Clifford Noise Reduction (CliNR)

• Clifford Noise Reduction (CliNR) is a suite of techniques that decomposes Clifford circuits and employs resource state verification to reduce logical errors.
• The protocols leverage recursive verification, efficient stabilizer measurements, and combinatorial optimization to enable scalable error reduction on mid-scale quantum systems.
• Regression-based mitigation and symmetric twirling extend CliNR to near-Clifford circuits, delivering superior performance compared to sampling-based approaches.

Clifford Noise Reduction (CliNR) encompasses a suite of circuit-level error suppression and mitigation techniques specifically targeted at Clifford circuits. These protocols achieve reduced logical error rates at moderate qubit and gate overhead, operating below the thresholds required for full quantum error correction and typically outperforming sampling-based error mitigation in resource efficiency. Modern developments include recursive extensions, combinatorial optimization of verification subroutines, and connections to regression-based learning approaches that leverage the efficient classical simulability of Clifford and near-Clifford circuits. CliNR methodologies have demonstrated significant logical error reductions and scalability on circuits and hardware spanning tens to hundreds of qubits.

1. Fundamental CliNR Protocols and Circuit Decomposition

The canonical CliNR protocol is defined for an $n$-qubit Clifford circuit $C$ of gate-count $s$ under a circuit-level independent Pauli error model, physical error rate $p$ per gate (Delfosse et al., 9 Jul 2024). $C$ is decomposed into $t$ subcircuits $\{C_i\}$, each realized by gate teleportation using ancilla blocks and resource state preparation. For each subcircuit:

• Prepare $n$ Bell pairs, apply $C_i$ to ancilla, and verify the resource state via $r$ random stabilizer measurements.
• If any verification fails (outcome $-1$), only the ancilla is restarted; input qubits remain undisturbed.
• Successful verification triggers gate teleportation through the checked resource to propagate logical computation.

Key overhead metrics include $3n+1$ qubits, $2s+o(s)$ gates in expectation, and a strictly nonnegative logical error rate that vanishes asymptotically if $nsp^2 \to 0$ (Delfosse et al., 9 Jul 2024). Resource constraints and probabilistic guarantees on logical errors are formalized in the main theorem:

$$p_L \le t \frac{g_p(3n+s_0) 2^{-r} + 2 g_p(2n+3) + g_p(5n)}{(1-p)^{m_0}}$$

where $g_p(x) = 1 - (1-p)^x$ and $m_0 = 3n + s_0 + (2n+3)r$.

Direct implementation succeeds only if $sp \to 0$ (logical error scales as $sp$), while CliNR pushes success into the regime $nsp^2 \ll 1$, drastically increasing feasible circuit sizes.

2. Recursive CliNR and Error-Threshold Relaxation

The recursive form of CliNR was introduced to surmount the size limitations of standard (depth-1) CliNR (Brodutch et al., 27 Nov 2025). When $s$ exceeds $O(1/(np^2))$, residual errors from resource-state injection remain undetected. Recursive CliNR nests CliNR$^1$ within higher-level verification trees:

• The circuit $C$ is represented as a tree structure of depth $D = \max\{1, \lceil \log_2(sp)\rceil + 1\}$.
• Each tree node corresponds to a subcircuit verified by CliNR$_{1,r}$, with ancilla reuse at each recursion level.
• Parameters $\mathbf{s}(v_{\ell,i})$ and $\mathbf{r}(v_{\ell,i})$ label the gate count and stabilizer checks, with an inductively constructed nested verification protocol.

Recursive CliNR achieves vanishing logical error for the relaxed threshold $np\to0$, bypassing the original quadratic constraint. The top-level error scales as

$p_{\rm logical} \le s\,p^{D+1}(\alpha n)^D$

for constant $\alpha$, with $p_{\rm logical}\ll 1/s$ whenever $np\to 0$. Resource overheads are: $$\text{qubits} = (2D+1)n + 1 = (2\lceil\log_2(sp)\rceil+3)n+1$$

$\text{gates} \le 24s\lceil(sp)^4\rceil$

Recursive CliNR thus expands the noise-reduction regime to $s=O(1/(np))$—an order of magnitude larger than the original protocol.

3. Verification Sequence Optimization and Symmetry Reduction

Recent advances optimize the verification sequence—the ordered list $V = (V_1,\dots,V_r)$ of stabilizer measurements—through combinatorial algorithms (Tham et al., 17 Apr 2025). The goal is to minimize the logical error rate $p_{\CliNR(C_R,V)}$, subject to the constraints of stabilizer independence. Optimization proceeds by:

• Defining a proxy function for the error rate, which is invariant under permutations and subgroup transformations, significantly reducing search space dimensionality.
• Employing metaheuristics (tabu search) on the Grassmann graph of rank-$r$ stabilizer subgroups, with further refinement using full Monte Carlo evaluation of $p_V$.
• Leveraging the symmetry group $GL_r(\mathbb{F}_2)$, achieving up to $20,160\times$ reduction in search size for $r=4$ and demonstrating 25% logical error reduction over baseline randomized sequences.

This approach enables scalable enumeration and identification of verification subroutines, which is essential for hardware implementations with restricted measurement capabilities.

4. Regression-Based Error Mitigation Using Clifford Data

CliNR includes regression-based mitigation for general (Clifford and near-Clifford) circuits, notably Clifford Data Regression (CDR) (Czarnik et al., 2020) and Clifford Perturbation Data Regression (CPDR) (Zhang et al., 12 Dec 2024). The protocol:

• Constructs a training set $\{(X_i^{\rm noisy}, X_i^{\rm exact})\}$ from circuits with up to $N$ non-Clifford gates, classically simulable via stabilizer or Sparse Pauli Dynamics methods.
• Fits a linear mapping: $X^{\rm exact} = a_1 X^{\rm noisy} + a_2$, where the coefficients are learned via least squares or ridge regression.
• Applies this learned mapping to mitigate noisy observables from arbitrary target circuits.
• CPDR extends the protocol by perturbing Clifford gates with small-angle rotations, increasing training diversity and regression accuracy.

Empirical results show order-of-magnitude reductions in mean squared error for circuits up to $64$ qubits and in experimental settings (IBM Eagle). Diversity of the training set and regression model selection are critical for generalization beyond low-depth, low-$N$ circuits.

5. Symmetric Clifford Twirling and Depolarizing Noise Engineering

Symmetric Clifford twirling transforms structured Pauli noise into nearly global depolarizing channels using symmetric Clifford subgroups (Tsubouchi et al., 13 May 2024). For a non-Clifford gate $U$, the symmetry subgroup $Q_U$ comprises all Paulis commuting with $U$. The symmetric Clifford group is defined as: $$\mathcal{G}_{n,Q_U} = \{C \in \mathcal{G}_n \mid \forall P \in Q_U, CP=PC\}$$ SC-twirling averages over this subgroup to yield: $$\mathscr{T}_{Q_U}(\mathcal{E})(\rho) = \sum_{P} \alpha'_P P \rho P$$ where $\alpha'_P$ is the uniform average over the orbit of $P$.

SC-twirling exponentially suppresses bias from single-qubit $X, Y$ errors to $O(2^{-n})$ and reduces sampling overhead to $O(e^{2p_{\rm tot}/\varepsilon^2})$ for total error $p_{\rm tot} = pL$ over $L$ layers. Sparse (local) twirling further achieves polynomial suppression with manageable gate overhead, strongly enabling cost-optimal mitigation in the early fault-tolerant regime.

6. Practical Implementation Regimes, Limitations, and Numerical Results

All forms of CliNR assume circuit-level independent Pauli noise, reliable stabilizer measurements, and ancilla reinitialization. Protocols are currently restricted to Clifford circuits (except for regression-based extensions) and performance bounds presuppose $np\to 0$ (recursive) or $nsp^2\to 0$ (standard). Idle noise or high-fidelity non-Clifford gates reduce the improvement from deeper recursion or more aggressive verification.

Validated by stabilizer, Monte Carlo, and Markov-chain simulations, standard and recursive CliNR protocols demonstrate threshold-like error reduction, with recursive schemes outperforming standard at fixed gate overhead when $n$ and $s$ are large. Optimized verification and regression-based extension amplify these gains and extend applicability. Hardware experiments on trapped-ion quantum systems display breakeven performance; minor improvements in gate fidelity promise below-threshold operation. Numerical benchmarks confirm substantial error suppression on realistic mid-scale systems.

7. Future Directions and Open Problems

• Developing rigorous, quantitative error bounds as functions of $n$, $p$, and circuit architecture remains open, both for verification-based and regression-based CliNR.
• Extending CliNR protocols to non-Clifford circuits, potentially via composite schemes or improved twirling constructions.
• Investigating nonparametric regression models and hybrid simulation approaches for deeper circuits with more non-Clifford content.
• Optimizing resource allocation and parallelization, particularly for large recursive trees in noisy hardware settings.
• Further symmetry reduction and combinatorial design of verification subroutines for advanced hardware architectures and restricted measurement regimes.

CliNR constitutes an efficient, architecturally compatible pathway to error reduction in large-scale Clifford workloads, with recursive schemes and optimization algorithms extending utility-scale computation into regimes previously inaccessible without full fault tolerance.