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Technical Physics Letters

Benchmarking a Long-Range Interacting Fermionic Model on IBM Heron r2 Processors: Zero-Noise Extrapolation versus Symmetry-Based Post-Selection on a 6-Qubit Ring Testbed

A. G. Kiiamov, Tayurskii Dmitrii Albertovich

Published Oct 1, 2026• 0 citations
Publisher

Abstract

We benchmark two error-mitigation strategies for digital quantum simulation of a long-range interacting fermionic model on IBM Heron r2 superconducting processors. The testbed is a minimal exactly solvable system—two spinless electrons on a 6-site ring with repulsive $${\text{1/}}r$$ chord-distance Coulomb interactions—whose ground state exhibits a smooth Wigner-dimer localization crossover quantified by the antipodal occupation probability $${P_3}(U)$$ . Circuit parameters of a particle-number-conserving variational ansatz are optimized classically; the quantum processors execute only final energy and configuration measurements. We compare (i) zero-noise extrapolation (ZNE) applied to the full Hamiltonian with (ii) a hybrid estimator in which the off-diagonal kinetic energy is measured with ZNE while the diagonal interaction energy is reconstructed from particle-number post-selected bitstrings. Across eight interaction strengths $$U{\text{/}}t \in [5,75]$$ , five measurement sessions on two devices (ibm_fez, ibm_marrakesh), and three circuit depths, the hybrid estimator achieves, in the interacting regime $$U{\text{/}}t \geqslant 15$$ , relative ground-state energy errors of 4.3–14% that decrease systematically with $$U$$ —an order of magnitude better than full-Hamiltonian ZNE (29–85%)—and reproduces the localization crossover in $${P_3}(U)$$ . At $$U{\text{/}}t = 5$$ , where $${E_0} \approx 0.08{\kern 1pt} t$$ renders the relative error ill-conditioned, the absolute hybrid error is $$0.62{\kern 1pt} t$$ , in line with the rest of the scan. A depth scan at fixed $$U{\text{/}}t = 45$$ shows that the shallowest circuit wins despite a 1.8% ansatz expressivity penalty, quantifying the expressivity–noise trade-off. Shot-noise uncertainties, recovered exactly from per-noise-factor data, are below one percentage point in relative energy error for the hybrid estimator (versus 7–14 percentage points for ZNE at equal shot budget) and an order of magnitude smaller than the measured inter-session calibration spread. The exact variational bound $${E_{{\text{exp}}}} \geqslant {E_{{\text{theory}}}}$$ is satisfied in all 27 full-Hamiltonian hardware energy measurements. All data were acquired with under nine minutes of QPU time on the free IBM Quantum Open Plan.

DOI
10.1134/s1063785026600687

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•Jan 1, 2026•0 citations
We benchmark two error-mitigation strategies for digital quantum simulation of a long-range interacting fermionic model on IBM Heron r2 superconducting processors. The testbed is a minimal exactly solvable system—two spinless electrons on a 6-site ring with repulsive 1/r{\text{1/}}r1/r chord-distance Coulomb interactions—whose ground state exhibits a smooth Wigner-dimer localization crossover quantified by the antipodal occupation probability P3(U){P_3}(U)P3​(U) . Circuit parameters of a particle-number-conserving variational ansatz are optimized classically; the quantum processors execute only final energy and configuration measurements. We compare (i) zero-noise extrapolation (ZNE) applied to the full Hamiltonian with (ii) a hybrid estimator in which the off-diagonal kinetic energy is measured with ZNE while the diagonal interaction energy is reconstructed from particle-number post-selected bitstrings. Across eight interaction strengths U/t∈[5,75]U{\text{/}}t \in [5,75]U/t∈[5,75] , five measurement sessions on two devices (ibm_fez, ibm_marrakesh), and three circuit depths, the hybrid estimator achieves, in the interacting regime U/t⩾15U{\text{/}}t \geqslant 15U/t⩾15 , relative ground-state energy errors of 4.3–14% that decrease systematically with UUU —an order of magnitude better than full-Hamiltonian ZNE (29–85%)—and reproduces the localization crossover in P3(U){P_3}(U)P3​(U) . At U/t=5U{\text{/}}t = 5U/t=5 , where E0≈0.08t{E_0} \approx 0.08{\kern 1pt} tE0​≈0.08t renders the relative error ill-conditioned, the absolute hybrid error is 0.62t0.62{\kern 1pt} t0.62t , in line with the rest of the scan. A depth scan at fixed U/t=45U{\text{/}}t = 45U/t=45 shows that the shallowest circuit wins despite a 1.8% ansatz expressivity penalty, quantifying the expressivity–noise trade-off. Shot-noise uncertainties, recovered exactly from per-noise-factor data, are below one percentage point in relative energy error for the hybrid estimator (versus 7–14 percentage points for ZNE at equal shot budget) and an order of magnitude smaller than the measured inter-session calibration spread. The exact variational bound Eexp⩾Etheory{E_{{\text{exp}}}} \geqslant {E_{{\text{theory}}}}Eexp​⩾Etheory​ is satisfied in all 27 full-Hamiltonian hardware energy measurements. All data were acquired with under nine minutes of QPU time on the free IBM Quantum Open Plan.
Variational Quantum Eigensolvers (VQE) promise molecular ground-state energies on today's noisy intermediate-scale quantum (NISQ) processors, but device noise biases the estimated energy severely — under a device-calibrated noise model, the hydrogen molecule's energy error grows from 0.000 mHa (noiseless) to 303.6 mHa (noisy), roughly 190× the "chemical accuracy" threshold of 1.6 mHa that computational chemistry requires. Error-mitigation techniques repair such biases in software, but published results usually report each technique in isolation, making it impossible to know which mitigation to buy with a fixed shot budget. This work answers that question with a controlled comparison. Under identical circuits, noise profiles and shot budgets, I measure the bias and variance of four estimator stacks — raw sampling, twirled-readout extinction (TREX-style calibrated readout inversion), zero-noise extrapolation (ZNE), and ZNE+TREX — across independent random seeds. Three findings emerge. (1) ZNE is a bias–variance tradeoff, not a free lunch: a 7-point Richardson extrapolant reduces the deterministic bias 3200-fold (303.6 → 0.094 mHa) but amplifies statistical noise by a measured factor of up to 25× per run, while TREX removes readout bias with no variance penalty. (2) ZNE has a finite operating window in circuit depth: error recovery falls 99.97% → 99.1% → 95.5% as CX count grows 56 → 112 → 168, and is exhausted entirely for a 1616-CX lithium-hydride circuit at near-term error rates. (3) A noise-bias-tailored repetition code corrects the dominant error axis with logical error rates matching theory (3p², 10p³, 35p⁴ for distances 3, 5, 7) at strictly linear gate overhead (gates = 6d − 6). The results reframe mitigation selection as budget allocation on an empirically measured bias–variance frontier and provide open-source tooling to reproduce every number.
We report on a high-fidelity digital quantum simulation of Wigner localisation in a quasi-one-dimensional (quasi-1D) electron system using a 6-qubit segment of the state-of-the-art \textbf{IBM\,Heron\,2} quantum processor. By mapping the Coulomb interaction Hamiltonian onto a 6-qubit ring lattice, we reconstruct the ground-state energy landscape for a 2-electron Wigner dimer across fifteen interaction regimes in the range U∈[5,75]U \in [5, 75]U∈[5,75]. This study serves as a rigorous \textbf{benchmarking} exercise, translating foundational experimental models originally developed for electrons on liquid helium into the domain of modern quantum computing. Leveraging the enhanced gate fidelity and tunable coupler architecture of the Heron 2, we demonstrate that the digital simulation accurately captures the energy minimisation trends associated with Wigner dimer formation, achieving a relative error below 7\% in the strong-interaction limit. Our results provide a crucial \textbf{proof-of-principle} validation for using superconducting quantum hardware to probe strongly correlated phases of matter with high precision, establishing a baseline for future simulations beyond the classical limit.
We introduce a Basis Adaptive (BA) algorithm for hybrid quantum-classical simulation of correlated quantum many-body systems. Starting from a small set of physically motivated bitstrings, the algorithm iteratively applies a single-step first-order Trotterized circuit on a quantum processor, filters the sampled configurations by enforcing U(1)U(1)U(1) spin conservation and lattice reflection symmetry, and classically diagonalizes the Hamiltonian in the resulting reduced Hilbert space. This design avoids the variational optimization overhead of VQE, the deep coherent circuits required by QPE, and the symmetry-violating subspaces that arise in SKQD. The ground-state energy error is bounded analytically by 8 ∥H∥(1−αDT)1/2\sqrt{8}\,\|H\|\left(1-\sqrt{α_{D_T}}\right)^{1/2}8​∥H∥(1−αDT​​​)1/2, where αDTα_{D_T}αDT​​ is the probability weight captured by the DTD_TDT​ sampled basis states. This bound connects algorithm performance directly to ground-state sparsity and explains the observed accuracy hierarchy across different phases. Benchmarked on the spin-1/21/21/2 Heisenberg XXZ chain (up to N=62N=62N=62 qubits on the IBM Heron processor), the algorithm achieves a 3.5%3.5\%3.5% energy error in the gapped Neel phase (Δ=2.0Δ=2.0Δ=2.0) and below 0.5%0.5\%0.5% at the ferromagnetic boundary (Δ=−1.0Δ=-1.0Δ=−1.0). The accuracy degrades to 28.7%28.7\%28.7% in the strongly quasi-long-range-ordered regime (Δ=0.5Δ=0.5Δ=0.5). Spin-spin correlation functions are reproduced across all regimes, confirming that symmetry-filtered real-time sampling provides a practical and noise-resilient pathway to ground-state properties on near-term quantum hardware.
Qubit layout selection determining which physical qubits a logical circuit occupies on a superconducting processor is fundamentally a noise-field sampling problem. On IBM Heron r1 and Heron r2 devices, two-qubit gate error varies by nearly one order of magnitude (7– $9\times $ ) across qubit pairs on the same chip, causing measured fidelity differences up to 0.903 from layout choice alone. We introduce the Gate Noise Sensitivity Index (GNSI), a calibration-weighted linear combination of per-qubit readout error and per-edge two-qubit gate error, derived analytically as a first-order approximation of circuit fidelity under independent, heterogeneous noise channels. For 12 backend–circuit combinations on three IBM processors (133-qubit Heron r1 and 156-qubit Heron r2), GNSI scores are computed for the full pool of 1,320 computationally enumerated candidate layouts; the top-ranked (GNSI-BEST) and bottom-ranked (GNSI-WORST) layouts are then executed on hardware. GNSI-BEST ranks first by score in all 12 instances (Hit@1 = 100% within the enumerated candidate pool) and consistently achieves higher measured fidelity than GNSI-WORST, with Spearman rank correlation ρ=0.926±0.016\rho = 0.926 \pm 0.016ρ=0.926±0.016 and statistically significant fidelity separation in 9 of 12 paired tests. Initial hardware comparisons on ibm_fez ( n=5n=5n=5 ) show GNSI-BEST matching or exceeding SABRE-selected fidelity. We further identify a depth-dependent regime transition: gate-error heterogeneity appears to govern ranking in shallow circuits (depth ≤15\leq 15≤15 ), while qubit-specific T1T_{1}T1​ variation may dominate for deeper circuits (depth ≥20\geq 20≥20 ). The analytic regime boundary d∗(b)d^{*}(b)d∗(b) , computable from calibration data alone, shows agreement within ±2 depth layers across all three backends in initial comparisons; GNSI-Extended incorporating the decoherence term receives an initial hardware comparison on two circuits, with full validation left to future work. These results suggest that layout-dependent fidelity variation may be governed by a calibration-predictable noise field, with direct implications for compiler design and noise-adaptive quantum circuit execution.
Near term quantum processors operate in a noise dominated regime, motivating error mitigation techniques that recover accurate expectation values without full fault tolerance. Zero Noise Extrapolation (ZNE) is a widely used but biased error mitigation method that lacks rigorous error bounds. Its effective application requires nontrivial technical choices, most notably the selection of noise scaling factors and extrapolation models, making ZNE sensitive to user expertise and often necessitating costly trial and error procedures. Here, we introduce Folding Free Zero Noise Extrapolation (FF-ZNE), a method that removes the need for noise factor selection by achieving effective noise amplification without circuit folding. FF-ZNE exploits isomorphic hardware layouts with distinct native noise profiles, such that executing a fixed circuit across these layouts induces controllable variations in the effective noise strength. Under a depolarizing noise model, we analytically show that the resulting extrapolation admits a fixed linear form, eliminating extrapolator choice and enabling a seamless, user independent mitigation procedure. We further propose two algorithms that identify sets of isomorphic hardware layouts on which a given circuit yields sufficiently distinct expectation values to enable reliable zero noise extrapolation. Experiments on a 133 qubit IBM Quantum device demonstrate that FF-ZNE yields mitigated expectation values with average deviations of ~6% and 4.5% for up to 50 qubit EfficientSU2 (sparse) and Hamiltonian simulation (dense) circuits, respectively. The method is thus scalable and applicable to a broad range of circuits. By eliminating noise factor and extrapolator selection, FF-ZNE transforms zero noise extrapolation from a technique requiring expert tuning into a practical, scalable, and broadly accessible error mitigation method for current quantum hardware.
Experimental measurement data of the QITE formalism (Quantum-Inspired Thermal Equilibrium) on IBM quantum processors of the Heron R2 generation with 156 physical qubits. Altogether 75+ jobs on two different backends (ibm_kingston and ibm_fez), over 830,000 shots, carried out on 8, 10, and 11 April 2026. Seven experiment series: (1) Static phases (R=0.5 / 1.0 / 2.0): three clearly separated Hamming-weight distributions (16.3σ significance), positive ZZ correlations under QITE (+0.318 at R=2.0) versus negative ones in the null experiment (−0.061). Controlled comparison experiment with null hypothesis (identical circuit structure with random parameters). (2) Real-time feedback loop (8 cycles): two targeted perturbations (R=3.0 and R=0.3); the system returns autonomously to equilibrium within 2 cycles. First demonstration of dynamic controllability of a QITE quantum collective on a real quantum computer. (3) Cross-backend reproducibility: identical results on ibm_kingston and ibm_fez — deviations within a few percent. (4) Zero-noise extrapolation (ZNE): circuit folding (1×, 3×, 5×) with Richardson extrapolation. Result: coherence 0.474 → 0.474 extrapolated (improvement: −0.0005). The optimized circuit (depth 13, 120 CZ gates) already operates at the hardware noise floor — ZNE cannot improve it further because the circuit is already near-optimal. (5) Scaling comparison 27 vs. 156 qubits: the 27-qubit version on the same backend (ibm_kingston) shows ZZ correlation of 0.788 against 0.318 at 156 qubits (Δ = +0.470). Circuit depth drops from 13 to 6, CZ gates from 120 to 13. Confirms: the QITE signal scales with gate quality. (6) Extended feedback loop (20 cycles): four extreme perturbations — R=3.0 (excess), R=0.3 (deficit), R=5.0 (5-fold excess), R=0.1 (10-fold deficit). Maximum coherence C=0.785 (highest value ever measured). The system reorganizes itself after each perturbation within 2–4 cycles. Long-term stability demonstrated across 20 cycles and 436 seconds of total runtime. (7) 3-minute stability demonstration (20 cycles, 262.7 seconds): dedicated long-term test with two targeted shocks (R=4.0 and R=0.2). Result: ones_fraction remains between 0.491 and 0.499 over 4.4 minutes (maximum deviation from equilibrium: 0.009). Drift across the entire runtime: +0.003. Complete recovery after both shocks. 18/18 non-shock cycles stable. First demonstration of a self-correcting quantum field on a real quantum computer over several minutes. Circuit optimization relative to 8 April: reduction of circuit depth from 295 to 13 (factor 22×), two-qubit gates from 1,946 to 120, by exclusive use of native Heron R2 gates (CZ, SX, RZ, X) and direct lambda encoding via RY(θ) with θ = 2·arctan(λ₂/λ₁). The system demonstrably produces controllable, correlated, self-organizing quantum states on real hardware — no simulation, no theory, but physical measurement data of a concrete technical system. All 59 IBM job IDs are contained in the raw data and verifiable on quantum.ibm.com. German patent applications Az. 10 2025 003 906.9 (QITE) and Az. 10 2025 004 891.2 (QISE). Final document (12 April 2026) — 21 sections, 9 experiment series, 372 individual quantum measurements. Extensions compared to the previous version: Section 19: 9-minute experiment (209 cycles, 156 qubits, 4096 shots) Phase A: null test (23 cycles, R=1.0) — baseline control measurement Phase B: 8 shocks (R=0.1 to R=8.0) incl. double shock — complete recovery after each shock Section 20: overall summary of all 9 experiment series (8–12 April 2026) Section 21: raw-data appendix with SHA-256 checksums of all 9 JSON files Core results: 372 quantum measurements on IBM Heron R2 (ibm_kingston, ibm_fez) Dynamic range R=0.1 to R=8.0 (factor 80) with complete self-correction The null test proves: recovery is an active process, not a hardware artefact No drift, no collapse across 209 consecutive measurements Optimized circuit: depth 13, 120 CZ gates, at the hardware noise floor (ZNE-confirmed)