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QVECTOR: an algorithm for device-tailored quantum error correction

Peter D. Johnson, Jonathan Romero, Jonathan Olson, Yudong Cao, Alán Aspuru-Guzik

arXiv:1711.02249v1quant-ph

TL;DR

The paper evaluates QVECTOR through simulated noise models and circuit-optimization procedures, while analyzing how approximate unitary 2-designs estimate average fidelity. Experimental implementations may be limited by sampling and SPAM errors affecting the fidelity cost function.

Abstract

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Current approaches to fault-tolerant quantum computation will not enable useful quantum computation on near-term devices of 50 to 100 qubits. Leading proposals, such as the color code and surface code schemes, must devote a large fraction of their physical quantum bits to quantum error correction. Building from recent quantum machine learning techniques, we propose an alternative approach to quantum error correction aimed at reducing this overhead, which can be implemented in existing quantum hardware and on a myriad of quantum computing architectures. This method aims to optimize the average fidelity of encoding and recovery circuits with respect to the actual noise in the device, as opposed to that of an artificial or approximate noise model. The quantum variational error corrector (QVECTOR) algorithm employs a quantum circuit with parameters that are variationally-optimized according to processed data originating from quantum sampling of the device, so as to learn encoding and error-recovery gate sequences. We develop this approach for the task of preserving quantum memory and analyze its performance with simulations. We find that, subject to phase damping noise, the simulated QVECTOR algorithm learns a three-qubit encoding and recovery which extend the effective T2 of a quantum memory six-fold. Subject to a continuous-time amplitude- plus phase-damping noise model on five qubits, the simulated QVECTOR algorithm learns encoding and decoding circuits which exploit the coherence among Pauli errors in the noise model to outperform the five-qubit stabilizer code and any other scheme that does not leverage such coherence. Both of these schemes can be implemented with existing hardware.

A Noise models

The simulations model phase-damping and amplitude-plus-phase-damping noise using Kraus operators, with parameters tied to T1, T2, gate errors, and wait times. The APD setting also compares the coherent channel with its Pauli-twirled approximation.

Register noise

  • The n-qubit noise process is constructed from tensor products of single-qubit Kraus operators under an independent-noise assumption.The resulting channel sums over all tuples of single-qubit Kraus indices.Source: A Noise models · p. 12

Noise channels

  • QVECTOR simulations use phase-damping (PD) and amplitude-plus-phase-damping (APD) channels represented through standard Kraus-operator noise models.The models are parameterized by decoherence times and process duration.Source: A Noise models · p. 12

Phase damping

  • The PD approximation uses p = (1 − e^−tstep/T2)/2, but the authors note that it does not fully capture the Xmon specifications motivating the simulations.They present it as a simple starting point for more sophisticated exploration.Source: A Noise models · p. 12

Three-qubit phase damping

  • The first simulation incorporates recovery-gate errors into three independent PD channels, yielding a per-qubit phase-error probability p = 0.091.Single- and two-qubit gate error rates determine the component probabilities p′ = 0.00110 and p′′ = 0.00113.Source: A Noise models · p. 12

Amplitude-plus-phase damping

  • The APD simulation uses T2 = 19 µs and T1 = 57 µs, making dephasing more dominant than amplitude damping and differing from isotropic depolarizing noise.The APD channel’s coherent Pauli-error superpositions are contrasted with the incoherent Pauli-twirled approximation.Source: A Noise models · pp. 12–13

B Numerical simulation of QVECTOR

QVECTOR simulations prepare logical states, encode them with programmable circuits, apply noise, recover or decode them, and estimate average fidelity. The circuits use hardware-compatible rotation and entangling layers, while numerical optimization and fidelity-estimation analyses address practical training and accuracy.

Simulation protocol

  • The simulation pipeline prepares six one-qubit stabilizer states, encodes them, applies noise, performs recovery, decodes, and computes average fidelity.The six states form a one-qubit 2-design; the phase-damping example uses n = 3, k = 2, and r = 2.Source: B Numerical simulation of QVECTOR · p. 13

Programmable circuits

  • Programmable circuits use fixed gate networks with optimized rotation angles, alternating single-qubit rotations and entangling blocks suited to current hardware.The phase-damping circuits use repeated controlled-Zθ, Xθ, and Zθ layers, while the APD encoder uses arbitrary single-qubit rotations.Source: B Numerical simulation of QVECTOR · pp. 13–14, Figure 4 · p. 14

Parameter optimization

  • The numerical procedure initializes 100 random parameter settings, selects the highest-fidelity one, and launches parallel L-BFGS optimizations because the cost function may contain local optima.The simulations use central finite differences with step size h = 10^-6.Source: B Numerical simulation of QVECTOR · p. 13

Experimental considerations

  • Experimental QVECTOR costs can be affected by sampling and SPAM errors, motivating noise-tolerant optimizers and randomized benchmarking for more reliable fidelity estimates.The authors specifically suggest SPSA, Basin-Hopping, and randomized benchmarking.Source: B Numerical simulation of QVECTOR · p. 14

Reference optimization

  • The bi-convex routine alternately optimizes decoding and encoding channels, increasing average fidelity until a threshold is reached but definitively establishing only a lower bound on the optimum.Results can vary with random initialization even though each optimization procedure is otherwise deterministic.Source: B Numerical simulation of QVECTOR · p. 14

C Accuracy of average fidelity estimate from approximate unitary 2-design

The paper analyzes fidelity estimation when efficiently sampled approximate unitary 2-designs replace exact designs, deriving the resulting estimator bias and sampling deviation.

Approximate 2-design sampling

  • The approximate design is used because average fidelity depends only on second moments, while Haar-random sampling is inefficient.The construction in provides an efficiently implementable alternative, with each measurement producing a binary sample from a biased estimator.Source: C Accuracy of average fidelity… · p. 15

Bias bound

  • The approximate 2-design estimator has bias p_ℓ|⟨F⟩−δ_ℓ|, with p_ℓ decreasing as the number of randomization repetitions grows.The analysis expresses the estimator mean as a convex combination of the actual mean and a channel-dependent term δ_ℓ.Source: C Accuracy of average fidelity… · p. 15
  • The bias is upper bounded by p_ℓ = d^(ℓ+1)/(d^(ℓ+1)+d^(ℓ−2)) after bounding an inner product of quantum states by 1.The bound follows from separability and partial-transpose properties of the auxiliary channel.Source: C Accuracy of average fidelity… · p. 16
  • For d = 2^k, the bias bound scales as O(1/d^ℓ), so increasing the number of approximate-design repetitions suppresses systematic error.The estimator therefore approaches the exact-design average as ℓ increases.Source: C Accuracy of average fidelity… · pp. 15–16

Sampling error

  • After N samples, the estimated fidelity also fluctuates around the biased estimator mean according to a sampling deviation determined by ⟨E_ℓ⟩(1−⟨E_ℓ⟩)/N.Thus total estimation error contains both approximate-design bias and finite-sample variation.Source: C Accuracy of average fidelity… · p. 16
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