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Physical Review A

Constructions for measuring error syndromes in Calderbank-Shor-Steane codes between Shor and Steane methods

Shilin Huang, Kenneth R. Brown

Published Aug 25, 2021• 6 citations

Abstract

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.

DOI: 10.1103/physreva.104.022429

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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.

Fault-tolerant quantum error correction requires the measurement of error syndromes in a way that minimizes correlated errors on the quantum data. Steane and Shor ancilla are two well-known methods for fault-tolerant syndrome extraction. In this Letter, we find a unifying construction that generates a family of ancilla blocks that interpolate between Shor and Steane. This family increases the complexity of ancilla construction in exchange for reducing the rounds of measurement required to fault tolerantly measure the error. We then apply this construction to the toric code of size L×L and find that blocks of size m×m can be used to decode errors in O(L/m) rounds of measurements. Our method can be applied to any Calderbank-Shor-Steane code and presents a new direction for optimizing fault-tolerant quantum computation.

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.

Fault-tolerant state preparation is essential for reliable quantum error correction, particularly in Steane-type error correction, which relies on robust ancilla states for syndrome readout. One method of fault-tolerant state preparation is to initialize multiple ancilla states and check them against each other to detect problematic errors. In the worst case, the number of states required for successful initialization grows polynomially with the code distance, but it has been shown that this can be reduced to constant ancilla overhead-in the best case, only four states are required. However, existing techniques for finding low-overhead initialization schemes are limited to codes with large symmetry groups, such as the Golay code. In this work, we propose a general, automated synthesis methodology for Steane-type fault-tolerant state preparation circuits that applies to arbitrary Calderbank-Shor-Steane (CSS) codes and does not rely on code symmetries. We apply the proposed methods to various CSS codes up to a distance of seven and simulate the successful fault-tolerant initialization of logical basis states under circuit-level depolarizing noise. The circuits synthesized using the proposed methodology provide an important step towards experimental realizations of high-fidelity ancilla states for near-term demonstration of fault-tolerant quantum computation.

Fault-tolerant quantum computation using quantum error-correcting codes requires fault-tolerant constructions of nontransversal gates. Shor proposed a fault-tolerant construction of a nontransversal gate, i.e., the Toffoli gate for a family of the Calderbank-Shor-Steane codes which has even-weight stabilizer generators and odd-weight logical operators. In fact, as shown in this paper, error propagations in the ancilla preparation of Shor's construction lead to logical errors. This work provides a solution to this problem through the insertion of bit flip error correction. Finally, error rates of a modified construction are evaluated with use of the concatenated Steane codes, and the same accuracy threshold as transversal gates is obtained in cases where ancillary systems can be prepared just in time.

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.