Abstract
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance. In this work, we propose a highly effective use of the surface-GKP code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-dimensional qubits. In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates. Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from $0.87\%$ to $0.36\%$ at a GKP squeezing of $12$dB, compared to the case where the simple closest-integer decoding is used. Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by $9.9$dB under the the assumption that finite-squeezing of the GKP states is the dominant noise source. More importantly, we show that a low logical failure rate $p_{L} < 10^{-7}$ can be achieved with moderate hardware requirements, e.g., $291$ modes and $97$ qubits at a GKP squeezing of $12$dB as opposed to $1457$ bare qubits for the standard rotated surface code at an equivalent noise level (i.e., $p=0.36\%$). Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface code decoder. Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds. We also show that a highly-squeezed GKP state of GKP squeezing $\gtrsim 12$dB can be experimentally realized by using a dissipative stabilization method, namely, the Big-small-Big method, with fairly conservative experimental parameters. Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
Highlights
Despite various opportunities offered by noisy intermediate-scale quantum (NISQ) devices [1], faulttolerant quantum-error-correction techniques [2] are essential for executing quantum algorithms intractable by classical computers such as integer factorization [3] and the simulation of real-time dynamics of large quantum systems [4]
Since the relevant noise variance is smaller in the teleportation-based GKP error correction scheme by a factor of 3/2, logical Pauli errors are less likely to occur in the teleportation-based method than in the Steane-type method, given the same GKP squeezing σG(dKBP)
Before analyzing the error-corrected logical CNOT and CZ gate for the GKP qubits, we remark that a noisy GKP error correction with finitely squeezed ancilla GKP states can be understood as an ideal GKP error correction preceded and followed by extra shift errors (−ξq(−), ξp(−)) and (ξq(+), ξp(+)), respectively, where ξq(−), ξp(−), ξq(+), ξp(+) ∼IID N (0, σG2KP) [see Figs. 3(a) and 3(b)]
Summary
Despite various opportunities offered by noisy intermediate-scale quantum (NISQ) devices [1], faulttolerant quantum-error-correction techniques [2] are essential for executing quantum algorithms intractable by classical computers such as integer factorization [3] and the simulation of real-time dynamics of large quantum systems [4]. Going beyond the discussion of fault-tolerant thresholds, we further demonstrate that our surface-GKP code can achieve low logical error rates with moderate resource costs at a reasonable value of GKP squeezing. To achieve a low logical error rate with only moderate resource requirements, we adopt the teleportation-based GKP error correction [40] in our surface-GKP code proposal, instead of the more widely used Steane-type GKP error correction [22].
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