Approximate Quantum Error Correction with 1D Log-Depth Circuits

Efficient and high-performance quantum error correction is essential for achieving fault-tolerant quantum computing. Low-depth random circuits offer a promising approach to identifying effective and practical encoding strategies. In this work, we construct quantum error-correcting codes based on one-dimensional, logarithmic-depth random Clifford encoding circuits with a two-layer circuit structure. We demonstrate that these random codes typically exhibit good approximate quantum error correction capability by proving that their encoding rate achieves the hashing bound for Pauli noise and the channel capacity for erasure errors. Moreover, we show that the error correction inaccuracy decays exponentially with circuit depth, resulting in negligible errors for these logarithmic-depth circuits. We also establish that these codes are optimal, proving that logarithmic depth is necessary to maintain a constant encoding rate and high error correction performance. To prove our results, we propose new decoupling theorems for one-dimensional, low-depth circuits. These results also imply the strong decoupling and rapid thermalization within low-depth random circuits and have potential applications in quantum information science.