![]() 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.Ī major challenge in quantum information science is to develop error-correction strategies that are relevant for near-term applications of quantum computers. ![]() 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 D = 2 dimensions. With such targeted measurements, we can achieve sublogarithmic depth in D ≥ 2 spatial dimensions below capacity without increasing the maximum weight of the check operators. 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. We find that the requisite depth scales like O ( log N ) only for dimensions D ≥ 2 and that random circuits require O ( N ) depth for D = 1. 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 N. Previous results on random circuits have only shown that O ( N 1 / D ) depth suffices or that O ( log 3 N ) depth suffices for all-to-all connectivity ( D → ∞). For random stabilizer codes and the erasure channel, we find strong evidence that a depth O ( log N ) 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 D. Here, we use ensembles of low-depth random circuits with local connectivity in D ≥ 1 spatial dimensions to generate quantum error-correcting codes. Random quantum circuits have played a central role in establishing the computational advantages of near-term quantum computers over their conventional counterparts.
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