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arXiv:2208.00581 · Physical Review A

Parallel syndrome extraction with shared flag qubits for Calderbank-Shor-Steane codes of distance three

Pei-Hao Liou, Ching–Yi Lai

Published Feb 17, 2023• 8 citations
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Abstract

To perform achieve fault-tolerant quantum computation, one can use flagged syndrome extraction with fewer ancilla qubits. However, it suffers from long circuit depth if one stabilizer is measured at a time. Previously, Reichardt showed that it is possible to measure multiple stabilizers with at most one shared flag qubit for certain small quantum codes. In this paper, we propose a procedure for general Calderbank-Shor-Steane codes of distance three so that multiple $Z$ stabilizers ($X$ stabilizers) can be fault tolerantly measured in parallel with one shared flag qubit. We simulate the memory and computation pseudothresholds for various code schemes. In particular, our parallel scheme based on Shor's nine-qubit code performs better than known seven- and nine-qubit schemes in the literature.

DOI
10.1103/physreva.107.022614

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To perform achieve fault-tolerant quantum computation, one can use flagged syndrome extraction with fewer ancilla qubits. However, it suffers from long circuit depth if one stabilizer is measured at a time. Previously, Reichardt showed that it is possible to measure multiple stabilizers with at most one shared flag qubit for certain small quantum codes. In this paper, we propose a procedure for general Calderbank-Shor-Steane codes of distance three so that multiple ZZZ stabilizers (XXX stabilizers) can be fault tolerantly measured in parallel with one shared flag qubit. We simulate the memory and computation pseudothresholds for various code schemes. In particular, our parallel scheme based on Shor's nine-qubit code performs better than known seven- and nine-qubit schemes in the literature.
Given that quantum error correction processes are unreliable, an efficient error syndrome extraction circuit should use fewer ancillary qubits, quantum gates, and measurements, while maintaining low circuit depth, to minimizing the circuit area, roughly defined as the product of circuit depth and the number of physical qubits. We propose to design parallel flagged syndrome extraction with shared flag qubits for quantum stabilizer codes. Versatile parallelization techniques are employed to minimize the required circuit area, thereby improving the error threshold and overall performance. Specifically, measurement outcomes across multiple rounds of syndrome extraction are integrated into a lookup table decoder, enabling parallelization of multiple stabilizer measurements with shared flag qubits. Additionally, we introduce an adaptive technique to reduce the overhead from excessive syndrome extraction. We present flag-sharing and fully parallel schemes for the[ ⁣[17,1,5] ⁣][\![17,1,5]\!][[17,1,5]],[ ⁣[19,1,5] ⁣][\![19,1,5]\!][[19,1,5]]Calderbank-Shor-Steane (CSS) codes and the[ ⁣[5,1,3] ⁣][\![5,1,3]\!][[5,1,3]]non-CSS code, where the[ ⁣[5,1,3] ⁣][\![5,1,3]\!][[5,1,3]]implementation achieves the minimum known circuit area. Numerical simulations have demonstrated improved pseudo-thresholds for these codes by up to an order of magnitude compared to previous schemes in the literature.
Flag qubits have recently been proposed in syndrome extraction circuits to detect high-weight errors arising from fewer faults. The use of flag qubits allows the construction of fault-tolerant protocols with the fewest number of ancillas known to date. In this work, we prove some critical properties of Calderbank-Shor-Steane (CSS) codes constructed from classical cyclic codes that enable the construction of a flag fault-tolerant error correction scheme. We then develop fault-tolerant protocols as well as a family of circuits for flag fault-tolerant error correction and operator measurement, requiring only four ancilla qubits and applicable to cyclic CSS codes of distance 3. The measurement protocol can be further used for logical Clifford gate implementation via quantum gate teleportation. We also provide examples of cyclic CSS codes with large encoding rates.
In this work, we present a low-depth flagged syndrome extraction strategy for Calderbank-Shor-Steane (CSS) codes of distance 3. We demonstrate that if one stabilizer generator of a distance-3 CSS code measured using the flagged circuit has and only has two distinct qubit supports with certain stabilizers of opposite types, then the number of stabilizers required for the second round of measurements can be reduced. The number of these stabilizers depends on the weight of the measured generator. This novel strategy utilizes past syndrome information and elements from the stabilizer group, enabling the implementation of a flag-FTEC protocol for distance-3 CSS codes with lower-depth syndrome extraction circuits compared to previously flag fault-tolerant protocol if the total weight of these stabilizers is less than that of stabilizer generators. Based on our strategy, we also improve the flag-FTEC protocol for cyclic CSS codes of distance 3, which requires fewer stabilizer measurements. We utilize the[[14,2,3]][[{14,2,3}]][[14,2,3]]CSS code,[[15,1,3]][[{15,1,3}]][[15,1,3]]quantum Reed-Muller code and[[15,7,3]][[{15,7,3}]][[15,7,3]]cyclic CSS code to demonstrate how the strategy operates. Furthermore, we also provide a low-depth flagged syndrome extraction strategy for a[[19,1,5]][[{19,1,5}]][[19,1,5]]CSS-type code.
In another work [S. Huang and K. R. Brown, Phys. Rev. Lett. 127, 090505 (2021)] we introduced syndrome extraction methods for Calderbank-Shor-Steane quantum error-correction codes that interpolate between the well-known Shor and Steane syndrome extraction methods. Here we provide detailed proofs of the main theorems and show that up to gate ordering there is a one-to-one correspondence between extraction gadgets and partitions of the parity check matrix. Operationally, all the circuits in our framework can be obtained by merging ancilla qubits of Shor error correction with certain rules, which enables us to design fault-tolerant syndrome extraction circuits for given specific codes. We then apply our construction to the toric code and provide a detailed analysis of the time overhead of fault tolerance. In particular, we construct a syndrome extraction family whose time overhead smoothly varies from Shor to Steane error correction. To understand the potential advantage, we consider two simplified error models: no errors on the ancilla block and uncorrelated errors on the ancilla block. We study the threshold behavior of the family for both error models and show its potential advantage for quantum architectures with long coherence times.
Abstract Steane’s seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. However, the two-qubit error-correction technique, known as ‘flagged’ syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane’s error-correction method, and also conforms to planar geometry constraints. We further show that Steane’s code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.
We reduce the extra qubits needed for two fault-tolerant quantum computing protocols: error correction, specifically syndrome bit measurement, and cat state preparation. For distance-three fault-tolerant syndrome extraction, we show an exponential reduction in qubit overhead over the previous best protocol. For a weight- w stabilizer, we demonstrate that stabilizer measurement tolerating one fault needs at most ⌈log2⁡w⌉+1 ancilla qubits. If qubits reset quickly, four ancillas suffice. We also study the preparation of entangled cat states, and prove that the overhead for distance-three fault tolerance is logarithmic in the cat state size. These results apply both to near-term experiments with a few qubits, and to the general study of the asymptotic resource requirements of syndrome measurement and state preparation.With a flag qubits, previous methods use O(a) flag patterns to identify faults. In order to use the same flag qubits more efficiently, we show how to use nearly all 2a possible flag patterns, by constructing maximal-length paths through the a -dimensional hypercube.