Direct fidelity estimation through joint fiducial grouping

Abstract: Fault-tolerant quantum computation hinges on the requirement for low physical error rates. Reaching below threshold regime requires the accounting of circuit dependent noise, that is inherent to the execution context in which a quantum gate is usually embedded. Direct fidelity estimation is a technique that offers natural context preservation as it solely requires the insertion of local Pauli preparation and measurement fiducials around the window of interest. However, each sampled input-output Pauli pair demands its own preparation and measurement setting, an overhead that grows rapidly once the target gate is no longer Clifford. We introduce joint fiducial grouping, which partitions Pauli pairs into sets with commuting input and output operators, allowing several Pauli-transfer coefficients to be estimated within the same preparation-measurement setting. We derive an unbiased grouped estimator and finite-sample guarantees showing that grouping always reduces the number of distinct input-output settings and can also reduce the required channel uses when the target weight is concentrated within compatible groups. We characterize these gains for the parametric two-qubit gate $\mathrm{fSim}(θ,\varphi)$, and use the grouped estimator as a context-sensitive reward for reinforcement-learning-based gate calibration. Our results provide a practical route to lower-overhead, context-preserving fidelity estimation for continuously parameterized quantum gates.

• The paper introduces joint fiducial grouping, which estimates multiple Pauli-transfer coefficients from one compatible preparation–measurement setting while preserving unbiasedness and finite-sample guarantees.
• The method guarantees setting compression and can reduce channel uses when target weight is concentrated within commuting groups, with fSim simulations reaching up to 2.70× setting compression and 1.43× shot improvement.
• The grouped estimator lowers estimation error in simulated noise and provides an inverse-free, context-sensitive reward for reinforcement-learning gate calibration, although hardware validation and practical runtime savings remain open.

Direct fidelity estimation (DFE) provides a context-preserving route to estimating the entanglement fidelity of a quantum channel, but its standard formulation assigns one preparation–measurement configuration per sampled input–output Pauli pair, an overhead that grows rapidly for non-Clifford targets. The paper under review introduces joint fiducial grouping, which partitions the support of the target Pauli transfer matrix into blocks whose input and output Paulis are simultaneously (qubit-wise) commuting, so that a single fiducial pair yields unbiased estimates of all Pauli-transfer coefficients in the block. The authors derive finite-sample guarantees showing that grouping never increases the number of distinct input–output settings and can reduce the required channel uses when target weight is concentrated within compatible groups. They validate the framework on the continuously parameterized two-qubit gate $\mathrm{fSim}(\theta,\varphi)$ and deploy the grouped estimator as a reward signal in reinforcement-learning-based gate calibration (2608.18548).

Background: direct channel fidelity estimation

For an $n$-qubit implemented channel $\mathcal{E}$ and unitary target $\mathcal{U}$ with $d=2^n$, the entanglement fidelity is the normalized Hilbert–Schmidt overlap of their superoperators,

$$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$

where $$\chi_{\mathcal{A}}(\alpha,\beta)=\frac{1}{d}\operatorname{Tr}[P_\alpha \mathcal{A}(P_\beta)]$$ are Pauli transfer coefficients in the phase-free Pauli basis. Standard DFE samples pairs from $\Omega=\{(\alpha,\beta):\chi_\mathcal{U}\neq 0\}$ with probability $$p_{\alpha,\beta}=\chi_\mathcal{U}(\alpha,\beta)^2/d^2$$; each sampled pair requires preparing eigenstates of $P_\beta$, applying $n$0, and measuring $n$1. Because only local Pauli fiducials bracket the circuit window, the protocol preserves the execution context — unlike randomized benchmarking, whose twirling destroys information about how a specific circuit shapes noise, and unlike context-aware fidelity estimation (CAFE), which requires physically implementing the reference inverse $n$2 and hence suffers a bootstrapping problem when the characterized gates are themselves imperfect.

The cost structure of DFE depends on the target's nonstabilizerness: Clifford targets have $n$3 and admit $n$4 sampling, whereas generic full-support non-Clifford targets approach $n$5. This makes fractional gates such as $n$6 — used on superconducting processors for Hamiltonian simulation — a natural but demanding testbed.

Joint fiducial grouping

A valid group $n$7 satisfies $n$8 and $n$9 for all members; in practice the stronger qubit-wise commuting (QWC) condition is imposed so that fiducials reduce to single-qubit rotations. Groups are constructed via a greedy clique-cover heuristic on a compatibility graph weighted by $\mathcal{E}$0, adapting measurement-grouping techniques from variational-algorithm observable estimation to channel certification.

The grouped estimator samples whole groups with probability $\mathcal{E}$1, where $\mathcal{E}$2 collects the target coefficients of the group. A single shot prepares a uniformly random common eigenbasis state of the commuting inputs, applies $\mathcal{E}$3 once, and measures all commuting output observables simultaneously; the resulting parities $\mathcal{E}$4 are unbiased estimators of each coefficient, so the aggregate $\mathcal{E}$5 is an unbiased single-shot estimator of $\mathcal{E}$6. Notably, the authors retain the empirical mean rather than median-of-means aggregation, since median-of-means introduces bias while Hoeffding's inequality suffices given bounded single-shot contributions.

Performance guarantees

The central result is a theorem establishing additive error $\mathcal{E}$7 with failure probability at most $\mathcal{E}$8 using $\mathcal{E}$9 sampled groups and per-group shot allocations derived from the target coefficients. The expected number of channel uses obeys

$\mathcal{U}$0

where the effective group size $\mathcal{U}$1 satisfies $\mathcal{U}$2 and $\mathcal{U}$3. Two consequences follow directly:

• Guaranteed setting compression: the number of distinct input–output settings drops from $\mathcal{U}$4 to $\mathcal{U}$5, giving compression factor $\mathcal{U}$6 independent of the implemented channel.
• Conditional shot compression: the improvement factor $\mathcal{U}$7 exceeds unity only when target weight within compatible groups is non-uniform.

The quantity $\mathcal{U}$8 admits an entropic interpretation as $\mathcal{U}$9, the Rényi-$d=2^n$0 effective support of the target weight conditioned on the group. This refines the known connection between DFE hardness and stabilizer Rényi entropies: what matters here is not the global nonstabilizerness of $d=2^n$1 but the concentration of weight inside each commuting block. A corollary sharpens the trade-off: for Clifford targets every conditional distribution is uniform, so $d=2^n$2 and no worst-case shot saving exists ($d=2^n$3), even though full configuration compression $d=2^n$4 remains available. Grouping therefore benefits structured non-Clifford dynamics most, and the paper is explicit that non-Cliffordness alone does not imply a shot advantage.

Numerical validation on fractional gates

Mapping the predicted advantages across the full $d=2^n$5 landscape of $d=2^n$6 (with a $d=2^n$7 coefficient cutoff), the leading shot improvement $d=2^n$8 ranges from 1 to 1.43, concentrated on ridges where target weight clusters within compatible groups, while broad regions remain near unity. The setting compression $d=2^n$9 ranges from 1.67 at Clifford corners up to 2.70, with a plateau near 2.5; along lines of constant support the counts are fixed (e.g., $$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$0 along $$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$1), even as coefficients vary continuously.

Fixed-point simulations at $$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$2 with $$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$3 compare grouped and standard DFE under ideal readout and symmetric 5% readout error with linear-inversion mitigation. Under both conditions the grouped estimator exhibits lower RMSE across the precision sweep, driven primarily by variance reduction rather than bias shift. The authors caution that this is an estimator-level comparison at common requested $$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$4, not an equal-total-budget comparison, and that the advantage is conditional on the grouping structure and allocation rule — linear-inversion REM amplifies finite-shot fluctuations by factors $$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$5, and grouping merely amortizes these fluctuations over shared configurations without making the assignment inversion intrinsically more accurate.

Inverse-free repeated-cycle estimation versus CAFE

Substituting DFE into CAFE's repeated-window construction yields an inverse-free alternative: local Pauli fiducials estimate $$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$6 at each depth without any two-design preparation or physical reference inverse, confining auxiliary errors to high-fidelity single-qubit primitives. Under a controlled Qiskit Aer noise model including coherent residuals and readout assignment (mitigation disabled), CAFE shows lower trial-to-trial variability in the fitted one-cycle fidelity (standard deviation $$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$7 vs. $$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$8 for grouped DFE at matched budgets), but larger bias and RMSE ($$F_e(\mathcal{U},\mathcal{E})=\frac{1}{d^2}\sum_{\alpha,\beta}\chi_{\mathcal{U}}(\alpha,\beta)\,\chi_{\mathcal{E}}(\alpha,\beta),$$9 vs. $$\chi_{\mathcal{A}}(\alpha,\beta)=\frac{1}{d}\operatorname{Tr}[P_\alpha \mathcal{A}(P_\beta)]$$0). The fitted depth-independent SPAM term is also larger for CAFE ($$\chi_{\mathcal{A}}(\alpha,\beta)=\frac{1}{d}\operatorname{Tr}[P_\alpha \mathcal{A}(P_\beta)]$$1 vs. $$\chi_{\mathcal{A}}(\alpha,\beta)=\frac{1}{d}\operatorname{Tr}[P_\alpha \mathcal{A}(P_\beta)]$$2), consistent with its entangling reference operations contributing error even at zero forward repetitions. Grouped DFE thus delivers the most accurate fitted cycle fidelity at matched budget, while CAFE concentrates estimates more tightly — a distinction between variance and accuracy the paper states plainly. A further practical advantage for parametric gates is that varying target parameters updates only the classical sampling distribution, not the physical fiducial template.

Reinforcement-learning-based calibration

Finally, the grouped estimator serves as the reward in a PPO-driven, model-free calibration of the five-parameter ansatz $$\chi_{\mathcal{A}}(\alpha,\beta)=\frac{1}{d}\operatorname{Tr}[P_\alpha \mathcal{A}(P_\beta)]$$3 against three representative fSim targets. Readout noise affects the estimated reward trajectory more than the final mean-policy infidelity, which remains comparable between ideal and mitigated conditions. The clearest benefit appears in cumulative executed configurations: grouping lowers the count by reusing compatible bases within each action evaluation, reducing compilation and controller-loading pressure — relevant because process-fidelity optimization can itself become circuit-prohibitive. The paper notes this count is a proxy for instruction overhead, not wall-clock runtime, which depends on the control stack.

Limitations and open questions

Several caveats are stated explicitly. All evidence is simulation-based; quantifying the resource accounting on real control hardware and compilation policies remains open, particularly whether FPGA-based runtime parameter selection converts setting compression into actual upload savings. Truncation of small Pauli-transfer coefficients introduces a bias bounded by $$\chi_{\mathcal{A}}(\alpha,\beta)=\frac{1}{d}\operatorname{Tr}[P_\alpha \mathcal{A}(P_\beta)]$$4, computable from the target alone but nonzero. Shot compression is absent for flat within-group weight distributions, and the REM-related RMSE advantage is conditional on grouping structure and allocation rule. The comparison against CAFE rests on a specific simulator noise model rather than calibrated device data. Open directions include a derandomized, target-biased grouped analogue of classical-shadow process tomography that could reuse grouped configurations across nearby calibration iterations, and extension to overlapping groups, which may further reduce sampling cost.

Conclusion

This work adapts measurement-grouping methodology from variational observable estimation to channel certification, yielding a grouped DFE estimator that preserves the finite-sample guarantees of the original protocol while provably compressing input–output settings and conditionally reducing channel uses according to a Rényi-$$\chi_{\mathcal{A}}(\alpha,\beta)=\frac{1}{d}\operatorname{Tr}[P_\alpha \mathcal{A}(P_\beta)]$$5 effective-support criterion. The fSim case studies delineate precisely where the gains arise — concentrated within-group weight, not mere non-Cliffordness — and demonstrate the estimator's utility as a context-sensitive, inverse-free reward for machine-learning-driven gate calibration. The framework offers an auditable resource trade-off for closed-loop characterization pipelines, pending experimental validation on hardware control stacks.