distributions follow a thermal form for the discussed approximations.The F-approximation is specifically described as asymmetric, while parity symmetry gives support only on even photon numbers for several approximations.
  • Logical Gates and Noise: Keeping approximate GKP states centered around the vacuum helps reduce error propagation and inaccuracies in logical gates.The paper identifies low average photon number as an overall strategy for mitigating these effects.

3. Circuit-QED Realizations of GKP Qubit Components

The paper surveys circuit-QED components for preparing, measuring, and coupling GKP qubits, using regular or bosonic ancillas and cavity-mediated interactions. Proposed implementations trade gate speed and tunability against dephasing, leakage, unwanted nonlinearities, and errors from large phase-space excursions.

  • Hardware platform: GKP qubits can be stored in low-loss 3D microwave cavities, with multiple coupled cavities providing a platform for scalable architectures.Coupling may use dipolar antenna interactions to a planar chip hosting a tunable coupler mode.
  • Coupling with regular qubits: A transmon can control GKP-mode displacements by converting dispersive cross-Kerr evolution into conditional rotations and then controlled displacements.The interaction maps b†b to Pauli Z in the regular-qubit subspace, while additional displacements and qubit flips complete the construction.
  • Coupling with regular qubits: A 50 ns controlled-displacement realization requires χ/2π = 25 MHz and a tunable cross-Kerr interaction that can be switched off during transmon preparation or measurement.Residual cross-Kerr evolution rotates and dephases the GKP grid state in the Fock basis; flux tuning also makes the resonator more anharmonic.
  • Coupling with regular qubits: An alternative uses weak coupling, χ/2π = 28 kHz, and cavity displacements to |β|^2 = 320 photons, but realizes the controlled displacement in 1.2 µs.The scheme cancels the controlled rotation by flipping the qubit and reversing the cavity displacement midway through the interaction.
  • Logical GKP measurement: Logical GKP measurement can use ancilla-controlled modular measurements of q, while finite squeezing gives a symmetric readout error q = 1/2(1 − e^(-π∆^2/4)).At ∆ = 0.3, the readout error is about 3.4%; repeated measurements with majority voting can reduce it at the cost of time and possible feedback error.
  • Logical GKP measurement: A proposed improved measurement releases the GKP state into a transmission line for phase-sensitive amplification, while CZ quality depends on suppressing unwanted Kerr, cross-Kerr, and rotating-wave-approximation errors.The relevant comparison is between unwanted nonlinear terms and the pump-activated interactions used to implement the gate.

4. Prospects for a GKP-Surface Code Architecture

The paper outlines GKP-surface-code architectures, emphasizing parity-check implementation, decoding assumptions, and the engineering trade-offs between GKP and regular-qubit ancillas.

  • Architecture: GKP surface-code cycles interleave individual GKP error correction with surface-code parity checks implemented on oscillator-encoded qubits.The proposed checks use CZ-based circuits and ancilla GKP measurements through cavity release, amplification, and quadrature detection.
  • Architecture: The All-GKP-Ancilla architecture requires each GKP oscillator to support CZ interactions with five GKP ancillas.Four ancillas serve surface-code checks and one serves the oscillator’s own GKP error correction.
  • Architectures and decoding: The paper reports a stochastic-displacement threshold σ0c ≈ 0.243 for the toric code, corresponding to ∆ = 0.34 or 9.3 dB, but notes this conversion is optimistic.Finite-squeezing simulations produce somewhat worse error rates than the stochastic model.
  • Architectures and decoding: The All-Regular-Qubit-Ancilla architecture maps GKP-induced errors to a phenomenological model with measurement-error probability q(∆), whose q = p threshold is 3.3%.The model assumes otherwise perfect surface-code cycles and treats GKP error correction as generating effective ancilla errors.
  • Architectures and decoding: 12% ancilla error occurs at ∆ = 0.3, while ∆ = 0.15 is needed to reach q(∆) = 3.4%.These values follow q(∆) = 1/2(1 − e^−π∆2) in the stated effective error model.
  • Engineering prospects: A practical alternative uses high-Q cavity data qubits with transmon, fluxonium, or cat-qubit ancillas, relying on high-on/off-ratio pump-activated CZ or controlled-displacement gates.The comparison with transmon architectures highlights gate fidelity, leakage, duration, measurement, coherence, and cross-talk as key engineering factors.

A. Fock State Representation of GKP Grid States

The appendix derives and numerically evaluates Fock coefficients for approximate GKP and sensor states, showing asymptotic behavior connected to geometric or thermal photon-number distributions.

  • Coefficient derivation: The appendix derives Fock coefficients for D-approximate GKP states and sensor grid states using position-space Hermite functions and theta-function identities.The resulting formulas support both asymptotic analysis and numerical evaluation.
  • Sensor state: The sensor state has photon-number support only at multiples of four because it is an eigenstate of exp(iπa†a/2).Its odd coefficients vanish, and the relevant coefficients are c4n.
  • Asymptotic analysis: The coefficient sequences are obtained from normalization functions whose analytic behavior near u → 1 enables a transfer-theorem analysis.The sequences converge to finite values that determine the asymptotic Fock coefficients.
  • Asymptotic results: The approximate GKP and sensor-state Fock coefficients are asymptotically equivalent to a geometric or thermal distribution.The equivalent thermal distribution is parameterized by average photon number n through u = n/(n + 1).
  • Asymptotic results: For the D-approximate states, the equivalent thermal average photon number is n = u/(1 − u) = e^−2∆2.Figures A1 and A2 compare the computed coefficients with thermal distributions for ∆ values including 0.3, 0.4, and 0.5.

B. Decoders For Repeated GKP Error Correction With Finite Squeezing

The section develops decoders for repeated GKP error correction with finitely squeezed ancillas, contrasting full wave-function tracking with classical path and stochastic approximations. It also describes decoder adaptations, assumptions, and numerical logical-error evaluation.

  • Classical decoder construction: The action separates into kinetic dynamics in position and a potential that pins the position to measured values.For small finite-squeezing parameter ∆, the dominant cosine terms motivate simplifying the action.
  • Finite-squeezing effects: The finite-squeezing action contains a pure imaginary term from Z-error feedback, allowing interference among paths and distinguishing wave-function dynamics from probability-level stochastic models.A single-path approximation retains this contribution as an additional phase, while the full path sum can change through interference.
  • Classical decoder construction: The decoder approximates wave-function dynamics by selecting a single optimal classical path in a quantum path integral.This avoids fully calculating the entire evolution and motivates classical or simplified Wigner-function tracking.
  • Efficient decoder construction: For efficient decoding, the method applies a stochastic diagonal approximation and estimates transition probabilities by sampling endpoints and weighting classical paths by e^-2Re(S).The normalization is unnecessary for decoding, and the forward probability can be estimated from sampled initial and final positions.
  • Decoder variants: The repeated-round decoder uses measured quadrature outcomes and final homodyne data to choose whether to flip the logical output under maximum-likelihood or forward strategies.A memoryless variant instead applies corrective shifts motivated by a stochastic-shift model while keeping photon number low.
  • Evaluation and assumptions: Logical-error rates for three decoders were numerically estimated versus the number of rounds for ∆=0.3 and ∆=0.4, with adapted corrective-displacement versions evaluated separately.The estimates assume comparable behavior for the alternate logical input and arbitrary superpositions without logical interference.

C. Hamiltonian Engineering Via Rotating Wave Approximations

This section explains rotating-wave approximations for Hamiltonian engineering using Schrieffer-Wolff and Magnus expansions. Their validity depends on perturbative strength, detuning, convergence conditions, and photon-number-dependent corrections relevant to CZ-gate accuracy.

  • Rotating-wave approximation: RWA terms can be justified by treating excitation-number-changing or sufficiently detuned terms as perturbations in a rotating frame.Schrieffer-Wolff perturbation theory projects onto nearly degenerate Fock-state blocks separated by frequency gaps.
  • Magnus expansion: The Magnus expansion instead makes off-diagonal terms rapidly time-dependent so their effects average out over sufficiently long times.It provides an effective representation for time-dependent Hamiltonian evolution in a rotating frame.
  • Perturbative scaling: The lowest-order effect of a rotating term decreases inversely with detuning and scales perturbatively with ||A||/∆.Higher-order terms involve increasingly complex commutators, while diagonal Fock-basis terms remain time independent.
  • Validity conditions: Both Schrieffer-Wolff and Magnus methods produce perturbative expansions whose validity depends on the perturbative parameter.A sufficient Magnus-convergence condition is an integrated Hamiltonian norm below π, although the series may still be useful asymptotically.
  • Driven Hamiltonians: For driven modes, replacing the driven operator by its coherent amplitude introduces explicit time dependence, making a Magnus treatment convenient for analyzing the effective dynamics.The drive can create a coherent state with ⟨c(t)⟩=Ee^-iω_pt before the rotating-frame analysis.
  • Scope and limitation: A more quantitative RWA error analysis is needed because approximation errors affect CZ-gate accuracy and depend on the photon number in a GKP mode.Higher photon number increases the strength of the relevant perturbation.

C.1. Time-Dependent Displacement Frame

The time-dependent displacement frame absorbs a drive into a coherent displacement, converting the driven evolution into evolution under a displaced Hamiltonian up to an irrelevant phase.

  • Displacement-frame construction: The frame transformation represents the driven oscillator state through a displacement D(β(t)) determined by the time-integrated drive envelope.When the total frame evolution returns to the identity at the final time, the transformed Hamiltonian describes the actual Schrödinger evolution over the full interval.
  • Effective Hamiltonian: In the displacement frame, the static Hamiltonian is evaluated with a→a+β and a†→a†+β*, while the remaining scalar drive term contributes only an irrelevant phase.This separates the coherent drive from the oscillator's internal Hamiltonian dynamics.
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