Abstract
Random quantum circuits have played a central role in establishing the computational advantages of near-term quantum computers over their conventional counterparts. Here, we use ensembles of low-depth random circuits with local connectivity in spatial dimensions to generate quantum error-correcting codes. For random stabilizer codes and the erasure channel, we find strong evidence that a depth random circuit is necessary and sufficient to converge (with high probability) to zero failure probability for any finite amount below the optimal erasure threshold, set by the channel capacity, for any . Previous results on random circuits have only shown that depth suffices or that depth suffices for all-to-all connectivity (). We then study the critical behavior of the erasure threshold in the so-called moderate deviation limit, where both the failure probability and the distance to the optimal threshold converge to zero with . We find that the requisite depth scales like only for dimensions and that random circuits require depth for . Finally, we introduce an “expurgation” algorithm that uses quantum measurements to remove logical operators that cause the code to fail by turning them into either additional stabilizers or into gauge operators in a subsystem code. With such targeted measurements, we can achieve sublogarithmic depth in spatial dimensions below capacity without increasing the maximum weight of the check operators. We find that for any rate beneath the capacity, high-performing codes with thousands of logical qubits are achievable with depth 4–8 expurgated random circuits in dimensions. These results indicate that finite-rate quantum codes are practically relevant for near-term devices and may significantly reduce the resource requirements to achieve fault tolerance for near-term applications.
4 More- Received 13 November 2020
- Revised 11 June 2021
- Accepted 12 July 2021
DOI:https://doi.org/10.1103/PhysRevX.11.031066
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.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
A major challenge in quantum information science is to develop error-correction strategies that are relevant for near-term applications of quantum computers. Many of the most promising codes require a demanding, high-depth encoding circuit, while those with low depth achieve a small number of logical qubits per physical qubits (a ratio known as the coding rate). Here, we show that low depth does not always entail a low coding rate. Rather, high performance with large coding rates is the typical behavior of quantum codes generated by random circuits at very low depths, thus offering a promising path to fault tolerance with minimal overhead.
While the study of random error-correcting codes has a long history, applications to quantum computing have only recently been investigated. Many prior works have focused on the depth required to achieve a very strong form of quantum error correction. Our approach to quantum code design is rooted in arguments from statistical physics and establishes several deep connections between quantum coding theory and critical phenomena in phase transitions. In addition, we introduce a method of targeted measurements to significantly improve random coding performance. These latter results provide interesting connections to the emerging topic of measurement-induced entanglement transitions.
Our work adds crucial evidence to an emerging consensus that quantum information processing with error correction may not be that far from practical reality. There are several obstacles still to overcome, most notably incorporating fault tolerance, so that error correction is robust when the encoding circuits have imperfect components. If these challenges can be met, then random circuits have the potential to be a boon to the field of quantum-error mitigation strategies in near-term devices that rely on quantum error-correcting codes.