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New class of quantum error-correcting codes for a bosonic mode

Marios H. Michael, Matti Silveri, R. T. Brierley, Victor V. Albert, Juha Salmilehto, Liang Jiang, S. M. Girvin

arXiv:1602.00008v3quant-ph

TL;DR

The paper addresses bosonic-mode error correction where photon loss can produce correlated multi-qubit errors that standard independent-error schemes cannot easily handle. It constructs binomial codes with explicit recovery, demonstrating protection against loss, gain, and dephasing to a chosen order while identifying recovery-fidelity and control trade-offs.

  • Problem

    Photon loss can become a correlated multi-qubit error model, so typical schemes based on independent single-qubit errors cannot be easily transferred to bosonic modes.

  • Method

    The paper constructs binomial codes and an explicit recovery process using error detection and unitary operations, with optional echo operations to reduce higher-order infidelity terms.

  • Results

    The codes provide approximate quantum error correction for continuous dissipative evolution under photon loss, gain, and dephasing, with uncorrectable loss dominated by losing L + 1 photons when protecting against L losses.

  • Takeaways & Limitations

    Binomial codes can offer smaller correctable and uncorrectable error rates and protection against photon gain and dephasing, but their full advantage requires sophisticated high-fidelity unitary control at a high rate.

  • Takeaways & Limitations

    The presented recovery is not optimal, and recovery adjustments cannot beat the code's overall accuracy limit, though they can reduce higher-order infidelity prefactors.

Abstract

from arXiv · show

We construct a new class of quantum error-correcting codes for a bosonic mode which are advantageous for applications in quantum memories, communication, and scalable computation. These 'binomial quantum codes' are formed from a finite superposition of Fock states weighted with binomial coefficients. The binomial codes can exactly correct errors that are polynomial up to a specific degree in bosonic creation and annihilation operators, including amplitude damping and displacement noise as well as boson addition and dephasing errors. For realistic continuous-time dissipative evolution, the codes can perform approximate quantum error correction to any given order in the timestep between error detection measurements. We present an explicit approximate quantum error recovery operation based on projective measurements and unitary operations. The binomial codes are tailored for detecting boson loss and gain errors by means of measurements of the generalized number parity. We discuss optimization of the binomial codes and demonstrate that by relaxing the parity structure, codes with even lower unrecoverable error rates can be achieved. The binomial codes are related to existing two-mode bosonic codes but offer the advantage of requiring only a single bosonic mode to correct amplitude damping as well as the ability to correct other errors. Our codes are similar in spirit to 'cat codes' based on superpositions of the coherent states, but offer several advantages such as smaller mean number, exact rather than approximate orthonormality of the code words, and an explicit unitary operation for repumping energy into the bosonic mode. The binomial quantum codes are realizable with current superconducting circuit technology and they should prove useful in other quantum technologies, including bosonic quantum memories, photonic quantum communication, and optical-to-microwave up- and down-conversion.

I. QUANTUM ERROR CORRECTION AGAINST PHOTON LOSS, GAIN AND DEPHASING ERRORS

The section introduces single-mode bosonic codes for photon loss, gain, and dephasing, using error syndromes and recovery operations to preserve logical information. It develops simple examples whose distinguishable error subspaces and matched photon-number moments satisfy quantum error-correction requirements.

  • General criteria: Quantum error correction requires logical code words whose error states satisfy the Knill–Laflamme conditions independently of the encoded logical state.These conditions ensure that detectable errors can be corrected without losing quantum information.
  • Motivation: A binary Fock-state encoding is unsuitable because one photon loss becomes a correlated multi-qubit error, while its loss rate grows exponentially with the number of encoded qubits.The construction therefore focuses on one logical qubit using a small number of cavity states.
  • Photon loss: Equal mean photon numbers make single-photon loss equally likely for both logical states, preventing the loss event from deforming their quantum information.For the example code, the common mean photon number is 2.
  • Photon loss: Photon loss is detected through parity, after which a conditional unitary transfers the error subspace back to the logical subspace.In microwave implementations, high-fidelity quantum non-demolition parity measurements provide the detection mechanism.
  • Multiple errors: A larger code protects against two photon losses and dephasing, using photon number modulo 3 for loss detection and projective measurements plus unitary recovery for dephasing.Dephasing error states are non-orthogonal, so they require an orthonormalized measurement basis.
  • Photon gain: The same construction can be modified to protect against photon gain instead of, or alongside, additional photon-loss errors.Photon gain and two-photon loss share the same change in photon number modulo 3 in the discussed code.

II. BINOMIAL QUANTUM CODES

Binomial codes use finite Fock-state superpositions with binomial coefficients to correct prescribed loss, gain, and dephasing errors. Their spacing enables generalized-parity syndrome detection, while their coefficients balance photon-number moments across logical states.

  • Code construction: The binomial error set includes up to L photon losses, G photon gains, and D dephasing events.The construction targets polynomial error operators involving a, a†, and n.
  • Code construction: The codes use spacing S = L + G and maximum order N = max {L, G, 2D}, with Fock-state amplitudes weighted by binomial coefficients.The two-parameter (N, S) space describes the resulting code family.
  • Error detection: Occupied Fock states are separated by S + 1, allowing loss and gain errors to be distinguished by measuring photon number modulo S + 1.The paper calls this syndrome measurement generalized parity.
  • Error correction: Equal photon-number moments through the required order ensure that detectable errors do not deform the logical state and admit unitary recovery.The Fock-state spacing identifies errors, while the coefficients balance the moments governing their rates.
  • Qudit generalization: The logical basis extends to qudits through extended binomial coefficients, with the same moment-balancing approach applicable beyond binary codes.The discussion concentrates on qubit codes but states that results extend to binomial qudit codes.
  • Relation to cat codes: Increasing N extends dephasing protection without a fixed limit, while the code family approaches multi-legged cat codes as N tends to infinity.Cat codes use related modulo measurements but generally satisfy approximate rather than exact correction conditions.

III. APPROXIMATE QUANTUM ERROR CORRECTION UNDER CONTINUOUS-TIME DISSIPATIVE EVOLUTION

The section treats continuous-time dissipation, where infinitely many errors occur over a finite timestep and exact correction is impossible. By expanding the evolution and conditioning recovery on measured error syndromes, binomial codes correct the dominant terms to a controlled order.

  • Continuous-time errors: Continuous-time evolution produces an infinite set of possible errors during a finite timestep, so exact correction of the full process is impossible.Approximate correction instead targets errors whose probabilities have the largest powers of κδt.
  • Error process: The photon-loss channel is represented by Kraus operators for exactly ℓ jumps together with no-jump evolution over δt.The jump-time integration yields an exact analytic Kraus representation for the damped oscillator.
  • No-jump evolution: No-jump evolution is itself an error because it preferentially suppresses higher-occupation Fock components and changes the encoded state.Its measurement backaction must therefore be corrected alongside photon jumps.
  • Multiple-loss recovery: The generalized recovery uses projections onto photon-number sectors modulo L + 1 and unitaries that transfer each detected error subspace to the logical subspace.The same operations correct photon loss and the associated no-jump evolution to the targeted accuracy.
  • First-order recovery: For the single-loss code, parity measurement distinguishes no-jump and one-jump sectors, followed by conditional recovery operations that restore the state to first order in κδt.The recovery can use parity projections and a unitary transfer between the error and logical subspaces.
  • Implementation: Projective recovery can suppress linear no-jump evolution through frequent measurements, but logical-subspace measurements are harder to realize with high fidelity than parity measurements.The measurement-based process is conceptually simpler but technologically more demanding.

B. Approximate quantum error correction to Lth order in κδt

The section proves that binomial codes correct continuous-time photon-loss evolution to the desired order by separating leading jump errors from subleading no-jump interference. Equal photon-number moments make the residual logical-state dependence sufficiently small.

  • Recovery construction: An explicit projective-measurement and unitary-recovery process extends the construction to multiple photon losses with accuracy (κδt)^L.The recovery is built from generalized-parity projections and state transfers between logical and error words.
  • Approximate error expansion: The continuous-time Kraus operators are truncated at order L because terms with more than L losses contribute only O[(κδt)^(L+1)].The remaining operators are separated into leading parts Bℓ and relevant subleading parts Cℓ.
  • Correction conditions: The leading error operators must satisfy exact correction conditions, while the subleading interference terms need only be independent of the logical code words to the target order.This separation is what permits approximate rather than exact correction of the continuous process.
  • Moment balancing: Binomial codes satisfy the required conditions because their logical states have equal expectation values of n^ℓ through ℓ = L.The relevant products Cℓ†Cℓ reduce to polynomials in n whose degree is at most L.
  • Result: The resulting recovery corrects both photon-loss jumps and no-jump evolution, with generalized parity identifying the number of losses up to L.The residual error terms depend on the code parameters S and N.
  • Scope: Single-mode binomial codes are approximate codes for continuous-time photon loss with accuracy (κδt)^L, while more general errors require more complicated recovery processes.The same framework is stated to extend to gain and dephasing channels.

IV. BINOMIAL CODE PERFORMANCE

For a single photon-loss channel, binomial-code performance is governed by the leading uncorrectable loss event and by recovery-timestep trade-offs. Higher-order codes help at shorter timesteps, while recovery imperfections and cavity nonlinearities constrain practical optimization.

  • Performance scaling: The dominant uncorrectable event for a code protecting L photon losses is losing L + 1 photons during one timestep.The mean photon number scales as 1/2(L + 1)^2 for S = N = L, increasing decay of higher-order code words.
  • Performance scaling: At small timesteps, entanglement-infidelity slopes closely follow the rate of the leading uncorrectable error.The plotted rate assumes a perfectly faithful recovery process.
  • Code optimization: The L = 1 binomial code outperforms naive encoding for δt ≲ 0.4κ^-1, while L > 1 becomes favorable for δt ≲ 0.2κ^-1.For a fixed timestep, an optimal finite L exists; larger codes are preferable at smaller timesteps.
  • Code optimization: Recovery-stage infidelity η contributes η/δt, favoring lower-order codes with longer optimal timesteps.The best code also depends on the detailed recovery process, because some recovery imperfections may be suppressed by later rounds.
  • Implementation constraints: A self-Kerr term produces jump-time-dependent phases and additional dephasing of order δt, yielding a net effect of order δt^2 when its strength is not much larger than κ.Increasing N can protect against additional dephasing, but raises error probabilities and requires correspondingly shorter correction timesteps.
  • Two-mode comparison: Two-mode codes have three two-photon-loss paths versus one for the one-mode code, making their uncorrectable rate three times larger when decay rates are equal.The one-mode code instead requires a no-jump correction, so the preferred architecture depends on recovery-operation fidelity.

B. Cat codes

Cat codes provide approximate protection for damped bosonic modes through coherent-state superpositions and photon-number-modulo measurements. Binomial codes share this structure but use finite Fock-state superpositions, enabling exact first-order loss suppression with lower mean photon number and explicit corrections.

  • Cat-code structure: Cat codes use 2(L + 1) coherent-state legs to protect against L photon losses and diagnose errors by measuring photon number modulo S + 1.They are related to binomial codes with spacing S = L.
  • Cat-code limitations: Finite cat codes generally satisfy approximate rather than exact quantum-error-correction conditions because their state normalizations differ.The normalization factors become equal only as |β| approaches infinity, where the codes can protect against dephasing to unlimited order.
  • Loss protection: Cat-code uncorrectable errors are O(κδt) and can be suppressed by increasing β, at the cost of increasing average photon number and error rate.The binomial codes with S = 1 exactly suppress all photon-loss errors to first order in κδt.
  • Relation between encodings: Binomial and cat Fock-state distributions approach a normal distribution as their average photon number increases, so the two encodings become asymptotically similar.The same asymptotic relation extends to qudit binomial and qudit cat encodings.
  • Recovery and resources: Binomial codes require an explicit correction gate every timestep, whereas cat codes need only repumping after no-jump evolution when jumps are detected.For approximate correction, the binomial code of Eq. (2) has mean photon number 2, compared with approximately 2.3 for the cat code minimizing its error expression.
  • Relation to permutation-invariant codes: Bosonic binomial codes differ from permutation-invariant qubit codes because bosonic error asymmetry and the large-system limit simplify error-operator action and expand code-design flexibility.The large-M limit suppresses the n^2 term and permits smaller bosonic codes without a finite-M permutation-invariant equivalent.

D. GKP codes

GKP codes use continuous-variable position and momentum structure, allowing correction of a continuous set of shifts and, in principle, photon loss, gain, and dephasing. Their experimentally realizable approximations require finite photon number and imperfectly squeezed states, while comparisons with binomial and cat codes remain incomplete.

  • GKP code structure: GKP code words are defined using continuous, non-normalizable position eigenstates rather than a discrete basis.Their ideal logical states form infinite position-space combs with spacing 2√π and effective logical spacing S = √π.
  • Correctable errors: Position and momentum shifts form a continuous set of correctable errors for ideal GKP codes.The same spacing structure applies in momentum space through Fourier transformation.
  • Correctable errors: GKP codes can in principle correct photon loss, gain, and dephasing when the timestep is sufficiently small.These errors are handled by expanding photon-loss and related operators in the basis of correctable shifts.
  • Experimental limitations: Ideal GKP states require infinite photon number and perfect squeezing, so practical implementations must filter photon number and replace sharp position eigenstates with wavepackets.Approximate constructions can use different filters and starting states, including proposals based on phase estimation.
  • Experimental limitations: Approximate GKP code words are not perfectly orthogonal, creating an additional error that requires sufficiently high photon number to suppress.An optimistic estimate gives a mean photon number of n̄ = 4 for the traditional approximate code construction.
  • Comparison: GKP code words protect against a larger error set than the minimal binomial code, but detailed comparisons among GKP, binomial, and cat codes remain unresolved.The comparison is complicated by different filters, starting states, and continuous versus discrete correctable error sets.

Appendix A: Conditional unitary control of the binomial code recovery process

The recovery process detects photon-loss errors through generalized photon-number parity and then applies conditional unitaries to restore logical code words. Extended binomial codes generalize the construction to qudits using roots of unity and extended binomial coefficients.

  • Conditional recovery: Generalized photon-number parity measurements detect changes that serve as a proxy for the number of photons lost during a short timestep.The measurement projects the cavity state onto a parity sector indexed modulo L+1.
  • Conditional recovery: A correction unitary transfers each detected error word |Bσ⟩ back to the corresponding logical code word |Wσ⟩.The recovery Kraus operator is R̂_k = Û_k Π̂_k mod L+1, and repeating over k realizes the full recovery.
  • Code conditions: Equal moments of the photon-number operator for both code words provide orthonormality and error-correction conditions up to the relevant order.The ℓ = 0 condition handles orthonormality, while nonzero ℓ conditions support correction of the targeted errors.
  • Qudit extension: Extended binomial codes use extended binomial coefficients and dth roots of unity to generalize the binomial construction from qubits to qudits.The code indices are evaluated modulo d, with S = L + G and N = max{L, G, 2D}.
  • Qudit extension: The extended construction is explicitly distinguished from quantum polynomial codes.The paper calls these states extended binomial codes because they are not to be confused with quantum polynomial codes.

1. Moments of ˆn for the extended binomial codes

The extended binomial code construction makes photon-number moments independent of the logical state, establishing orthogonality, normalization, and the moment conditions needed for error correction. The associated loss-channel Kraus representation organizes errors by photon number lost and satisfies completeness.

  • Moment conditions: Equal photon-number moments across extended-binomial code words lead to the required diagonal error-correction conditions.The proof uses derivatives of the polynomial expansion and the root-of-unity identities.
  • Orthogonality and normalization: The extended-binomial code words are orthogonal because the relevant polynomial sum vanishes for nonzero logical-index differences.For zero difference, the same identity gives d^(N+1), establishing normalization.
  • Photon-number scaling: The coefficient α1, equal to the mean photon number, scales linearly with spacing S, qudit dimension d, and the maximum number N of correctable errors of one type.This scaling characterizes the photon-number cost of the extended construction.
  • Loss-channel representation: The loss-channel Kraus representation groups errors according to the number of photons lost, while incorporating no-jump evolution between jumps.Thus the ℓ-photon-loss operator is Êℓ rather than simply âℓ.
  • Loss-channel representation: The Kraus operators satisfy completeness because their sum acts as the identity on every Fock state.Applying the operator sum to |m⟩ produces a binomial expansion equal to |m⟩ for every m.

Appendix E: Lindblad evolution correctable by binomial codes

For Lindblad evolution, the paper expands dissipative dynamics into quantum trajectories and imposes approximate QEC conditions on the error operators relevant to a chosen timestep order. Worst-case operator combinations determine binomial-code parameters.

  • Trajectory expansion: Lindblad evolution over a short interval δt can be unraveled into continuous no-jump evolution and jumps drawn from operators √κ_i Â_i.The expansion parameters are ε_i = κ_iδt, and x_i specifies the desired suppression order for each error operator.
  • Approximate QEC conditions: Approximate QEC requires the code words to satisfy error-correction conditions for the time-evolution Kraus operators to the chosen order in the ε_i.Mixed products of error operators are included through the trajectory expansion.
  • Code selection: Normal ordering reduces the conditions to worst-case operator combinations, allowing a binomial code to be chosen for the highest-order errors.The relevant cases maximize either the number of excitation-increasing and decreasing operators or their difference.
  • Example: For the example errors Â1 = n̂â + ↠with x1 = 1 and Â2 = â with x2 = 2, minimal parameters are L = 2, G = 1, and N = 3.These values satisfy the required loss, gain, and number-operator constraints for the example.

2. Including no-jump evolution

Including no-jump evolution can add conjugate-error terms and change the required loss and gain protection, although some physical cases retain the same conditions. Higher-order codes improve performance only when recovery infidelity is sufficiently small.

  • No-jump effects: No-jump evolution inserts additional terms between jumps, so the Kraus operators contain corrections proportional to the timestep expansion parameters.These terms must be included when imposing approximate QEC conditions.
  • Physical cases: No-jump evolution can introduce conjugate error operators that change excitation number differently from jump events and therefore alter L and G.This effect does not necessarily change N because the total number of ladder operators at the relevant order can remain unchanged.
  • Physical cases: When Â_i†Â_i is a function of n̂ or the error is Hermitian, no-jump evolution produces the same error conditions as the jump-only analysis.For the stated example, this leaves the conditions unchanged from the analysis without no-jump evolution.
  • Performance and limitation: The largest unrecoverable error for an L-loss-protected code is losing L + 1 photons during a recovery timestep.Total error also includes recovery infidelity η from unfaithful gates and imprecise measurements.
  • Performance and limitation: Higher-order codes provide a performance benefit only when recovery infidelity η is small.The optimal timestep balances unrecoverable-error rates against recovery-process infidelity.

Appendix G: Hardware proposal for the two-mode codes

The proposed two-mode hardware uses two cavities or two modes of one cavity with dispersive coupling to a common transmon. Controlled drives and commutator-based Hamiltonian synthesis provide universal control of the multimode system.

  • Hardware configuration: A two-cavity system with a dispersively coupled common transmon is sufficient to realize the two-mode codes.Two distinct modes of one cavity provide an alternative configuration.
  • Hamiltonian resources: The dispersive coupling, cavity drives, and qubit drives are externally controlled ingredients of the hardware Hamiltonian.The dispersive interaction couples each mode's annihilation operator to the qubit Pauli operator.
  • Hamiltonian synthesis: Approximate Hamiltonian identities generate sums, commutators, and higher-order nested commutators from the available interactions.These identities are applied repeatedly to synthesize more complex effective Hamiltonians.
  • Universal control: Single-mode universal control together with a beamsplitter interaction is sufficient for universal control of the multimode system.The required beamsplitter interaction is x_j p_k − x_k p_j, equivalent to a_j a_k† interactions.
  • Universal control: Additional separately controlled qubits may simplify control pulses, but they are not necessary in principle.The single-qubit, two-cavity configuration therefore supplies the needed control resources.

Appendix H: Optimized bosonic codes

The optimized bosonic codes relax the binomial codes' sparse parity structure to reduce uncorrectable-error rates while retaining exact correction of selected discrete loss errors. The trade-off is more sophisticated, potentially lower-fidelity recovery control, and most solutions are numerical.

  • Motivation: The binomial code's performance is dominated by the rate of losing L + 1 photons, the largest uncorrectable-error rate.This scaling is the cost of the code words' definite generalized photon-number parity.
  • Construction: Optimized code words minimize the dominant L + 1-photon-loss rate while satisfying quantum error-correction criteria for {I, a, a^2, …, a^L}.The construction first enforces exact correction for discrete loss errors through L losses, then optimizes the leading uncorrectable term.
  • Limitations: Most optimized solutions are numerical, and their detailed exploration and classification are left for future work.This is the main stated scope boundary of the optimization study.
  • Results: The optimized Ē1 code improves both correctable and uncorrectable error rates relative to the corresponding binomial code.The comparison uses P2/κδt = 2κδt for the corresponding binomial code.
  • Trade-off: Relaxing definite parity requires general projective loss-detection measurements instead of straightforward parity measurements.Such projections may be feasible with current superconducting technology but likely at lower fidelity than parity measurements.

1. Approximate quantum error correction under continuous-time dissipative evolution

For continuous-time photon loss, the optimized code uses projective measurements and unitary recovery to correct the channel approximately. The correction order is κδt for L = 1, while higher-loss protections can require timestep-dependent code words.

  • Error mechanism: The recovery error is of order (κδt)^2 because no-jump evolution and photon-loss errors can overlap and be misidentified.The corresponding error probability is reported as approximately (κδt)^2.
  • Syndrome detection: Breaking the binomial parity structure prevents straightforward parity detection and necessitates general projections for photon-loss identification.This connects the optimized-code recovery procedure to its altered syndrome measurement.
  • Recovery operation: For the optimized code, photon-loss recovery uses a projective test for the loss-error subspace followed by conditional unitary state transfers.The recovery is R = {U0(I − P1), U1P1}, with U1 repumping the photon-loss error words.
  • Correction order: The optimized code corrects continuous-time photon loss to order O(κδt) for L = 1.For L = 1, the approximate continuous-time conditions are equivalent to the discrete-loss QEC conditions.
  • Higher-order protection: For L > 1, code words optimized for continuous-time protection to order (κδt)^L generally depend on the timestep.The continuous-time error operators can be written as approximately a^ℓE0, reducing the problem to bare loss-error QEC conditions with timestep-dependent normalization.
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