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arXiv:1209.0794 · Quantum Information and Computation

Optimal Bacon-Shor codes

John C. Napp, John P. Preskill

Published May 1, 2013• 32 citations
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Abstract

We study the performance of Bacon-Shor codes, quantum subsystem codes which are well suited for applications to fault-tolerant quantum memory because the error syndrome can be extracted by performing two-qubit measurements. Assuming independent noise, we find the optimal block size in terms of the bit-flip error probability $p_X$ and the phase error probability $p_Z$, and determine how the probability of a logical error depends on $p_X$ and $p_Z$. We show that a single Bacon-Shor code block, used by itself without concatenation, can provide very effective protection against logical errors if the noise is highly biased ($p_Z/p_X\gg 1)$ and the physical error rate $p_Z$ is a few percent or below. We also derive an upper bound on the logical error rate for the case where the syndrome data is noisy.

DOI
10.26421/qic13.5-6-9

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We study the performance of Bacon-Shor codes, quantum subsystem codes which are well suited for applications to fault-tolerant quantum memory because the error syndrome can be extracted by performing two-qubit measurements. Assuming independent noise, we find the optimal block size in terms of the bit-flip error probability pXp_XpX​ and the phase error probability pZp_ZpZ​, and determine how the probability of a logical error depends on pXp_XpX​ and pZp_ZpZ​. We show that a single Bacon-Shor code block, used by itself without concatenation, can provide very effective protection against logical errors if the noise is highly biased (pZ/pX≫1)p_Z/p_X\gg 1)pZ​/pX​≫1) and the physical error rate pZp_ZpZ​ is a few percent or below. We also derive an upper bound on the logical error rate for the case where the syndrome data is noisy.
We develop a scheme for fault-tolerant quantum computation based on asymmetric Bacon-Shor codes, which works effectively against highly biased noise dominated by dephasing. We find the optimal Bacon-Shor block size as a function of the noise strength and the noise bias, and estimate the logical error rate and overhead cost achieved by this optimal code. Our fault-tolerant gadgets, based on gate teleportation, are well suited for hardware platforms with geometrically local gates in two dimensions.
Quantum error correction is necessary to perform large-scale quantum computations in the presence of noise and decoherence. As a result, several aspects of quantum error correction have already been explored. These have been primarily studies of quantum memory[1, 2], an important first step towards quantum computation, where the objective is to increase the lifetime of the encoded quantum information. Additionally, several works have explored the implementation of logical gates[3-5]. In this work we study a next step - fault-tolerantly implementing quantum circuits. We choose the [[4,1,2]][[4, 1, 2]][[4,1,2]] Bacon-Shor subsystem code, which has a particularly simple error-detection circuit. Through both numerics and site-counting arguments, we compute pseudo-thresholds for the Pauli error rate ppp in a depolarizing noise model, below which the encoded circuits outperform the unencoded circuits. These pseudo-threshold values are shown to be as high as p=3%p=3\%p=3% for short circuits, and p=0.6%p=0.6\%p=0.6% for circuits of moderate depth. Additionally, we see that multiple rounds of stabilizer measurements give an improvement over performing a single round at the end. This provides a concrete suggestion for a small-scale fault-tolerant demonstration of a quantum algorithm that could be accessible with existing hardware.
Inspired by Knill's scheme for message passing error detection, here we develop a scheme for message passing error correction for the nine-qubit Bacon-Shor code. We show that for two levels of concatenated error correction, where classical information obtained at the first level is used to help interpret the syndrome at the second level, our scheme will correct all cases with four physical errors. This results in a reduction of the logical failure rate relative to conventional error correction by a factor proportional to the reciprocal of the physical error rate.
We optimize fault-tolerant quantum error correction to reduce the number of syndrome bit measurements. Speeding up error correction will also speed up an encoded quantum computation, and should reduce its effective error rate. We give both code-specific and general methods, using a variety of techniques and in a variety of settings. We design new quantum error-correcting codes specifically for efficient error correction, e.g., allowing single-shot error correction. For codes with multiple logical qubits, we give methods for combining error correction with partial logical measurements. There are tradeoffs in choosing a code and error-correction technique. While to date most work has concentrated on optimizing the syndrome-extraction procedure, we show that there are also substantial benefits to optimizing how the measured syndromes are chosen and used. As an example, we design single-shot measurement sequences for fault-tolerant quantum error correction with the 16-qubit extended Hamming code. Our scheme uses 10 syndrome bit measurements, compared to 40 measurements with the Shor scheme. We design single-shot logical measurements as well: any logical Z measurement can be made together with fault-tolerant error correction using only 11 measurements. For comparison, using the Shor scheme a basic implementation of such a non-destructive logical measurement uses 63 measurements. We also offer ten open problems, the solutions of which could lead to substantial improvements of fault-tolerant error correction.
The Bacon-Shor code is a quantum error correcting subsystem code composed of weight 2 check operators that admits a single logical qubit, and has distance ddd on a d×dd \times dd×d square lattice. We show that when viewed as a Floquet code, by choosing an appropriate measurement schedule of the check operators, it can additionally host several dynamical logical qubits. Specifically, we identify a period 4 measurement schedule of the check operators that preserves logical information between the instantaneous stabilizer groups. Such a schedule measures not only the usual stabilizers of the Bacon-Shor code, but also additional stabilizers that protect the dynamical logical qubits against errors. We show that the code distance of these Floquet-Bacon-Shor codes scales as Θ(d/k)Θ(d/\sqrt{k})Θ(d/k​) on an n=d×dn = d \times dn=d×d lattice with kkk dynamical logical qubits, along with the logical qubit of the parent subsystem code. Unlike the usual Bacon-Shor code, the Floquet-Bacon-Shor code family introduced here can therefore saturate the subsystem bound kd=O(n)kd = O(n)kd=O(n). Moreover, several errors are shown to be self-corrected purely by the measurement schedule itself. This work provides insights into the design space for dynamical codes and expands the known approaches for constructing Floquet codes.