Adaptive Syndrome Measurements For Shor-style Error Correction

Theerapat Tansuwannont^1,2, Balint Pato^1,2, and Kenneth R. Brown^1,2,3,4

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 $t=lfloor(d-1)/2rfloor$ 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.

Featured image: The plots of the logical error rate (left) and the average number of syndrome measurement rounds (right) at each physical error rate when the traditional Shor FTEC scheme (blue), the adaptive scheme satisfying the strong FTEC conditions (orange), and the adaptive scheme satisfying the weak FTEC conditions (green) are applied to the hexagonal color code of distance 9 that can correct up to $t=4$ errors. With the adaptive strong and the adaptive weak schemes, the pseudothreshold is improved from $1.19 times 10^{-4}$ to $2.75 times 10^{-4}$ and $2.99 times 10^{-4}$, respectively. At the high physical error rate regime, the average numbers of rounds for the adaptive strong and the adaptive weak schemes reach $2t+1$ and $2t$, while the number for the traditional Shor scheme reaches $(t+1)^2$.

Popular summary

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.

► BibTeX data

@article{Tansuwannont2023adaptivesyndrome, doi = {10.22331/q-2023-08-08-1075}, url = {https://doi.org/10.22331/q-2023-08-08-1075}, title = {Adaptive syndrome measurements for {S}hor-style error correction}, author = {Tansuwannont, Theerapat and Pato, Balint and Brown, Kenneth R.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1075}, month = aug, year = {2023}<br /> }

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Cited by

[1] Joschka Roffe, Lawrence Z. Cohen, Armanda O. Quintavalle, Daryus Chandra, and Earl T. Campbell, “Bias-tailored quantum LDPC codes”, Quantum 7, 1005 (2023).

[2] Lior Ella, Lorenzo Leandro, Oded Wertheim, Yoav Romach, Ramon Szmuk, Yoel Knol, Nissim Ofek, Itamar Sivan, and Yonatan Cohen, “Quantum-classical processing and benchmarking at the pulse-level”, arXiv:2303.03816, (2023).

[3] Andrew Nemec, “Quantum Data-Syndrome Codes: Subsystem and Impure Code Constructions”, arXiv:2302.01527, (2023).

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

[5] Balint Pato, Theerapat Tansuwannont, Shilin Huang, and Kenneth R. Brown, “Optimization tools for distance-preserving flag fault-tolerant error correction”, arXiv:2306.12862, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-08-09 01:48:39). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2023-08-09 01:48:38).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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Source: https://quantum-journal.org/papers/q-2023-08-08-1075/

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Adaptive syndrome measurements for Shor-style error correction

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