Statevector-to-Hardware Reconstruction of a Four-Qubit ZZ Quantum Kernel: A Single-Backend Case Study of Three Execution Jobs

AI summary

Overview

Research area: Quantum machine learning on near-term hardware — specifically, the fidelity with which a quantum kernel (Gram) matrix generated by a ZZ feature map can be reconstructed on a real superconducting device.

Technical level: Intermediate. The paper is written for readers comfortable with quantum circuits, kernel methods, and standard similarity metrics (correlation, CKA, KTA), but its central question is conceptual rather than mathematical: how much does execution on hardware change a designed kernel?

Scope (one sentence): A single-backend, fixed-subset case study comparing three execution configurations (baseline, dynamical decoupling, gate twirling) of one frozen four-qubit ZZ feature map against an exact statevector reference, on N = 24 indoor air-quality windows executed on ibm_fez at 1024 shots per circuit.

What This Paper Is About

A quantum kernel compares two data points by measuring the squared overlap of their quantum-encoded states, producing a Gram matrix that every downstream kernel method depends on. On real hardware, noise and finite sampling distort the measured overlaps, typically pushing entries toward a common value and flattening the matrix. This paper asks a deliberately narrow, logically prior question: for one fixed four-qubit ZZ feature map, how far do hardware-reconstructed Gram matrices depart from the exact statevector matrix, and how do three runtime mitigation configurations compare on that distortion — without making any claim about classifier accuracy, forecasting skill, or quantum advantage.

Key Contributions

1. A statevector-referenced hardware reconstruction protocol for one frozen kernel. The authors execute the fixed ZZ4 kernel (Qiskit's ZZFeatureMap, four qubits, two repetitions, linear entanglement, scaling α = 2) on ibm_fez for a checksum-locked subset of N = 24 indoor air-quality windows, covering 300 unordered kernel evaluations (276 unique off-diagonal pairs), with the statevector matrix used as the intended object of comparison rather than the hardware matrix.

2. A three-configuration descriptive comparison from single, non-interleaved jobs. Baseline, dynamical decoupling alone (XX sequence), and gate twirling alone (active-accum, 16 randomizations per circuit) are each submitted as one backend job, and compared on rank-order, linear, entrywise, alignment, spectral, and positive-semidefiniteness diagnostics.

3. Explicit finite-shot reference scales to bound what sampling alone can explain. Two reference scales are defined at S = 1024 shots — a conservative global scale of 1/√(2S) ≈ 0.022097 and an entry-resolved matrix-aware scale of ≈ 0.0083 — and used to test whether the observed hardware–statevector discrepancy is attributable to finite sampling.

4. A documented CKA/KTA tension treated as a distortion diagnostic. Centered kernel alignment (CKA) against the statevector reference and centered kernel–target alignment (KTA) against labels are paired as separate implementation-fidelity and task-relevance axes, with fixed-seed permutation nulls (B = 5000) and a count-level finite-shot resampling null (20,000 replicates) as references.

Main Findings

• All three reconstructions completed and were well-formed. Each job returned 300 PUB results, no retrieval failures, and complete, finite, positive-semidefinite 24 × 24 matrices (uncorrected λ_min = 0.429, 0.462, 0.232 versus statevector 0.0565), so no reported metric depends on PSD replacement. The measured diagonal has mean ≈ 0.94 and was retained rather than forced to one.

• Gate twirling deviated least on every reported geometry axis. Off-diagonal RMSE against the statevector reference was 0.0878 (baseline), 0.0864 (dynamical decoupling), and 0.0427 (gate twirling); MAE was 0.0490, 0.0473, and 0.0257; Spearman was 0.741, 0.775, and 0.944; Pearson was 0.827, 0.843, and 0.986.

• Centered alignment was high overall and highest under twirling. Full-matrix CKA ranged 0.933, 0.937, and 0.989; the post hoc diagonal-excluded (U-centered) CKA was 0.8156, 0.8373, and 0.9863. Worst-case entrywise error reached 0.569 (baseline) and 0.564 (dynamical decoupling) versus 0.264 under twirling.

• Dynamical decoupling was not separated from the baseline. No baseline-versus-decoupling contrast was deletion-stable on any persisted metric (|z_desc| ≤ 1.96). The gate-twirled contrasts against baseline and decoupling were deletion-stable for Spearman, MAE, RMSE, and full-matrix CKA (|z_desc| ≈ 2.8–5), while the Pearson twirling-versus-baseline contrast (1.92) and the U-centered twirling-versus-baseline contrast (1.95) fall just below the |z_desc| ≥ 2 reading convention.

• The hardware kernels compressed the off-diagonal spread. Hardware matrices retained 28.9%, 28.3%, and 52.3% of the statevector off-diagonal variance (0.01866), the direction expected under concentration. Effective rank was inflated on hardware (21.18, 21.22, 19.79 versus 17.97 statevector), least under twirling.

• The observed error exceeded both finite-shot reference scales. The three RMSE values are about 4.0, 3.9, and 1.9 times the global scale (≈ 0.0221) and about 10.6, 10.5, and 5.0 times the matrix-aware scale (≈ 0.0083). A direct sampling-only simulation at the executed shot count gave RMSE ≈ 0.0082 on average (99th percentile ≈ 0.0094) and full-matrix CKA ≈ 0.9994, so under those sampling-only models, sampling does not explain the discrepancy.

• The statevector kernel did not align with the labels beyond chance. Its centered KTA was 0.1585, below the permutation-null mean of 0.1710 with upper-tail p = 0.5988 (seed-stable across a 16-seed envelope of [0.587, 0.608]); a split-preserving variant agreed (0.562). Two post hoc classical comparators on the same inputs also showed no above-chance alignment (linear 0.0076, p = 0.796; median-bandwidth RBF 0.0680, p = 0.589).

• Hardware KTA uplift ran opposite to geometric fidelity. ΔKTA was +0.0248, +0.0230, and +0.0125 for baseline, decoupling, and twirling, so the most faithful job carried the lowest hardware KTA (0.1710, closest to the statevector 0.1585). The uplift lies outside a count-level finite-shot resampling null (0/20,000 replicates; null SD 0.0022) but every hardware kernel sits at or below its own label-permutation mean (p = 0.670, 0.711, 0.639), so it is not captured label signal. A numerator–denominator attribution locates it arithmetically: the label-directed numerator y⊤Ky falls by 3.5% to 5.6% on hardware, while the centered kernel norm falls substantially more (−16.6%, −16.8%, −12.5%). All KTA contrasts between configurations are unresolved (|z_desc| ≤ 0.87), and the uplift deletion probe is unresolved (z_desc ≈ 1.1–1.2).

Methodology in Plain English

The team first froze a data subset before any hardware access was authorized: 24 windows from a real indoor air-quality duplicate-sensor dataset (16 training, 8 test, split evenly between 12 event-onset and 12 non-onset windows), spanning 2024-12-22 to 2026-04-20 with median spacing ≈ 4.7 days. Four one-hour pollutant summaries (pm25, pm10, hcho, tvoc means) were min-max scaled to [0, π] under a train-only policy.

They then computed the exact statevector Gram matrix for all 24 windows using a standard four-qubit ZZ feature map — this is the "intended" object. For hardware, each kernel entry was estimated by running a compute–uncompute circuit (encode point j, invert, encode point i) and reading the probability of the all-zero outcome, which equals the squared overlap in the noiseless limit. Each entry is the ratio of observed all-zero counts to 1024 shots.

All 300 pair circuits were submitted as three separate jobs on ibm_fez (a 156-qubit IBM device), one job per configuration, run sequentially within a roughly 13.3-minute window on 2026-05-09 UTC, totaling 921,600 shot samples and 244 billed QPU seconds. The three regimes differ only in Sampler-level runtime options; the logical circuits are shared. Compiled depth was at most 102, with at most 22 two-qubit gates at optimization_level=1, and the planned 4096 shots per circuit were reduced to a budget-safe 1024.

Analysis compares each hardware matrix to the statevector reference using rank-order (Spearman), linear (Pearson), and entrywise error metrics on the off-diagonal domain (552 directed entries), plus centered CKA and effective rank. Uncertainty is handled with leave-one-window-out jackknife standard errors and a descriptive ratio z_desc = Δ / SE_JK(Δ), with |z_desc| ≥ 2 used only as a readability convention called "deletion-stable" — explicitly not a significance test. Label relevance is measured with centered KTA against fixed-seed permutation nulls, and the finite-shot question is answered by comparing observed RMSE to two analytically derived reference scales.

Why This Matters

Impact on research. The paper separates two things that hardware quantum-ML studies often report as one: how faithfully a designed kernel was executed, and whether that kernel is relevant to the task. Because CKA and centered KTA are both invariant to affine (scale-and-shift) changes, a purely depolarizing contraction cannot move them — so a positive KTA uplift certifies a non-affine distortion component, and greater fidelity to the intended structure need not coincide with higher task alignment. The result argues that hardware QML papers should report both axes, since reporting either alone invites overclaiming in one direction or the other.

Real-world applications (as framed by the paper):

• Indoor air-quality monitoring with low-cost duplicate sensors, where noisy, partially redundant multivariate streams make similarity-based learning difficult.
• Redundancy- and missingness-aware (RMA) sensor modeling — the broader project context, though the paper explicitly does not execute any RMA kernel on hardware.
• Benchmarking and validating quantum-kernel implementations before they are used in any downstream pipeline.
• Characterization of runtime mitigation options (dynamical decoupling, gate twirling) as they affect a specific kernel's geometry on a specific device.

Industry relevance. As quantum cloud providers expose mitigation features as runtime-selectable options, users need device- and kernel-specific evidence about what those options change. This study's finding that the twirled job was most faithful on this backend, while decoupling was indistinguishable from baseline, is directly relevant to configuration choices — though the paper stresses that the ordering describes three realized jobs, with job-to-job variation and calibration drift confounded with configuration, and makes no causal mitigation-efficacy claim.

Future Directions

• Interleaved, repeated execution to separate configuration from drift. The authors identify the single non-interleaved design as the most consequential limitation: the three jobs were submitted about eleven minutes apart in fixed order, so drift over that interval is a competing explanation the recorded data cannot exclude. Repeating and interleaving configurations would address this.

• Restoring the planned shot budget and persisting per-randomization counts. Execution used 1024 shots rather than the planned 4096, and for the twirled job the 1024 shots were pooled across 16 randomizations without per-randomization counts, leaving between-randomization variability unestimated. The margin over the reference scales was narrowest for the twirled job, which is where the larger shot budget would have mattered most.

• Physical attribution of the residual discrepancy. The finite-shot comparisons are reference scales, not fitted noise models. The near-identical numerator–denominator attribution identifies an arithmetic locus for the KTA uplift, not a physical mechanism, and the paper leaves physical attribution open.

• Kernel selection on training data with independent hardware validation. Because the map was deliberately frozen, no alternative encoding, bandwidth, or entanglement structure was explored, and any selection on the same 24 windows would be circular. A future design would select on training data, freeze, and validate independently on hardware; a quantitative spectral task–model alignment profile and shot-budget-aware acquisition are named as complementary directions.

• A stopped technical validation attempt. A separate prospectively specified validation attempt stopped at its mandatory acceptance gate because the tested Runtime result representation did not expose separately addressable data for the 16 twirling randomizations; no validation block was submitted and no primary endpoint was computed, so the attempt provides no replication evidence.

Target Audience

Researchers and practitioners working on quantum kernel methods and near-term quantum machine learning who need to know how much a designed kernel changes when executed on real hardware; quantum-computing methodologists interested in reconstruction-fidelity and alignment diagnostics (CKA, centered KTA, effective rank, PSD checks) and in how to report them without overclaiming; users of IBM Quantum backends evaluating runtime mitigation options; and applied data scientists in environmental sensing who want a realistic sense of what current hardware kernel execution does and does not deliver — the paper is explicit that it makes no forecasting, classifier-superiority, or quantum-advantage claim.

Authors’ abstract

Hardware noise and finite sampling perturb the fidelity estimates forming a quantum-kernel Gram matrix. We measured how far three hardware-reconstructed Gram matrices depart from an exact statevector reference for one frozen four-qubit ZZ feature map on N=24 indoor air-quality windows, executed on ibm_fez at 1024 shots per circuit in three single, non-interleaved jobs: baseline, dynamical decoupling alone, and gate twirling alone. All were complete, finite, and positive-semidefinite. Off-diagonal root-mean-squared error (RMSE) against the reference was 0.0878, 0.0864, and 0.0427; full-matrix centered kernel alignment (CKA) ranged 0.933-0.989 and the post hoc diagonal-excluded (U-centered) CKA 0.816-0.986. The gate-twirled job deviated least on every reported geometry axis; its baseline contrasts are deletion-stable for the Spearman, mean-absolute-error, RMSE, and full-matrix CKA diagnostics, while the Pearson and diagonal-excluded contrasts fall just below that convention. Dynamical decoupling was not separated from the baseline. The observed error exceeded both finite-shot reference scales, so, under those sampling-only models, sampling does not explain it. Centered kernel-target alignment did not track reconstruction fidelity and stayed at or below each label-permutation reference: implementation fidelity and task relevance are distinct diagnostic axes. All configuration-level statements describe three realized jobs on one backend; no mitigation-efficacy, classifier-superiority, forecasting, or quantum-advantage claim is made.