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High-threshold fault-tolerant quantum computation with analog quantum error correction

Kosuke Fukui, Akihisa Tomita, Atsushi Okamoto, Keisuke Fujii

arXiv:1712.00294v3quant-ph

TL;DR

Continuous-variable fault-tolerant quantum computation requires GKP qubits, but experimentally achievable squeezing remains below existing theoretical requirements. This work combines analog quantum error correction with surface-code topological measurement-based computation and postselected cluster-state construction, numerically reducing the required squeezing level to less than 10 dB.

  • Problem

    Existing continuous-variable fault-tolerant quantum computation requires squeezing beyond currently achievable levels, while analog errors accumulate during computation.

  • Method

    The paper applies analog quantum error correction to the surface code and uses postselected measurements with fusion gates to construct cluster states with low error accumulation.

  • Results

    Less than 10 dB of squeezing is numerically sufficient for the proposed scheme, with a 6.2 dB improvement over an existing continuous-variable fault-tolerant computation scheme.

  • Takeaways & Limitations

    The approach alleviates the squeezing requirement for practical large-scale measurement-based quantum computation with GKP qubits.

Abstract

from arXiv · show

To implement fault-tolerant quantum computation with continuous variables, the Gottesman-Kitaev-Preskill (GKP) qubit has been recognized as an important technological element. However,it is still challenging to experimentally generate the GKP qubit with the required squeezing level, 14.8 dB, of the existing fault-tolerant quantum computation. To reduce this requirement, we propose a high-threshold fault-tolerant quantum computation with GKP qubits using topologically protected measurement-based quantum computation with the surface code. By harnessing analog information contained in the GKP qubits, we apply analog quantum error correction to the surface code.Furthermore, we develop a method to prevent the squeezing level from decreasing during the construction of the large scale cluster states for the topologically protected measurement based quantum computation. We numerically show that the required squeezing level can be relaxed to less than 10 dB, which is within the reach of the current experimental technology. Hence, this work can considerably alleviate this experimental requirement and take a step closer to the realization of large scale quantum computation.

I. INTRODUCTION

GKP qubits enable fault-tolerant continuous-variable computation, but existing schemes require squeezing beyond near-term experimental capabilities. The paper proposes analog-error-informed surface-code MBQC and low-error cluster-state construction to reduce this requirement.

  • Motivation: GKP encoding digitizes continuous variables so standard quantum error correction can address analog errors such as photon-loss-induced deviations.GKP qubits also retain optical implementation advantages, including beam-splitter entanglement and Gaussian implementations of qubit-level Clifford gates.
  • Motivation: 14.8–20.5 dB squeezing is required by existing CV-FTQC schemes, whereas experimentally promising GKP-qubit proposals target around 10 dB.The paper identifies this gap as a major barrier to large-scale CV-FTQC.
  • Proposed approach: The proposed high-threshold FTQC applies analog QEC to the surface code and uses postselected measurements to construct cluster states with low error accumulation.The postselected construction is designed to prevent qubit-level errors from propagating as the number of entangling gates increases.
  • Proposed approach: The method targets topologically protected measurement-based quantum computation on a 3D cluster state while harnessing analog information contained in GKP measurement outcomes.Analog deviations reflect noise and can improve error tolerance through likelihood-based processing.
  • Evaluation: The paper evaluates the required squeezing through leading-order unheralded-error calculations and simulations of analog QEC on fusion-gate-built 3D cluster states.The paper structure separates model review, proposal, threshold calculation, and discussion.

A. The GKP qubit

The GKP qubit encodes a qubit in oscillator quadratures using approximate grid states. Finite squeezing makes the code states nonorthogonal, while Gaussian-channel displacement noise increases quadrature variance.

  • Encoding: GKP encoding stores a qubit in an oscillator’s q and p quadratures, enabling correction of small displacement errors and their superpositions.The code basis consists of regularly spaced Gaussian peaks.
  • Encoding: Finite squeezing produces approximate, nonorthogonal code states, unlike the orthogonal basis obtained in the infinite-squeezing limit.The peaks have width σ, separation √π, and an envelope width 1/σ.
  • Measurement errors: A measurement can misidentify the encoded bit when finite squeezing causes the observed quadrature to deviate from its peak.Correct identification is associated with deviations within the decision interval around the peak.
  • Noise model: The Gaussian quantum channel models independent q- and p-quadrature displacements using zero-mean Gaussian variables with variance ξ^2.The channel preserves Gaussian-peak positions in measurement distributions while increasing their variance.
  • Noise model: The analysis uses a code-capacity noise model with one variance σ^2 combining initial GKP squeezing and Gaussian-channel degradation.This variance parameterizes the effective noise evaluated in the subsequent QEC analysis.

B. Analog quantum error correction

Analog QEC retains information about the magnitude of GKP measurement deviations instead of using only binary decisions. Likelihood-based decoding improves error correction, including for a single block code and the surface code.

  • Analog likelihoods: Analog QEC uses the GKP measurement outcome qm to choose the bit value k that minimizes the deviation from the nearest code peak.The outcome is represented as qm = qk + Δm, with qk determined by the nearest peak.
  • Analog likelihoods: Digital QEC calculates likelihoods from binary outcomes, whereas analog QEC uses Gaussian likelihoods based on the measured deviation magnitude.The analog likelihood distinguishes correct and incorrect bit decisions using the deviation information.
  • Likelihood model: The likelihood approximation uses simple Gaussian functions rather than the full periodic sum because neighboring Gaussian tails are sufficiently small to neglect.The exact likelihood would account for the periodic superposition of Gaussian peaks.
  • Decoding: Joint likelihoods over multiple qubits can reduce whole-codeword decision errors by selecting the most likely candidate.Unlike prior digital-QEC improvement under the code-capacity model, analog QEC can improve a single block code such as the three-qubit flip code.
  • Performance: The C4/C6 concatenated code with analog QEC reaches a standard deviation of approximately 0.607, matching the reported hashing bound for Gaussian-channel quantum capacity.The passage describes this as optimal performance against the Gaussian quantum channel.

C. Analog QEC with a surface code

The surface-code simulations test whether analog QEC improves fault tolerance under ideal and noisy syndrome measurements. Analog information lowers logical errors and raises the phenomenological-noise threshold relative to digital QEC.

  • Code-capacity noise: Analog QEC is evaluated on the surface code under code-capacity noise with ideal syndrome measurements.The simulations use minimum-weight perfect matching decoding and code distances d = 5,7,9,….
  • Code-capacity noise: The simulations compare logical error probabilities for digital and analog QEC across surface-code distances.Figure 2 uses 50000 samples for digital QEC and 50000 or 10000 samples for analog QEC, depending on distance.
  • Code-capacity noise: ∼0.607 is achieved with analog QEC, compared with ∼0.542 using only binary information in digital QEC.The analog-QEC value is close to the hashing bound of the Gaussian quantum channel.
  • Phenomenological noise: Analog QEC also suppresses errors under phenomenological noise in topologically protected measurement-based quantum computation.Figure 3 considers noisy syndrome measurements on a 3D cluster state.
  • Phenomenological noise: 0.41 to 0.47 is the threshold improvement under phenomenological noise, corresponding to squeezing improvement from 4.7 dB to 3.5 dB.This reduces the required squeezing level by 1.2 dB relative to digital QEC.

A. The accumulation of errors during the construction of the 3D cluster state

Constructing the 3D cluster state using only CZ gates propagates quadrature deviations and accumulates correlated errors. Because each cluster-state qubit has four neighbors, the resulting variance substantially worsens the squeezing requirement.

  • Error propagation: The paper defines the CZ-induced error propagation as a correlated error in the finite-squeezing construction model.This model describes a 3D cluster state built from single GKP qubits using only CZ gates.
  • Error propagation: CZ-gate operations displace q and p quadrature deviations, changing the p-quadrature variance of control and target qubits.The q-quadrature variance does not change, while p-quadrature error propagation increases bit-value misidentification.
  • Accumulation: 5σ^2 is the p-quadrature variance produced by straightforward construction when each cluster-state qubit connects to four neighbors.The variance assumes a single GKP-qubit variance of σ^2.
  • Accumulation: 13.7 dB replaces 4.7 dB as the required squeezing level when moving from phenomenological to correlated noise.The paper proposes postselected measurement to avoid this error accumulation.

B. The postselected measurement

The postselected measurement uses analog measurement outcomes to discard results with excessive deviations before entanglement generation. This reduces measurement errors at the cost of nondeterministic operation and lower success probability.

  • Measurement rule: Postselected measurement sets an upper limit vup on the measured deviation and discards outcomes exceeding that limit.The conventional measurement accepts outcomes using a broader decision boundary, whereas postselection restricts accepted deviations.
  • Measurement rule: The measurement is correct when the true deviation remains within the correct region, while deviations beyond the error boundary cause bit-value misidentification.The error probability decreases as vup increases, at the cost of measurement success probability.
  • Performance trade-off: Epost and PSuc quantify, respectively, the postselected measurement error probability and success probability.For a qubit with variance 3σ^2, both quantities are plotted against squeezing level for several vup values.
  • Performance trade-off: Both Epost and PSuc decrease as the squeezing level changes in the variance-3σ^2 example.The example represents a qubit frequently occurring in Bell measurements during cluster-state construction.
  • Fusion-gate construction: vup = 2√π/5 is applied during 3D cluster-state construction to prevent fusion-gate deviations from propagating qubit-level errors.The resulting fusion operation is nondeterministic and is handled using a divide-and-conquer approach.

C. The 3D cluster state construction

The construction uses single-qubit error correction and postselected fusion measurements to build hexagonal and then 3D cluster states while controlling error propagation and squeezing degradation.

  • 3-tree preparation: The initial 3-tree cluster state contains one node and two leaf qubits prepared with variance σ^2 in both quadratures and entangled by a CZ gate.The p-quadrature variances increase to 3σ^2 for the node and 2σ^2 for the leaves.
  • Single-qubit QEC: Single-qubit QEC entangles the node with an ancilla through a CNOT gate and postselects the ancilla measurement to reduce node-qubit errors.Outcomes beyond vup are discarded and the procedure is restarted.
  • Single-qubit QEC: The single-qubit QEC reduces the node’s p-quadrature variance from 3σ^2 to σ^2.The q-quadrature variance increases from σ^2 to 2σ^2, but the text states this does not affect the threshold value.
  • Hexagonal cluster construction: Postselected fusion combines 3-tree and 4-tree cluster states into larger trees, then constructs the hexagonal cluster state from six 5-tree cluster states.Fusion avoids increasing qubit deviations, while postselection prevents qubit-level errors from propagating during construction.
  • 3D cluster construction: The 3D cluster state is generated deterministically by fusing neighboring hexagonal cluster states without postselected measurement.This final step introduces an unheralded error associated with a qubit of variance 3σ^2.
  • Squeezing preservation: The proposed construction avoids squeezing degradation compared with a CZ-only construction, which yields node variance 5σ^2.The conventional method’s corresponding node squeezing level is −10log10(10σ^2).

IV. THRESHOLD CALCULATION FOR TOPOLOGICALLY PROTECTED MBQC

The threshold calculation models per-node unheralded errors, first estimates the required squeezing analytically, and then validates the result through surface-code simulations with analog QEC.

  • Error model: The per-node unheralded error Etot comprises node-qubit error, postselected-measurement errors, and deterministic-fusion error.The calculation treats these contributions under a phenomenological noise model characterized by σ^2 when correlated errors are negligible.
  • Leading-order threshold: 10.5 dB is the required squeezing level from the leading-order calculation without analog QEC for Etot = 3.0%.The calculation uses vup = 2√π/5.
  • Analog QEC: 9.3 dB is the leading-order required squeezing level with analog QEC, representing a 1.2 dB improvement over digital QEC.The improvement was obtained from phenomenological-noise simulations of topologically protected MBQC.
  • Detailed simulation: The simulated standard-deviation threshold improves from 0.208 to 0.228, corresponding to squeezing improvement from 10.6 dB to 9.8 dB.The detailed calculation uses minimum-weight perfect matching and an independent error model.
  • Comparison with existing schemes: The proposed analog-QEC scheme with postselected measurement improves the required squeezing level by 6.2 dB relative to the existing CV-FTQC scheme.The numerical setting is vup = 2√π/5.
  • Resource estimate: 9.8 dB requires an estimated 9.2 × 10^6 3-tree cluster states per hexagonal cluster state.At this squeezing level, PSuc(3σ^2) and PSuc(4σ^2) are 34.6% and 30.2%, respectively.

V. DISCUSSION AND CONCLUSION

The paper presents analog QEC and postselected cluster-state construction as a route toward lower-squeezing continuous-variable fault-tolerant quantum computation, while identifying implementation assumptions and broader analog-QEC performance.

  • Discussion and conclusion: The proposed method combines analog QEC on the surface code with postselected measurements that limit error accumulation during cluster-state construction.The authors report a required squeezing level below 10 dB for the resulting 3D cluster states.
  • Discussion and conclusion: Analog QEC reaches approximately 0.607 for the Gaussian quantum channel, close to its hashing bound.The paper describes this as evidence of strong analog-QEC performance under ideal syndrome measurements.
  • Novelty: Analog information had been used in classical error correction and qubit readout but was previously left unexploited for QEC performance with superposition preservation.The paper identifies analog QEC as improving performance toward high-threshold FTQC.
  • Experimental relevance: A promising circuit-QED proposal can prepare GKP qubits around 10 dB, the experimental regime targeted by the proposed scheme.The paper connects this regime to progress toward large-scale quantum computation.
  • Physical implementation: The implementation assumes architectures capable of handling nondeterministic fusion gates for topologically protected MBQC.The authors state that their method can be implemented straightforwardly in such architectures.
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