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.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.

arXiv:1209.0794Quantum Information and ComputationMay 1, 201332 citations

Short Shor-style syndrome sequences

Nicolas Delfosse, Ben W. Reichardt

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.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.

arXiv:2008.05051Aug 12, 20204 citations

Quantum error correction with bosonic-Bacon-Shor codes

Stefanus Tanuarta, Arne L. Grimsmo

Bulletin of the American Physical SocietyMar 15, 20210 citations

Improved performance of the Bacon-Shor code with Steane's syndrome extraction method

Guillermo Escobar-Arrieta, Mauricio Gutiérrez

We compare Steane's and Shor's syndrome extraction methods on the Bacon-Shor code. We propose a straightforward strategy based on postselection to prepare the logical ${|0\ensuremath{\rangle}}_{L}$ and ${|+\ensuremath{\rangle}}_{L}$ states of the Bacon-Shor code by using flaglike qubits to verify their constituent Greenberger-Horne-Zeilinger states. We perform stabilizer simulations with a circuit-level, depolarizing Pauli error model and find that Steane's method significantly outperforms Shor's. Not only does Steane's method result in pseudothresholds that are about one order of magnitude higher than Shor's, but also its advantage increases monotonically as we go from a distance-3 to a distance-9 Bacon-Shor code. The advantage of Steane's method is the greatest in the regime where gate errors dominate over measurement errors. Some of the circuit constructions we propose for Steane's method are not formally fault tolerant yet outperform the formally fault-tolerant Shor's protocols for experimentally relevant physical error rates. This suggests that constructing formally fault-tolerant circuits that maintain the full code distance is not strictly necessary to guarantee the usefulness of a quantum error-correcting protocol. Despite relying on postselection, we find that our methods can be efficient. These protocols would be naturally implementable on a platform with long-range qubit interactions like trapped ions or neutral atoms.We compare Steane's and Shor's syndrome extraction methods on the Bacon-Shor code. We propose a straightforward strategy based on postselection to prepare the logical ${|0\ensuremath{\rangle}}_{L}$ and ${|+\ensuremath{\rangle}}_{L}$ states of the Bacon-Shor code by using flaglike qubits to verify their constituent Greenberger-Horne-Zeilinger states. We perform stabilizer simulations with a circuit-level, depolarizing Pauli error model and find that Steane's method significantly outperforms Shor's. Not only does Steane's method result in pseudothresholds that are about one order of magnitude higher than Shor's, but also its advantage increases monotonically as we go from a distance-3 to a distance-9 Bacon-Shor code. The advantage of Steane's method is the greatest in the regime where gate errors dominate over measurement errors. Some of the circuit constructions we propose for Steane's method are not formally fault tolerant yet outperform the formally fault-tolerant Shor's protocols for experimentally relevant physical error rates. This suggests that constructing formally fault-tolerant circuits that maintain the full code distance is not strictly necessary to guarantee the usefulness of a quantum error-correcting protocol. Despite relying on postselection, we find that our methods can be efficient. These protocols would be naturally implementable on a platform with long-range qubit interactions like trapped ions or neutral atoms.

arXiv:2403.01659Physical Review AMar 25, 20254 citations

Efficient quotient codes decodable in polynomial time for quantum error correction and cryptography

Mitsuru Hamada

Quantum error-correcting codes that achieve high rates in the Shannon theoretic sense and that are efficiently decodable are presented.

International Symposium on Information Theory and its ApplicationsDec 1, 20081 citation

Fault-tolerant quantum computation with asymmetric Bacon-Shor codes

Peter Brooks, John P. Preskill

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.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.

arXiv:1211.1400Physical Review AMar 7, 201351 citations

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.