Low-cost noise reduction for Clifford circuits

TL;DR

The paper introduces CliNR, a low-cost noise-reduction scheme for Clifford circuits that uses gate teleportation to implement Clifford subcircuits and offline stabilizer fault-detection to identify faults in resource states. By partitioning circuits into subcircuits and performing a small number of random stabilizer checks, CliNR achieves a vanishing logical error rate in regimes where the circuit size satisfies $s n p^2 \to 0$, outperforming direct implementations that require $s = o(1/p)$. The method consumes only $3n+1$ qubits and an average of $2s+o(s)$ gates with zero restart rate, offering a practical alternative to full quantum error correction for near-term devices. Numerical results show substantial reductions in logical error rate for random Clifford circuits under realistic noise models, and the framework includes variants such as graph-state injections for CZ sequences. CliNR thus provides a scalable, hardware-efficient approach to reducing noise in Clifford-based quantum computations, with clear pathways to broader applicability and universality via Clifford sub-circuit compilation or magic-state techniques.

Abstract

We propose a Clifford noise reduction (CliNR) scheme that provides a reduction of the logical error rate of Clifford circuit with lower overhead than error correction and without the exponential sampling overhead of error mitigation. CliNR implements Clifford circuits by splitting them into sub-circuits that are performed using gate teleportation. A few random stabilizer measurements are used to detect errors in the resources states consumed by the gate teleportation. This can be seen as a teleported version of the CPC scheme, with offline fault-detection making it scalable. We prove that CliNR achieves a vanishing logical error rate for families of $n$-qubit Clifford circuits with size $s$ such that $nsp^2$ goes to 0, where $p$ is the physical error rate, meaning that it reaches the regime $ns = o(1/p^2)$ whereas the direct implementation is limited to $s = o(1/p)$. Moreover, CliNR uses only $3n+1$ qubits, $2s + o(s)$ gates and has zero rejection rate. This small overhead makes it more practical than quantum error correction in the near term and our numerical simulations show that CliNR provides a reduction of the logical error rate in relevant noise regimes.

Figures (7)

• Figure 1: The CliNR circuit (with $t=1$) implements a $n$-qubit Clifford circuit $C$ by gate teleportation using $2n+1$ ancilla qubits. We insert $r$ stabilizer measurements before the CNOTs, to detect errors in the application of $C$ on the ancilla qubits. Here $r=1$ and we measure the stabilizer $P$. If a stabilizer measurement returns a non-trivial outcome, the circuit restarts. Here $Q$ is a $n$-qubit Pauli operation that depends on the outcome of the measurement of the first $2n$ qubits.
• Figure 2: Averge logical error rate of the CliNR implementation of random Clifford circuits in two different noise regimes. We chose $r = \left \lfloor \log( \frac{s}{n} ) \right \rfloor$, as in \ref{['eq:proof:snp_tradeoff:def_r_t']}, and $t$ is the smallest integer such that the ${\omega_{\mathop{\mathrm{G}}\nolimits}} \leq 2$ for CliNR $2 \times$ and ${\omega_{\mathop{\mathrm{G}}\nolimits}} \leq 4$ for CliNR $4 \times$. The value of $t$ varies with $n$.
• Figure 3: (a) Gate teleportation circuit implementing $U$gottesman1999quantum. When $U = I$, we recover the original teleportation circuit bennett1993teleporting. (b) One-bit teleportation circuit zhou2000methodology.
• Figure 4: Improvement of the logical error rate of random Clifford circuits under CliNR for the circuit-level noise model of \ref{['sec:numerics']} with $p_{2}=p_{1}/10=10^{-4}$ as functions of the qubit count $n$ and the circuit shape $\alpha$ (for circuit size $s=n^{\alpha}$). Gray dashed lines are guides to the eye that delineates between regions where direct implementation is favored (white, upper-left) vs where CliNR yields lower noise (lower-right).
• Figure 5: Middle: A graph $G$ with five vertices and five edges. The sequence of $CZ$ gates corresponding to the edges of $G$ defines a unitary $U_G$. Left: Teleportation of five qubits, followed by the application of $U_G$ (dashed box) to the teleported qubits. Right: The graph state injection circuit associated with $G$. Commuting the sequence of $CZ$ gates through the circuit, we see that these two circuits are equivalent.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2: $snp^2$ threshold
• proof
• Corollary 1: size-$n^\alpha$ threshold
• Corollary 2: Clifford unitary threshold
• Lemma 1
• proof
• proof : Proof of \ref{['theorem:CliNR_s0_r']}
• Lemma 2
• proof