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Quantum computing with rotation-symmetric bosonic codes

Arne L. Grimsmo, Joshua Combes, Ben Q. Baragiola

arXiv:1901.08071v3quant-ph

TL;DR

The paper asks how bosonic encodings with rotation symmetry can support universal and fault-tolerant quantum computing while correcting loss and dephasing. It introduces universal operations and software-tracked teleportation error correction for number-phase codes, then shows strong numerical performance and a Bacon-Shor concatenation route to fault tolerance.

  • Problem

    The work addresses how bosonic rotation-symmetric codes can provide universal fault-tolerant computation and effective error correction despite nonlinear logical operations and realistic noise channels.

  • Method

    The paper develops cross-Kerr controlled-rotation gates, dual-basis measurements, teleportation-based error correction, and concatenation of number-phase codes with Bacon-Shor subsystem codes.

  • Results

    Cat and binomial codes surpass break even by several orders of magnitude under simultaneous loss and dephasing, while the proposed recovery is nearly optimal for noiseless ancillae and idealized measurements.

  • Takeaways & Limitations

    The scheme supports universal computation across different rotation-code encodings and integrates bosonic and qubit-level error correction without additional bosonic syndrome-measurement resources.

Abstract

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Bosonic rotation codes, introduced here, are a broad class of bosonic error-correcting codes based on phase-space rotation symmetry. We present a universal quantum computing scheme applicable to a subset of this class--number-phase codes--which includes the well-known cat and binomial codes, among many others. The entangling gate in our scheme is code-agnostic and can be used to interface different rotation-symmetric encodings. In addition to a universal set of operations, we propose a teleportation-based error correction scheme that allows recoveries to be tracked entirely in software. Focusing on cat and binomial codes as examples, we compute average gate fidelities for error correction under simultaneous loss and dephasing noise and show numerically that the error-correction scheme is close to optimal for error-free ancillae and ideal measurements. Finally, we present a scheme for fault-tolerant, universal quantum computing based on concatenation of number-phase codes and Bacon-Shor subsystem codes.

I. INTRODUCTION

The paper introduces bosonic rotation codes and develops universal, fault-tolerant quantum computing for the number-phase subset, including cat and binomial codes. It also proposes software-tracked teleportation error correction and finds strong numerical performance under simultaneous loss and dephasing.

  • Bosonic rotation codes: Bosonic rotation codes encode qubits in subspaces with discrete N-fold phase-space rotation symmetry, with the rotation operator acting as logical Z.The code parameter N also determines Fock-space support and detectable number shifts.
  • Number-phase codes: Number-phase codes are a subset with small modular phase uncertainty, including cat and binomial codes and analogous shift-resistant qudit codes.In the vanishing-uncertainty limit, number and phase play roles analogous to position and momentum in ideal GKP codes.
  • Universal quantum computation: A cross-Kerr controlled-rotation gate implements logical controlled-Z and entangles number-phase codes even when their rotation orders or code types differ.A self-Kerr interaction supplies the single-qubit phase gate, while dual-basis preparation, measurement, and magic-state injection complete universality.
  • Error correction: Teleportation-based error correction transfers encoded information to a fresh ancilla, allowing recovery operations to be tracked entirely in software.This replaces difficult nonlinear codespace-restoration operations with preparation of logical basis-state ancillae.
  • Numerical results: Cat and binomial codes surpass break even by several orders of magnitude for N = 2–4 and κt = 10^-3–10^-2 under simultaneous loss and dephasing.The teleportation-based scheme performs nearly as well as the numerically optimal recovery for noiseless ancillae and idealized measurements.
  • Fault tolerance: Concatenating number-phase codes with Bacon-Shor subsystem codes yields a route to fault-tolerant universal quantum computing tailored to bosonic-code strengths and weaknesses.The concatenation addresses difficult state-preparation errors, noisy measurements, and errors too large for the bosonic code.

1. The phase and Fock grids

Rotation codes organize encoded states simultaneously on a Fock grid and a phase grid. Increasing N separates states more in Fock space but brings their phase-space rotations closer, creating a distance trade-off.

  • The phase and Fock grids: For an order-N rotation code, the Fock grid contains states {|kN⟩}, while the phase grid contains angles {mdθ} for m = 0, 1, …, 2N − 1.These grids characterize the error types to which the code is naturally resilient.
  • The phase and Fock grids: Increasing N increases separation in Fock space but decreases separation between phase-space rotations, trading detectable number shifts against rotation errors.The rotational distance is π/N, while the number distance determines detectable number shifts.
  • The phase and Fock grids: A number-shift error smaller than d_n, including loss or gain of fewer than N excitations, is detectable in principle.For cat and binomial codes, the spacing parameter satisfies S = d_n − 1 and is the largest detectable number error.
  • The phase and Fock grids: For N = 2, computational codewords occupy alternating Fock-state families, while dual codewords differ by a phase-space rotation of π/2.Rotation errors small compared with dθ = π/2 are detectable with code-dependent uncertainty.

B. Distinguishing the codewords

The codewords are distinguished through number measurements in the computational basis or phase estimation in the dual basis. Phase decoding resolves rotations modulo 2π/N, with performance governed by measurement uncertainty and decoder assumptions.

  • Number measurements: Number measurements distinguish computational-basis codewords because even and odd multiples of N identify the two logical states.Under pure loss, the decoder should round outcomes upward to the nearest multiple of N.
  • Phase measurements: Phase estimation distinguishes dual-basis codewords by identifying θ modulo 2π/N, with 0 corresponding to |+N⟩ and π/N to |−N⟩.The canonical POVM is an optimal no-prior-information phase-estimation measurement for a specified fiducial codeword.
  • Phase measurements: The modular phase uncertainty ∆N(θ) quantifies uncertainty in estimating eiNθ and depends on the code’s Fock-grid coefficients.It vanishes when the coefficient magnitudes are completely flat, although that ideal state is not normalizable.
  • Measurement limitations: Canonical phase measurements may be sub-optimal when prior information about θ or additional noise is available, while heterodyne measurements have larger uncertainty but practical appeal.A biased decoder can account for asymmetric clockwise and counter-clockwise rotation noise.
  • Phase measurements: Dual-basis decoding is faithful when measurement fluctuations remain small relative to the rotational distance dθ = π/N.For N = 2, dθ = π/2; rotations near dθ/2 can cause logical misidentification.

A. Ideal number-phase codes

Ideal number-phase codes combine rotation symmetry with vanishing modular phase uncertainty, providing complementary number and phase structure. Cat, binomial, and Pegg-Barnett families approach this ideal only in appropriate excitation or truncation limits.

  • A. Ideal number-phase codes: Ideal number-phase codes are defined by phase-state-like superpositions with exact N-fold rotation symmetry and nonnormalizable infinite excitation.Their codewords are supported on Fock states separated by N and are stabilized by the rotation operator.
  • A. Ideal number-phase codes: Vanishing ∆N(θ) makes the dual-basis codewords eigenstates of the logical-X operator while preserving the rotation-code structure.For positive real Fock-grid coefficients, the phase-uncertainty limit is equivalent to the relevant logical-X and stabilizer relations.
  • B. Approximate number-phase codes: Number-phase duality separates dual-basis codewords by π/N in phase while separating computational-basis codewords by N in Fock space.The two bases therefore share complementary distance measures for rotation and number-shift errors.
  • B. Approximate number-phase codes: Cat, binomial, and Pegg-Barnett codes are examples whose embedded phase uncertainty approaches zero in large-parameter limits.The relevant limits involve large cat amplitude, large binomial truncation, or large Pegg-Barnett truncation.
  • B. Approximate number-phase codes: The quantum-computing scheme uses rotation symmetry for unitary gates and approximate number-phase symmetry for robust logical-X measurements.This combination supports Clifford operations, while state preparation and measurement remain code- and implementation-dependent.

2. Teleported gates

Teleportation completes a universal gate set for rotation-symmetric encodings using code-agnostic entangling gates, logical measurements, and prepared ancilla states. Conditional corrections can be tracked in a Pauli frame, while state preparation remains a central practical challenge.

  • 2. Teleported gates: The controlled-rotation gate and logical-X measurements enable teleported logical Hadamard and T gates, completing universal computation with ancillae.The Clifford group comes from CZ, H, and S, while T is supplied through magic-state teleportation.
  • 2. Teleported gates: Outcome-dependent Pauli corrections can be tracked entirely in software rather than physically applied after teleportation.Repeating the protocol until a desired measurement outcome is obtained is an alternative.
  • 2. Teleported gates: Injecting arbitrary ancilla states from the trivial encoding is not fault-tolerant because ancilla errors propagate into the rotation code.Concatenation and state distillation are proposed to address this issue.
  • 2. Teleported gates: Logical |+N⟩ preparation is required for gate teleportation and state injection, but optimal preparation depends on the code and is not treated in detail.A breeding protocol can raise rotation symmetry from order N to order 2N, although its success probability decreases exponentially with the number of rounds.

V. MODULAR NUMBER MEASUREMENT

Controlled rotations enable nondestructive measurement of excitation number modulo N, which can detect number shifts and conditionally prepare rotation-symmetric codewords. Ancilla order trades reduced propagated rotation errors against greater phase-measurement uncertainty.

  • V. MODULAR NUMBER MEASUREMENT: The controlled-rotation circuit measures n mod N by rotating an ancilla according to the data rail’s modular excitation number.The ancilla phase is measured destructively, with resolution determined by its embedded phase uncertainty.
  • V. MODULAR NUMBER MEASUREMENT: The modular measurement projects the data rail into an N-fold rotation-symmetric subspace and can conditionally prepare codewords from a primitive state.Measuring n mod 2N on |Θ⟩ prepares |0N,Θ⟩ for the appropriate outcome.
  • V. MODULAR NUMBER MEASUREMENT: Ancilla errors that do not commute with n, including loss and gain, can induce rotation errors on the data rail during the measurement.For M = 1, an unknown-time single loss can completely randomize the data rail’s phase.
  • V. MODULAR NUMBER MEASUREMENT: The maximum rotation error from one ancilla loss or gain scales as 1/M, whereas larger-M ancillae increase phase-measurement uncertainty.This creates a direct trade-off in choosing the ancilla’s rotation-code order.
  • V. MODULAR NUMBER MEASUREMENT: Small number shifts and rotations are approximately correctable when they remain below the code’s number and rotational distances.For number-phase codes, shifts with 0 < |k| < N are detectable, while pure rotations below π/N are formally detectable for ideal codes.

B. Error propagation

The paper analyzes how number-shift and rotation errors propagate through gates, finding quadratic gates preserve approximate correctability while higher-order gates can amplify and spread errors dangerously.

  • Error model: The error-propagation analysis uses E_k(θ) to track number-shift and rotation errors through the proposed gates.The gates generated by quadratic powers of the number operator are contrasted with the highly nonlinear T_N gate.
  • Quadratic gates: ZN propagates a general E_k(θ) error only into a phase factor, without amplifying the error.This result follows from the ℓ=1 commutation relation, whose prefactor is a global phase.
  • Code dependence: The effect of a small propagated error depends on the code’s ability to correct phase and number-shift errors, so error types cannot be ranked universally.The paper explicitly notes that rotation and number-shift errors may have different consequences for different codes.
  • Quadratic gates: For SN and crot, a number-shift error produces a rotation error proportional to k, with θ_k = πk/NM and N = M for SN.The figure identifies the output as a pure rotation error; the corresponding phase angle remains small for small k relative to the code distances.
  • Quadratic gates: If |k| < N/2, the rotation error introduced on the second mode by crot is smaller than π/(2M), preserving approximate correctability.The error spreads from mode a to mode b, but remains small relative to mode b’s angular distance.
  • Higher-order gates: Gates generated by n^ℓ for ℓ > 2, including T_N, amplify and spread errors through additional linear and nonlinear rotation errors.The paper therefore avoids using T_N directly and instead relies on gate teleportation for a logical T gate.

A. Error correction by teleportation

The paper presents teleportation-based error correction for number-phase codes, using crot gates, phase measurements, and fresh ancillae to restore the codespace while tracking recoveries in software. Numerical studies under simultaneous loss and dephasing show near-optimal performance under idealized ancilla and measurement assumptions, while finite protection motivates concatenation for fault tolerance.

  • Teleportation-based error correction: Telecorrection uses crot gates, phase measurements, and |+N⟩ ancilla preparation to teleport encoded information into a fresh output mode.The circuit is equivalent to two consecutive one-bit teleportations and attempts to distinguish damaged logical codewords in the dual basis.
  • Teleportation-based error correction: An error on the data rail spreads to a rotation on the second rail, but commuting with crot prevents that rotation from reaching the output rail.Correct dual-basis measurements can therefore remove the propagated error during teleportation.
  • Measurement condition: For number-phase codes, phase-measurement distinguishability is the condition for successful teleportation, while non-number-phase codes may require other measurements.The scheme’s effectiveness is therefore tied to the code’s phase resolution.
  • Software-tracked recovery: After correction, the remaining error channel acts entirely within the logical subspace, allowing the most likely Pauli recovery to be tracked in a software Pauli frame.This replaces potentially difficult nonlinear operations that would otherwise restore states shifted off the Fock grid.
  • Numerical performance: The pretty good and phase schemes approach optimal error correction for large average excitation, with pretty good measurements near optimal across almost all excitation numbers at small noise strengths.At small excitation numbers, canonical phase measurements can have a significant gap because they distinguish codewords poorly.
  • Numerical performance: Both cat and binomial codes achieve break-even pseudo-thresholds in the 1–10% noise-strength range, but increasing rotation order gives diminishing gains and cannot make infidelity arbitrarily small.The authors therefore identify realistic noise, faulty ancillae, encoding, measurements, and gates as important follow-up conditions, with concatenation needed for further suppression.

VIII. ROADMAP TO FAULT TOLERANCE

The paper develops a path to fault-tolerant universal computing by concatenating number-phase bosonic codes with repetition or Bacon-Shor codes, integrating bosonic and qubit-level correction.

  • VIII. ROADMAP TO FAULT TOLERANCE: Fault-tolerance requires handling state-preparation errors, noisy destructive measurements, and errors too large for the bosonic code.These challenges motivate concatenation with a conventional qubit code.
  • VIII. ROADMAP TO FAULT TOLERANCE: A Bacon-Shor construction integrates bosonic correction into qubit-code correction while using bosonic operations designed to limit error amplification and spread.The scheme assumes universal control of two-level ancillae and cross-Kerr gates between ancillae and rotation codes.
  • VIII. ROADMAP TO FAULT TOLERANCE: A dual-basis repetition code suppresses measurement errors through majority voting across independent phase measurements.The construction concatenates a length-n repetition code with the bosonic encoding.
  • VIII. ROADMAP TO FAULT TOLERANCE: The concatenated scheme protects against both rotation and loss errors because the bosonic code handles both error types.Bosonic and repetition-code correction are integrated into one error-correction step.
  • VIII. ROADMAP TO FAULT TOLERANCE: Two ancilla loss errors can induce a π/2 rotation error that an N = 2 number-phase code cannot handle, requiring n and r to depend on N and the physical loss rate.This establishes a design constraint for repetition-code length and ancilla repetition count.
  • VIII. ROADMAP TO FAULT TOLERANCE: Replacing repetition coding with Bacon-Shor coding adds protection against loss and gain errors through a second repetition-code dimension.The ratio m/n can be optimized to the underlying bosonic code and noise bias.

C. Universality

The paper combines code-agnostic Kerr-based entangling gates, dual-basis measurements, and teleportation to support universal operations and error correction for number-phase codes.

  • C. Universality: State injection of noisy |+i⟩ and |T⟩ states supplies the non-Clifford resource needed for universality.The injected states are teleported into Bacon-Shor or repetition-code blocks using a cross-Kerr interaction.
  • C. Universality: Compared with GKP codes, number-phase codes have a smaller set of natural unitary gates and greater nonlinearity, although Kerr interactions can be natural in some platforms.Number-phase codes are analogous to approximate GKP codes with number and phase replacing position and momentum.
  • C. Universality: Robust dual-basis phase measurements are central to both quantum-computing operations and teleportation-based error correction.The paper identifies heterodyne, homodyne, and adaptive homodyne measurements as implementations.
  • C. Universality: The crot gate is a cross-Kerr controlled-rotation that can entangle different rotation-code types and interface rotation codes with GKP codes.Its Kerr interaction amplifies errors only in a limited way, supporting fault-tolerant use.
  • C. Universality: For large average excitation number, teleportation-based recovery approaches the optimal recovery-map performance allowed by quantum mechanics.At small excitation number, the main gap arises from inaccurate phase discrimination between logical codewords.
  • C. Universality: Logical recoveries can be tracked entirely in software, avoiding explicit highly nonlinear operations to restore the codespace.This is a central simplification of the teleportation-based correction scheme.

Appendix A: Dual-basis primitives

The appendix constructs dual-basis primitives by combining rotated versions of a primitive state, with code-family-specific normalization and primitive choices determining the resulting codewords.

  • Appendix A: Dual-basis primitives: Dual-basis codewords are built from primitives rotated by N equally spaced angles θ = 2mπ/N for m = 0, . . . , N −1.The rotations are generated by exp(iθn̂).
  • Appendix A: Dual-basis primitives: The dual primitive is a weighted superposition of an original primitive and a copy rotated by π/N.For a fixed primitive, the resulting dual primitive generally depends on N.
  • Appendix A: Dual-basis primitives: When N0 = N1, computational- and dual-basis primitives coincide.This equality holds for the listed number-phase codes in the large average excitation-number limit, including cat codes as α →∞.
  • Appendix A: Dual-basis primitives: Rotation-code families differ through their primitive |Θ⟩ and Fock-grid coefficients {f_k^N}.Binomial-code primitives depend on N, whereas other listed families use the same primitive for all N.
  • Appendix A: Dual-basis primitives: The cat-code primitive is a displaced squeezed state |α, r, φ = 0⟩ with real positive α, whose computational codewords are obtained from its Fock-space representation.The squeezed-vacuum coefficients vanish for odd photon number.
  • Appendix A: Dual-basis primitives: Cat codes arise when squeezing vanishes, r = 0, while squeezed-vacuum codes arise when displacement vanishes, α = 0.The smallest cat code approaches the trivial encoding as α →0.

2. Binomial codes

Binomial codes form a family of rotation-symmetric bosonic codes parameterized by rotational order N and truncation K, with explicit codewords and associated primitives. The appendix also relates GKP codes to discrete rotation symmetry and identifies lattice-dependent logical Clifford operations.

  • Binomial-code construction: Binomial codes exactly correct loss, gain, and dephasing errors up to a specified order.They are introduced as a code family with error-correction capability determined by the chosen parameters.
  • Binomial-code construction: Each binomial code is specified by rotational order N and truncation parameter K, which sets its Fock-space truncation.Different parameter choices produce different code primitives.
  • Binomial-code construction: The smallest K = 1 binomial codes yield the 0_N code, including the trivial encoding when N = 1.The corresponding codeword is |+_N⟩ = (|0⟩ + |N⟩)/√2.
  • Other rotation-code constructions: Pegg-Barnett phase states provide another rotation-code primitive, with truncation s − 1 required to be at least N.When s = p × 2^N, rotated primitives are automatically orthogonal and the codes connect to shift-resistant qudit codes.
  • GKP relations: GKP codes have two-fold discrete rotation symmetry, but they are not rotation codes under the paper’s definition because R_2 does not act as a Pauli operator.Square- and hexagonal-lattice GKP codes can nevertheless support additional rotation-induced logical Clifford operations.

Appendix D: Pretty Good Measurements

The appendix motivates the Pretty Good Measurement because canonical phase measurement can be suboptimal for distinguishing noisy or even ideal orthogonal codewords. The proposed POVM incorporates the noise channel explicitly.

  • Motivation: Canonical phase measurement may fail to perfectly distinguish orthogonal codewords because of embedded phase uncertainty.The issue also persists when damaged codewords are compared against ideal codewords.
  • Measurement construction: The Pretty Good Measurement is designed to distinguish orthogonal states perfectly and states with small nonzero overlaps fairly well.Its construction is applied to the states after an arbitrary noise channel.
  • Measurement construction: The POVM uses the noisy projector σ = N(P), where P projects onto the subspace spanned by the states being distinguished.The measurement elements are normalized using the support of σ, with an additional complement element completing the measurement.

Appendix E: Hybrid Steane/Knill error correction

Hybrid error correction combines Steane-style syndrome extraction with Knill-style teleportation for number-phase codes. The circuit restores the codespace while allowing residual rotations and logical corrections to be tracked or decoded.

  • Circuit structure: Hybrid-EC uses Steane-EC for number-shift syndromes and Knill-EC for dephasing syndromes.The first controlled-rotation and top-rail measurement implement the Steane-like step, while the second controlled-rotation and data measurement implement one-bit teleportation.
  • Circuit operation: A phase measurement estimates the residual rotation index k, enabling recovery on the bottom rail or software tracking of the correction.The relevant rotation errors are proportional to k.
  • Circuit operation: The circuit teleports data to a fresh ancilla, restoring the codespace up to a known rotation error.The first circuit step nondestructively measures excitation number modulo N before teleportation.
  • Decoding and recovery: Maximum-likelihood decoding can infer the most likely output rotation and logical Clifford error.Repeating the circuit twice reduces the most likely residual logical error to a Pauli correction that can be tracked in a Pauli frame.
  • Performance: Numerical simulations found hybrid-EC performance identical to the referenced Knill-EC scheme.With noiseless ancillae, an unencoded upper ancilla can provide slightly better phase resolution than an M = 1 cat code at the same α.

a. Relation to recent experiments

Hybrid error correction generalizes experimental syndrome-extraction protocols for cat and binomial codes, while the appendix represents simultaneous loss and dephasing through a channel expansion. The earlier protocol detects number-shift parity but does not address dephasing or fully restore the Fock grid.

  • Relation to experiments: Hybrid-EC generalizes experiments that implemented error correction for N = 2 cat and binomial codes.Those experiments used an unencoded |+⟩ ancilla and a controlled-rotation gate with order N = 2.
  • Relation to experiments: The experimental syndrome circuit imprinted a data number-shift error k as an ancilla phase for measurement.For N = 2, odd k modulo 2 flips |+⟩ to |−⟩, whereas even k modulo 2 leaves |+⟩ unchanged.
  • Relation to experiments: The earlier N = 2 syndrome did not detect dephasing errors or restore the state to the Fock grid.For cat codes, loss events could be tracked in software, removing the need for some recoveries.
  • Noise model: The simultaneous loss-and-dephasing channel uses loss and dephasing Liouvillians, with pure dephasing represented through a Gaussian phase distribution.The Gaussian has zero mean and variance κ_φ t.
  • Noise model: A generalized Dyson expansion separates jump and no-jump evolution and enables the jump superoperators to be moved through the no-jump factors.This provides the channel representation used for the noise analysis.
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