Quantum error correction in the noisy intermediate-scale quantum regime for sequential quantum computing

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Quantum error correction in the noisy intermediate-scale quantum regime for sequential quantum computing

Arvid Rolander, Adam Kinos, and Andreas Walther

Phys. Rev. A 105, 062604 – Published 10 June 2022

Abstract

We use density-matrix simulations to study the performance of three distance three quantum error correction (QEC) codes in the context of the rare-earth (RE) ion-doped crystal platform for quantum computing. We analyze pseudothresholds for these codes when parallel operations are not available, and examine the behavior both with and without resting errors. In RE systems, resting errors can be mitigated by extending the system's ground-state coherence time. For the codes we study, we find that if the ground-state coherence time is roughly 100 times larger than the excited-state coherence time, resting errors become small enough to be negligible compared to other error sources. This leads us to the conclusion that beneficial QEC could be achieved in the RE system with the expected gate fidelities available in the noisy intermediate-scale quantum regime. However, for codes using more qubits and operations, a factor of more than 100 would be required. Furthermore, we investigate how often QEC should be performed in a circuit. We find that for early experiments in RE systems, the minimal $[[5, 1, 3]]$ would be most suitable as it has a high threshold error and uses few qubits. However, when more qubits are available the $[[9, 1, 3]]$ surface code might be a better option due to its higher circuit performance. Our findings are important for steering experiments to an efficient path for realizing beneficial quantum error correcting codes in early RE systems where resources are limited.

Received 8 December 2021
Revised 18 May 2022
Accepted 24 May 2022

DOI:https://doi.org/10.1103/PhysRevA.105.062604

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum computation Quantum error correction

Rare-earth doped crystals

Numerical approximation & analysis

Quantum Information, Science & Technology

Authors & Affiliations

Arvid Rolander , Adam Kinos , and Andreas Walther ^*

Department of Physics, Lund University, SE-22100 Lund, Sweden

^*andreas.walther@fysik.lth.se

Article Text

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Issue

Vol. 105, Iss. 6 — June 2022

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Images

Figure 1
Pseudothresholds for our chosen codes with resting errors either included (a) or excluded (b). The logical error rates were calculated as the average for starting states corresponding to the six axes of the Bloch sphere. In panel (a), we have assumed that the resting errors for the different operations scale with $p_{TQG}$ . This assumes that if we are able to improve $p_{TQG}$ , we will be able to improve the resting errors by the same factor. Black, vertical lines mark the points where the physical two-qubit error rate is $p_{TQG} = 10^{- 3}$ . This is approximately the error rate where a physical system, with today's protocols [2], could be expected to lie [30]. The purple (top), dotted lines show $p_{TQG}$ , and the green (bottom) dotted lines show $p_{TQG}^{2}$ .
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Figure 2
Logical error rate as a function of the ground-state coherence time $T_{2, spin}$ which varies between 1 ms and 1 h. The $x$ axis is scaled by the excited-state coherence time $T_{2, opt} = 2.5$ ms. In this simulation we have used a different model of the resting error compared to the one used in Fig. 1. Here, we assume a constant value of the two-qubit error rate, $p_{TQG} = 10^{- 3}$ , but the resting errors still change due to changes in $T_{2, spin}$ (see Table 2). This model therefore assumes that it is possible to reduce resting errors even though the two-qubit error rate is kept constant. The models give the same values for all errors when $p_{TQG} = 10^{- 3}$ and $T_{2, spin} = 2.5$ ms, which are reasonable values [30] for a RE system using current protocols [2]. These points are marked with black, vertical lines in both Figs. 1 and 2. Furthermore, in a RE system it is possible to reach values for $T_{2, spin}$ of hours [4]. This corresponds to points to the far right in Fig. 2, where the resting errors are small enough compared to other error sources to be neglected. The errors here have approximately the same values as in Fig. 1 when $p_{TQG} = 10^{- 3}$ , which is marked with a black line in Fig. 1. This shows that the resting errors can be reduced enough to become negligible.
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Figure 3
Gain, defined as the physical error when not using error correction divided by the logical error when using error correction calculated from Eq. (4) with $C = 1$ and $N = 75$ . Here, we have assumed that all operations have the same error rate, $p_{TQG}$ . The number of logical gates performed before error correction, as well as the number of physical gates, is given by $n_{g}$ . The red line marks the value of $n_{g}$ which maximizes the gain for every given $p_{TQG}$ . As is shown in Eq. (4), below the threshold the maximum gain is achieved for a constant value of $n_{g}$ , independent of $p_{TQG}$ .
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Figure 4
Gain of the logical circuit with error correction compared to the physical circuit. The $[[5, 1, 3]]$ code achieves maximum gain at $n_{g} = 99$ , the Steane code achieves maximum gain at $n_{g} = 245$ , and Surface-9(1) achieves maximum gain at $n_{g} = 142$ .
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Figure 5
Total error of the logical circuit with QEC compared to the physical circuit for the optimal $n_{g}$ which maximizes the gain shown in Fig. 4. The $[[5, 1, 3]]$ code uses $n_{g} = 99$ , the Steane code uses $n_{g} = 245$ , and Surface-9(1) uses $n_{g} = 142$ .
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