Non-Clifford Benchmarking via Ensemble Feature Selection

Abstract: We propose an Ensemble Feature Selection (EFS) method for fast estimation of process infidelity of involutory multi-qubit gates, including non-Clifford targets, for which standard Clifford-based benchmarking does not apply. The method selects a compact set of experimentally executable circuit measurements from a candidate pool through offline training on a physically motivated ensemble of noisy channels, and combines them into a linear estimator with weights learned by ridge regression. The training ensemble is an explicit and tunable component of the protocol, incorporating prior knowledge about dominant hardware noise mechanisms. The estimator is validated on ibm_kingston using two Clifford validation benchmarks structurally related to the transpiled CCZ circuit, against independent Interleaved Randomized Benchmarking (IRB). Both show close EFS-IRB agreement across a wide range of process infidelities, with an estimation precision of approximately 0.01 over a process infidelity range of 0.02-0.2. EFS is subsequently applied directly to CCZ on the same device.

• The paper introduces an Ensemble Feature Selection method that benchmarks non-Clifford gates by estimating process infidelity with an RMSE of approximately 0.01.
• It employs a tunable training ensemble and ridge regression to select informative features, ensuring unbiased and scalable error estimation.
• Experimental validation on a 156-qubit superconducting platform shows strong agreement with interleaved randomized benchmarking, confirming both efficiency and adaptability.

Ensemble Feature Selection for Non-Clifford Quantum Gate Benchmarking

Introduction

Accurate characterization of non-Clifford multi-qubit gates, such as CCZ, is essential for advancing universal quantum computation. However, conventional randomized benchmarking techniques—primarily tailored for Clifford gates—do not straightforwardly extend to the non-Clifford regime, presenting a substantial bottleneck for scalable quantum processor evaluation. This paper introduces an Ensemble Feature Selection (EFS) protocol enabling fast, experimentally efficient estimation of process infidelity for involutory, including non-Clifford, gates. The approach maintains strong accuracy relative to independent Interleaved Randomized Benchmarking (IRB), achieving estimation precision on the order of 0.01 over a process infidelity range of 0.02–0.20, with demonstrated scalability and minimal experimental overhead (2607.01180).

Context and Motivation

Non-Clifford operations, though indispensable for universality, are challenging targets for benchmarking due to the absence of the Clifford group structure that underpins the exponential decay models in traditional RB and IRB protocols. Full channel characterization methods (e.g., quantum process/gate set tomography) are prohibitively resource-intensive for growing system sizes, scaling exponentially with qubit number. Emerging alternatives—including machine learning and featuremetric techniques—seek to balance partial information loss against dramatic gains in scalability and reduced experimental cost. For practical implementations and real-time calibration, a robust yet efficient estimator of gate error is more valuable than exhaustive channel tomography.

Methodology

Ensemble Feature Selection (EFS) Architecture

EFS selects a compact, informative subset of potential circuit-based measurements (features) from a larger candidate pool. The feature selection and weight optimization are performed through offline training on a physically motivated ensemble of noisy quantum channels, specified to reflect the dominant hardware noise mechanisms. The ensemble is tunable, enabling a priori incorporation of knowledge about hardware-specific noise, such as dominant error channels for superconducting qubits.

The estimator maintains a linear form in the selected features, with weights trained via ridge regression subject to an anchor constraint (ensuring zero process infidelity for the identity channel). This linearity is justified by the structure of the process infidelity metric in the Pauli Transfer Matrix (PTM) representation. The candidate measurement pool exploits virtual Z rotation gates (frame updates), which provide 'analysis mixers' for coherent error redistribution in the X/Y sectors, enriching the feature set without additional physical error introduction.

Training Ensemble and Noise Modeling

The training ensemble is generated by composing coherent (unitary) and incoherent (dissipative) noise channels with the target gate. Coherent noise is modeled by random Pauli Hamiltonians, spectrally weighted to reflect hardware-specific error profiles (e.g., dominance of conditional-Z errors). Dissipative errors follow independently parameterized amplitude damping and dephasing channels. The approach decomposes each ensemble member’s error contribution into anticommuting and commuting components relative to the involutory target structure, and further separates coherent and incoherent parts—enabling fine-grained characterization and ensemble diversity.

Feature Selection and Weight Optimization

From the initial pool (8192 candidates for three-qubit gates), greedy forward selection is used to identify a subset (typically $M=24$) that minimizes a shot-noise-aware RMSE criterion. The associated regression weights are determined via regularized least-squares. Proper anchor constraints ensure that process infidelity estimates are unbiased for identity gates, and the general form of the estimator is unbiased towards Clifford or non-Clifford targets.

Experimental Validation

Clifford and Non-Clifford Targets

Experimental validation was performed on the ibm_kingston 156-qubit superconducting platform. Two Clifford gates structurally analogous to the transpiled CCZ (I CCZ, which removes virtual $R_z(\pm \pi/4)$ gates from CCZ, and CZ(0,2)) were used as validation benchmarks, each possessing comparable nontrivial error channels. Both EFS and independent IRB measurements were executed spanning 204 distinct qubit triples, enabling direct comparison across a broad infidelity range.

Numerical Results

• EFS–IRB agreement: For both I CCZ and CZ(0,2), EFS closely tracked IRB with a fitted slope of 0.93–0.94 and an intercept around 0.012, yielding $r=0.96$ and a calibrated RMSE of approximately 0.01 over $\epsilon$ in $[0.02, 0.20]$.
• Scaling with Feature Set Size: Increasing $M$ beyond 24 provides negligible further reduction in RMSE, demonstrating practical saturation.
• Transfer to Non-Clifford Regime: EFS applied directly to the non-Clifford CCZ exhibited strong correlation with the estimator for I CCZ, with 96–97% of qubit triples showing close agreement. This supports the principle that the ensemble-based estimator’s accuracy—validated on Clifford circuits—transfers to their non-Clifford counterparts provided the structural noise features are analogous.

Sensitivity to Coherent Error Redistribution

Virtual $R_z(\pm \pi/4)$ rotations present in CCZ, implemented as near-ideal frame updates, do not introduce additional noise but redistribute coherent error components, providing a stringent stress test for estimator sensitivity. Comparison of EFS estimates between CCZ and I CCZ for identical hardware confirmed that the estimator is not sensitive to such coherent error redistribution when diagonal (dephasing/relaxation) noise dominates; approximately 3–4% of qubit triples manifested measurable non-diagonal error structure, detectable only through these X/Y redistributions.

Implications and Future Developments

The EFS methodology establishes that fast, hardware-informed estimation of non-Clifford gate infidelity is practicable on near-term devices, achieving high empirical accuracy with $\mathcal{O}(50)$ circuit executions. The explicit, tunable nature of the ensemble allows users to incorporate evolving hardware noise profiles, offering a flexible platform for benchmarking and calibration across diverse architectures and gate structures.

The approach provides actionable diagnostic feedback ideal for experimental workflows emphasizing rapid feedback, such as iterative calibration, error mitigation, and error-correction protocols. By decoupling estimator architecture from Clifford-specific properties, EFS bridges the gap to routine benchmarking of arbitrary target gates.

Prospective directions include optimizing the ensemble noise model and feature pool, deploying advanced machine learning feature selection techniques, and incorporating adaptive experimental design to further compress measurement resources. Extension to higher-qubit non-Clifford gates, systematic validation across platforms, and integration into automated hardware tuning pipelines represent natural next steps.

Conclusion

Ensemble Feature Selection offers a practically validated, resource-efficient solution for non-Clifford gate benchmarking on current quantum hardware. The method leverages linear modeling and physically motivated training ensembles for accurate process infidelity estimation, sidestepping the limitations of Clifford-based approaches and full tomography. Its adaptability and minimal experimental cost will be instrumental in scaling quantum hardware characterization as system sizes and circuit complexities continue to grow (2607.01180).