We prove that random 1D Clifford brickwork circuits form (in expectation) good approximate quantum error correction codes in logarithmic depth. Our proof makes use of the statistical mechanics techniques for random circuits developed by Dalzell et al. [PRX Quantum 3, 010333], adapted extensively to our own purpose. We also consider exact error correction, where we give matching upper and lower bounds for the required depth in which random 1D Clifford brickwork circuits become error correcting.
Comments
24 pages, see also the related work of Liu et al. (specifically the journal version [PRX Quantum 7, 010331 (2026)])
We prove that random 1D Clifford brickwork circuits form (in expectation) good approximate quantum error correction codes in logarithmic depth. Our proof makes use of the statistical mechanics techniques for random circuits developed by Dalzell et al. [PRX Quantum 3, 010333], adapted extensively to our own purpose. We also consider exact error correction, where we give matching upper and lower bounds for the required depth in which random 1D Clifford brickwork circuits become error correcting.
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 rigorously prove through information-theoretic analysis that one-dimensional logarithmic-depth random Clifford encoding circuits can achieve high quantum error correction performance. 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. We show that the error correction inaccuracy decays once a threshold of logarithmic depth is exceeded, resulting in negligible recovery errors. This threshold is shown to be lower than that of the simple separate block encoding, and the decay rate is higher. We further establish that these codes are optimal by proving that logarithmic depth is necessary to maintain a constant encoding rate and high error correction performance. To prove our results, we propose decoupling theorems tailored for one-dimensional low-depth circuits. These results also imply strong decoupling and rapid thermalization properties in low-depth random circuits and have potential applications in quantum information science and physics.
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 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
The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary t -designs. Unitary t -designs are probability distributions that mimic Haar randomness up to t th moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth O(nt10.5) are approximate unitary t -designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by Ω(n−1t−9.5) . We improve this lower bound to Ω(n−1t−4−o(1)) , where the o(1) term goes to 0 as t→∞ . A direct consequence of this scaling is that random quantum circuits generate approximate unitary t -designs in depth O(nt5+o(1)) . Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.
We introduce magic-augmented Clifford circuits—architectures in which Clifford circuits are preceded and/or followed by constant-depth circuits of non-Clifford (“magic”) gates—as a resource-efficient way to realize approximate k -designs, with reduced circuit depth and usage of magic. We prove that shallow-Clifford circuits, when augmented with constant-depth circuits of magic gates, can generate approximate unitary and state k -designs with ε relative error. The total circuit depth for these constructions on N qubits is O ( log ( N / ε ) ) + 2 O ( k log k ) in one dimension and O ( log log ( N / ε ) ) + 2 O ( k log k ) in all-to-all circuits using ancillas, which improves upon previous results for small k ≥ 4 . Furthermore, our construction of relative-error state k -designs only involves states with strictly local magic. The required number of magic gates is parametrically reduced when considering k -designs with bounded additive error. As an example, we show that shallow-Clifford circuits followed by O ( k 2 ) single-qubit magic gates, independent of system size, can generate an additive-error state k -design. We develop a classical statistical mechanics description of our random circuit architectures, which provides a quantitative understanding of the required depth and number of magic gates for additive-error state k -designs. We also prove no-go theorems for various architectures to generate designs with bounded relative error.
We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S gate in the rotated surface code and a depth-one intra-block CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]] where β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S, SHS=X, and CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.