Good Stabilizer Codes from Shallow Clifford Circuits with Random Matchings

Abstract: 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 $O(\log³ n)$. We show that the same asymptotic tradeoff is attained in optimal $O(\log n)$ depth under a gate distribution with a more restricted support. For every fixed $δ>0$ and sufficiently large $n$, if $\frac kn < 1 - H(\frac{d}{n}) - \frac{d}{n}\log_2 3 - δ$, we can construct random circuits of depth $O(\log n)$ which define, with high probability, an $[n,k]$ stabilizer code of distance at least $d+1$, which matches the $Ω(\log n)$ 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.

• The paper shows that random matching Clifford circuits using twirled CNOT gates produce stabilizer codes meeting the quantum Gilbert–Varshamov rate–distance tradeoff at depth O(log n), with failure probability that is polynomially and exponentially small in n.
• The construction uses no work ancillas and restricts entanglement to CNOT with local Clifford twirls, while reducing the proof to low-weight hitting probabilities in a reversible Markov chain describing Pauli-support growth.
• The O(log n) depth and O(n log n) two-qubit gate count are optimal within the independent random-matching model, although derandomization, efficient decoding, and hardware-level noise remain open challenges.

Overview and main result

This paper, by Anand, Gorokhovsky, Hritz, and Sun (2608.18536), shows that random Clifford circuits built from restricted two-qubit gate sets produce asymptotically good stabilizer codes in optimal depth. Specifically, for any fixed $\delta > 0$ and sufficiently large even $n$, whenever

$$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$

a random matching circuit of depth $O(\log n)$ defines an $[n,k]$ stabilizer code of distance at least $d+1$ with probability at least $1 - n^{-m} - 2^{-\Omega_\delta(n)}$. This matches the quantum Gilbert–Varshamov (GV) rate–distance tradeoff achieved by Brown and Fawzi (Brown et al., 2013), but improves their depth from $O(\log^3 n)$ to the light-cone-optimal $O(\log n)$, while simultaneously answering an open question of Brown–Fawzi by restricting the entangling gate to CNOT with only local Clifford twirls.

The construction uses no work ancillas, in contrast to the exact unitary 2-design encoders of Cleve et al. (Cleve et al., 2015), which achieve the GV tradeoff but require $\widetilde O(n)$ ancillas and $n$0 depth.

The circuit ensemble

The paper's central object is a random matching circuit: each layer samples a uniformly random ordered perfect matching of $n$1 and applies an independent two-qubit Clifford gate drawn from a distribution $n$2 on $n$3 to each pair in parallel. A depth-$n$4 encoder is the product of $n$5 independent such layers.

The main theorem holds for any $n$6 satisfying three regularity conditions:

• Bi-invariance: there is a subgroup $n$7 acting transitively on $n$8 up to sign such that $n$9 is invariant under left/right multiplication by $$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$0;
• Positive entanglement: $$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$1 assigns positive probability to some entangling gate;
• Reversibility: $$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$2.

The principal instantiation is the Clifford-twirled CNOT layer, where each matched pair receives $$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$3 with $$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$4 i.i.d. uniform single-qubit Cliffords. Notably, the local twirls may be drawn from the three-element set $$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$5, since conjugation by $$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$6 cyclically permutes $$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$7 up to sign. Each depth-$$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$8 circuit thus uses exactly $$\frac{k}{n} < 1 - H\!\left(\frac{d}{n}\right) - \frac{d}{n}\log_2 3 - \delta,$$9 CNOT gates and $O(\log n)$0 one-qubit Clifford gates total — a substantially more hardware-friendly primitive than sampling from the full $O(\log n)$1, and closely paralleling recent trapped-ion experiments that implemented 98-qubit random Clifford layers with random pairings and a fixed maximally entangling gate (Ransford et al., 7 Nov 2025).

Proof architecture: reduction to a Pauli weight chain

The proof follows the Brown–Fawzi strategy of controlling second moments, but exploits the restricted structure more aggressively. For a Clifford encoder, distance $O(\log n)$2 is equivalent to every nontrivial logical Pauli having zero overlap with all physical Paulis of weight at most $O(\log n)$3; since traces are in $O(\log n)$4, failure probabilities equal second-moment transition probabilities $O(\log n)$5.

The key structural observation is that bi-invariance collapses the second-moment dynamics onto binary support strings. Local twirling maps any pair $O(\log n)$6 to the uniform average $O(\log n)$7 over Paulis with the same support, so the full $O(\log n)$8-state chain reduces to a Markov chain $O(\log n)$9 on $[n,k]$0: each step pairs bits via a random matching and evolves each two-bit block according to a kernel $[n,k]$1 recording how $[n,k]$2 moves nonidentity Pauli supports on two qubits.

Two lemmas connect the gate distribution to this chain:

• Reversibility and stationarity (Lemma 2.5): under the three regularity conditions, $[n,k]$3 (the $[n,k]$4 corner on supports $[n,k]$5) is reversible with respect to $[n,k]$6, and $[n,k]$7. The lower bound follows from stationarity; strict positivity of $[n,k]$8 follows from a Skolem–Noether argument showing that any gate mapping every $[n,k]$9 to a product Pauli must itself be a product gate.
• Consequently the string chain $d+1$0 is irreducible, aperiodic, and reversible with stationary distribution $d+1$1, and the induced weight chain $d+1$2 has stationary distribution $d+1$3, peaked near $d+1$4.

Hitting-time analysis of the weight chain

The technical core is a logarithmic hitting-time bound: from any starting weight $d+1$5, the chain reaches a narrow window around $d+1$6 within $d+1$7 steps with probability $d+1$8.

The argument proceeds in stages. First, an Azuma–Hoeffding inequality applied to a martingale revealing matched pairs one at a time gives one-step concentration $d+1$9, together with an explicit expected drift $1 - n^{-m} - 2^{-\Omega_\delta(n)}$0 whose attracting fixed point is $1 - n^{-m} - 2^{-\Omega_\delta(n)}$1. From low weight ($1 - n^{-m} - 2^{-\Omega_\delta(n)}$2), the drift exceeds $1 - n^{-m} - 2^{-\Omega_\delta(n)}$3, so the chain grows exponentially fast with high probability; a supermartingale argument on $1 - n^{-m} - 2^{-\Omega_\delta(n)}$4 yields escape times of $1 - n^{-m} - 2^{-\Omega_\delta(n)}$5 from every interval $1 - n^{-m} - 2^{-\Omega_\delta(n)}$6, with failure probability $1 - n^{-m} - 2^{-\Omega_\delta(n)}$7. Reversibility then converts escape-time bounds into hitting-time bounds via time-reversal of trajectories. From linear weight, contraction toward $1 - n^{-m} - 2^{-\Omega_\delta(n)}$8 at geometric rate $1 - n^{-m} - 2^{-\Omega_\delta(n)}$9 reaches the target window in a constant number of steps, and rare-downcrossing estimates ensure the chain does not repeatedly fall back below $O(\log^3 n)$0.

Combining these pieces, for $O(\log^3 n)$1,

$O(\log^3 n)$2

which is precisely the uniform low-weight estimate needed for the union bound over all $O(\log^3 n)$3. Substituting into the code-probability bound and summing over input strings yields the main theorem, with the entropy term $O(\log^3 n)$4 governing the GV tradeoff.

A notable methodological point: the argument requires only control of low-weight return probabilities in the second moment, not convergence of the full second-moment channel to a Haar-like or design form. This is why the result holds without unitary-design guarantees and with a severely restricted gate set.

Optimality within the model

Depth $O(\log^3 n)$5 is unconditionally necessary: a qubit's forward light cone grows by at most a constant factor per layer, so linear distance forces $O(\log^3 n)$6 depth. Within the independent random-matching model, the paper proves a stronger gate-count lower bound: if layer $O(\log^3 n)$7 places $O(\log^3 n)$8 two-qubit gates on uniformly random disjoint pairs and the encoder achieves distance $O(\log^3 n)$9 with probability at least $O(\log n)$0, then

$O(\log n)$1

so constant success probability with linear distance forces $O(\log n)$2 and $O(\log n)$3. The construction therefore exhausts what the independent-matching model permits; achieving the naive $O(\log n)$4 gate-count lower bound would require structured or correlated layers. This resolves the depth question posed explicitly by Brown–Fawzi while identifying the residual gap as inherent to the model rather than to the technique.

Limitations and open questions

The paper concedes several restrictions plainly. First, bi-invariance and reversibility are used essentially in the proof but are not obviously necessary; positive entanglement alone is clearly required. In particular, the current framework accommodates the cyclic twirl $O(\log n)$5 but not other natural three-element sets such as $O(\log n)$6, and relaxing these conditions remains open. Second, the result is existential: derandomizing the matching sequence and twirls to obtain an explicit good-code family is not addressed. Third, the paper studies encoding complexity only — no efficient decoder for the resulting ensemble is provided, and constructing one is left open. Finally, the bounds are logical circuit bounds and do not account for routing overhead, noise during encoding, fault-tolerant state preparation, or decoding on physical devices.

Conclusion

This paper establishes that $O(\log n)$7-depth random matching circuits using CNOT as the sole entangling gate, twirled by as few as three single-qubit Cliffords, attain the quantum Gilbert–Varshamov rate–distance tradeoff with high probability — matching Brown–Fawzi's parameters at optimal depth with a strictly smaller gate ensemble and no ancillas. The proof reduces the coding problem to hitting-time and reversibility arguments for a reversible Markov chain on Pauli support strings, demonstrating that full second-moment (design-level) convergence is unnecessary for GV-distance codes. The remaining gaps — necessity of the regularity conditions, derandomization, and efficient decoding — define the natural continuation of this line of work.