Recent work [M. J. Gullans , ] has shown that quantum error correcting codes defined by random Clifford encoding circuits can achieve a nonzero encoding rate in correcting errors even if the random circuits on n qubits, embedded in one spatial dimension (1D), have a logarithmic depth d = O ( log n ) . However, this was demonstrated only for a simple erasure noise model. In this work, we discover that, for the same class of codes, this desired property indeed holds for the conventional Pauli noise model. Specifically, we numerically demonstrate that the hashing bound, i.e., a rate known to be achieved with d = O ( n ) -depth random encoding circuits, can be attained for the above codes even when the circuit depth is restricted to d = O ( log n ) in 1D for depolarizing noise of various strengths. This analysis is made possible with our development of a tensor-network maximum-likelihood decoding algorithm that works efficiently for log -depth encoding circuits in 1D. Published by the American Physical Society 2024
Encoding quantum information with low circuit overhead is a fundamental challenge in fault-tolerant quantum computation. Random circuits provide a natural mechanism for rapidly spreading logical information through simple gates applied in parallel. Brown and Fawzi showed that random Clifford circuits on two-qubit Clifford gates provide such encoders that achieve the quantum Gilbert-Varshamov rate-distance tradeoff with depth
O(log3n). We show that the same asymptotic tradeoff is attained in optimal
O(logn) depth under a gate distribution with a more restricted support. For every fixed
δ>0 and sufficiently large
n, if
nk<1−H(nd)−ndlog23−δ, we can construct random circuits of depth
O(logn) which define, with high probability, an
[n,k] stabilizer code of distance at least
d+1, which matches the
Ω(logn) light-cone lower bound for linear distance encoders. Our ensemble employs a random matching circuit architecture consisting of
T independent permutation-invariant layers. In each layer, the qubits are paired up by a uniformly random perfect matching, and a random independent two-qubit Clifford gate is applied to each pair. The gate distribution need not be uniform over, or even have full support on, the two-qubit Clifford group; rather, we allow for very general distributions on Clifford gates satisfying three regularity conditions. In particular, the construction can be implemented using
n/2 CNOT gates on randomly matched pairs in each layer, with parallel one-qubit Clifford twirls. These regularity conditions allow us to reduce the second-moment dynamics of our random circuits to a reversible Markov chain on binary support strings. We establish logarithmic hitting-time bounds for this Markov chain and comparisons of its stationary distribution to prove the coding properties of the circuits.
Recent years have seen rapid development in the subject of quantum coding theory, with breakthroughs on many exciting classes of codes, including quantum LDPC codes, quantum locally testable codes, and quantum codes with interesting transversal gates. However, a natural class of quantum codes, which has been well-studied classically, has not yet been treated: those which can be quickly encoded and decoded. This problem concerns the channel capacity setting, where a noise channel sits between perfect encoding and unencoding/decoding operations; this is the setting that is relevant for communication between fault-tolerant quantum computers. In this work, we construct asymptotically good quantum codes that can be encoded and unencoded by quantum circuits of logarithmic depth and consisting of a linear total number of gates. The classical decoding algorithms also run in logarithmic depth and use
O(nlogn) gates, or alternatively a linear number of gates but with higher depth. We further construct explicit and asymptotically good quantum codes whose encoding, unencoding and decoding all use a linear number of gates, and additionally whose encoding and unencoding may be run in logarithmic depth.
We study error correction type protocols in which a quantum channel encodes logical information into an enlarged Hilbert space. Specifically, we consider channels realized by one dimensional random noisy quantum circuits with spatially local interaction gates. We analyze both noise acting after the encoding and noise affecting the encoding circuit itself. Using the coherent information as a metric, we show that in both cases the infinite depth limit is governed by random matrix theory, which predicts a universal phase transition at a critical noise rate. This critical point separates an error correcting phase, in which encoded information is preserved, from a phase in which it is irretrievably lost. Going beyond the infinite depth limit, we characterize the systematic finite depth deviations from random matrix universality. In particular, we show that these deviations behave parametrically differently depending on whether the noise acts after the encoding or also affects the encoding itself. For noiseless encoders, the approach is exponential in circuit depth, although boundary effects can delay perfect encoding relative to the circuit design time. For noisy encoders, we find that the circuit fidelity effectively replaces the Hashing bound, and perfect encoding is approached polynomially with depth.
In this work, we study the task of encoding logical information via a noisy quantum circuit. It is known that at superlogarithmic depth, the output of any noisy circuit without reset gates or intermediate measurements becomes indistinguishable from the maximally mixed state, implying that all input information is destroyed. This raises the question of whether there is a low-depth regime where information is preserved as it is encoded into an error-correcting codespace by the circuit. When considering noisy random encoding circuits, our numerical simulations show that there is a sharp phase transition at a critical depth of order
p−1, where
p is the noise rate, such that below this depth threshold quantum information is preserved, whereas after this threshold it is lost. Furthermore, we rigorously prove that this is the best achievable trade-off between depth and noise rate for any noisy circuit encoding a constant rate of information. Thus, random circuits are optimal noisy encoders in this sense.