Adaptive syndrome measurements for Shor-style error correction

Theerapat Tansuwannont1,2, Balint Pato1,2, and Kenneth R. Brown1,2,3,4

1Duke Quantum Center, Duke University, Durham, NC 27701, USA
2Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA
3Department of Physics, Duke University, Durham, NC 27708, USA
4Department of Chemistry, Duke University, Durham, NC 27708, USA

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Abstract

The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to prevent propagation of errors caused by gate faults. For a stabilizer code of distance $d$ that can correct up to $t=\lfloor(d-1)/2\rfloor$ errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated $t+1$ times in a row; in the worst-case scenario, $(t+1)^2$ rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions require no more than $(t+3)^2/4-1$ rounds and $(t+3)^2/4-2$ rounds, respectively, and are applicable to any stabilizer code. Our simulations of FTEC protocols with the adaptive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance $d$, our FTEC protocols with the adaptive schemes require no more than $d$ rounds on average.

To build a large-scale quantum computer, one needs to ensure that errors in a quantum circuit are under control. One way to do so is by using a quantum error-correcting code to encode the quantum data, performing syndrome measurements to determine potential errors, then applying an error correction operator to remove the errors. However, quantum operations such as gates and qubit measurements during the syndrome measurements can be faulty and introduce new errors. The Shor fault-tolerant error correction (FTEC) scheme handles this problem by using circuits with transversal gates and a heavily entangled 'cat' state to repeatedly perform syndrome measurements. In the traditional scheme, the measurements are done until the outcomes are the same for a certain fixed number of rounds.

In this work, we introduce an adaptive technique for syndrome measurements that can improve the Shor FTEC scheme while maintaining its applicability. Here we continuously observe a pattern of the syndrome measurement outcomes, then estimate the minimum number of faults that occurred in the syndrome measurement process and caused the pattern. With this information, we can reduce the number of rounds required to ensure that the repeated syndrome is suitable for error correction. Our simulations on small quantum error-correcting codes show that the adaptive FTEC schemes can significantly reduce the average number of rounds compared to the traditional Shor FTEC scheme, leading to fewer required quantum resources. The adaptive scheme can also improve the fault-tolerant pseudothreshold, the error probability below which the encoded qubit has better fidelity than the unencoded one, making FTEC more practical.

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[7] Shilin Huang, Kenneth R. Brown, and Marko Cetina, "Comparing Shor and Steane Error Correction Using the Bacon-Shor Code", arXiv:2312.10851, (2023).

[8] Benjamin Anker and Milad Marvian, "Flag Gadgets based on Classical Codes", arXiv:2212.10738, (2022).

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