Large circuit execution for NMR spectroscopy simulation on NISQ quantum hardware

Abstract: With the latest advances in quantum computing technology, we are gradually moving from the noisy intermediate-scale quantum (NISQ) era characterized by hardware limited in the number of qubits and plagued with quantum noise, to the age of quantum utility where both the newest hardware and software methods allow for tackling problems which have been deemed difficult or intractable with conventional classical methods. One of these difficult problems is the simulation of one-dimensional (1D) nuclear magnetic resonance (NMR) spectra, a major tool to learn about the structure of molecules, helping the design of new materials or drugs. Using advanced error mitigation and error suppression techniques from Q-CTRL together with the latest commercially available superconducting-qubit quantum computer from IBM and trapped-ion quantum computer from IonQ, we present the quantum Hamiltonian simulation of liquid-state 1D NMR spectra in the high-field regime for spin systems up to 34 spins. Our pipeline has a major impact on the ability to execute deep quantum circuits with the reduction of quantum noise, improving mean square error by a factor of 22. It allows for the execution of deep quantum circuits and obtaining salient features of the 1D NMR spectra for both 16-spin and 22-spin systems, as well as a 34-spin system, which lies beyond the regime where unrestricted full Liouvillespace simulations are practical (32 spins, the Liouville limit). Our work is a step toward near-term quantum utility in NMR spectroscopy.

• The paper demonstrates a robust quantum simulation pipeline using Trotterized evolution and Q-CTRL error mitigation to execute deep circuits for NMR spectra on systems up to 34 qubits.
• The paper shows significant circuit depth reductions (28–50%) and mean-squared error improvements (up to 22x), enabling accurate spectral recovery compared to classical benchmarks.
• The paper establishes that hardware noise is now subdominant, with algorithmic Trotter errors limiting accuracy and suggesting future paths for enhanced quantum simulation protocols.

Large Circuit Execution for NMR Spectroscopy Simulation on NISQ Quantum Hardware

Motivation and Context

This work addresses quantum simulation of nuclear magnetic resonance (NMR) spectra—a prime candidate for quantum utility given the exponential scaling of Hilbert/Liouville space with the spin number and the direct mapping of spin-1/2 Hamiltonians to qubit operations. Classical full-space NMR simulations are limited to $\sim$20 spins, with tensor network approaches extending this to 32 spins (the "Liouville limit") (Elenewski et al., 2024). Simulating realistic NMR spectra with strong spin-spin couplings in the high-field liquid regime remains intractable for classical methods beyond this threshold and is essential for advancing quantum simulation as a practical tool in chemistry and material sciences.

The paper demonstrates that by leveraging noise-suppressed quantum circuit execution pipelines—centered on Q-CTRL Fire Opal—and using both IBM superconducting qubit (Heron) and IonQ trapped-ion platforms, one can execute circuits deep enough to simulate spin systems up to 34 qubits. The focus is not on demonstrating quantum advantage per se, but on the feasibility of deep quantum circuit execution with controlled error and on extracting salient spectral features for NMR systems notably beyond the reach of unrestricted classical simulation.

Computation Pipeline and Methodology

The pipeline initiates with a molecular description, from which the spin system Hamiltonian is defined in the rotating frame, capturing isotropic chemical shifts and $J$ couplings appropriate for high-field liquid-state NMR. Time evolution of the spin system is simulated via Trotterized product formulas—specifically, a single Trotter step per time point is used for tractability on current hardware. Each nuclear spin in the system is mapped to a qubit. Observable magnetization operators ($M_X$, $M_Y$) are measured at each time point to construct the free induction decay (FID), which is post-processed by FFT (after zero padding) to synthesize the 1D NMR spectrum.

Figure 1: The computation pipeline integrates molecular input, Trotterized quantum simulation, Q-CTRL-based error suppression, measurement and FID construction, and final Fourier spectral analysis.

The Q-CTRL Fire Opal error suppression and mitigation workflow addresses the main bottlenecks in executing long-depth circuits on NISQ devices:

• Aggressive transpilation respecting hardware connectivity and native gate sets
• Error-aware hardware mapping, crucial for devices lacking all-to-all connectivity (notably IBM Heron)
• Optimized dynamical decoupling, especially to mitigate $T_2$ and $ZZ$ crosstalk errors, increasing single-qubit gates but reducing two-qubit and overall noise errors
• Optimized block-structured gate consolidation to reduce circuit depth
• Post-processing with scalable readout error mitigation strategies akin to M3 [Phys. Rev. A 103, 042605 (2021)]

Practical choices in experimental design (shots per time point, number of time points) were informed by empirical tests (see Appendix protocol).

Quantum Circuit Implementation and Depth Reduction

For all target molecules (anti-3,4-difluoroheptane (DFH, 16 spins), symmetric P-H (22 spins), and B[ACR9]3 phosphorus cluster (34 spins)), the error suppression pipeline consistently yielded significant two-qubit circuit depth reduction (typically 28–50%). Depth distributions across circuits for each molecule demonstrate a pronounced shift to lower average depth with the Q-CTRL pipeline.

Figure 2: Quantum circuit structure (left) and the circuit depth histograms (right panels) highlight two-qubit gate depth reduction using the error mitigation pipeline across several molecular targets.

This reduction enables the execution of circuits of depth up to 250 on IBM Heron and up to 69 effective gates on the IonQ Forte platform. Notably, IonQ’s all-to-all connectivity permits even further depth compaction by obviating SWAPs and similar overheads.

Error Reduction in NMR Signal Computation

Applying the Q-CTRL pipeline leads to a dramatic reduction in mean-squared error (MSE) between the quantum and ideal noiseless signals. For the symmetric P-H molecule (22 spins), MSE is reduced by a factor of 12 on the full FID, up to 22x on the early time points; cosine similarity increases from $C=0.51$ (raw) to $C=0.99$ (mitigated) on 800 time points with 8192 shots each.

Figure 3: Real part of the FID for symm_H showing substantial error reduction and signal fidelity restoration using error suppression/mitigation.

These improvements directly translate to an ability to reliably extract spectral features from quantum hardware, even for deep circuit executions. Notably, these MSE reductions are achieved at minimal computational overhead (i.e., no significant increase in circuit width or sampling budget).

NMR Spectral Simulation Results

Quantum-computed NMR spectra, as processed through the described pipeline, reproduce the main features and peak positions of the SPINACH-generated reference spectra for all molecules, including the phosphorous cluster at 34 spins—surpassing the practical limit of classical full Liouville space codes.

Figure 4: Molecular structures (left column) and computed spectra with quantum (red) versus classical (blue, SPINACH) for DFH, symmetric P-H, and the phosphorus cluster.

Sub-peak structure is observable in the quantum spectra, particularly for the largest system, though resolution for fine details is limited by the single Trotter step employed and intrinsic hardware noise. In direct IBM Heron/IonQ Forte comparisons for the 22-spin symmetric P-H, similar spectral quality is observed, with the IonQ results exhibiting fewer spurious side peaks.

Figure 5: Comparison of quantum-computed symm_P spectra between IonQ Forte (solid green) and IBM Heron (dotted red), illustrating hardware-dependent performance.

Execution Time and Hardware Comparison

Runtime analysis reveals a substantial advantage for superconducting (IBM Heron) hardware in terms of overall throughput (e.g., 4–12 hours per spectrum on IBM Heron at 4096–8192 shots, versus >100 hours for comparable datasets on IonQ Forte). This reflects the much faster native gate and measurement times in superconducting qubit hardware despite increased noise and connectivity limitations.

Figure 6: Hardware comparison for the FID computation: IonQ (700 shots) delivers smoother magnetization traces with less hardware-specific noise than IBM (4096 shots).

Across all platforms, employing deep error-suppressed circuits was essential for any meaningful agreement with classical computations at these system sizes.

Algorithmic Limits and Path Forward

A comparative analysis of quantum-simulated spectra versus both noiseless quantum circuits and SPINACH reference (see Appendix, Fig. 8) reveals that hardware noise is now subdominant to the algorithmic error introduced by the single Trotter step approximation. MSE between hardware-executed and noiseless quantum circuits is an order of magnitude lower than the mismatch to the SPINACH solution.

Figure 7: Quantum hardware noise is now much less limiting than algorithmic Trotter error; further advances depend on better Hamiltonian simulation protocols.

Emerging approaches offer immediate paths for improvement: more accurate Hamiltonian simulation protocols (qubitization [Low & Chuang, Quantum 3:163 (2019)], truncated Taylor/Dyson series [Phys. Rev. Lett. 114, 090502 (2015); Phys. Rev. A 99, 042314 (2019)], and spin echo approaches (Zhang et al., 22 Oct 2025, Abanin et al., 11 Jun 2025)) promise a reduction in Trotterization error without a linear increase in circuit depth. Applying such approaches in conjunction with scalable error mitigation could bring full spectral agreement, including fine substructure, within reach as hardware matures.

Implications and Future Directions

This work establishes a practical, scalable workflow for NMR spectral simulation at and beyond the classical tractability frontier. The deployment of deep error-mitigated circuits on current NISQ devices demonstrates that the limitation is now predominantly in the stochastic error due to circuit sampling and algorithmic error from low-order time evolution approximations. The implications are:

• Utility-scale quantum simulation for problems of immediate chemical and material science relevance is feasible with continued incremental hardware and algorithmic advances.
• Once higher-order Hamiltonian simulation and integration of robust error mitigation pipelines become standard, detailed, high-resolution NMR spectra for systems beyond the Liouville limit will be accessible, enabling novel applications in drug discovery and condensed matter.
• Techniques developed here will be transferable to other quantum simulation domains, including electronic structure, quantum dynamics in open systems, and beyond.

Exploration of open-system (bath-coupled) NMR simulation, integration with alternative error correction/mitigation schemes (e.g., QESEM (Aharonov et al., 14 Aug 2025)), and demonstration of complete qubitization-based NMR simulation pipelines are natural next steps.

Conclusion

The paper demonstrates that large-scale, high-depth quantum circuit execution for NMR spectra simulation on NISQ hardware, with advanced error suppression and mitigation, enables the extraction of relevant chemical information for spin systems up to 34 qubits—surpassing classical simulation thresholds. Error-suppressed circuits yield 12–22x improvement in MSE for FID recovery and successful reproduction of spectral features in systems previously inaccessible to quantum hardware. Limitations are now algorithmic rather than hardware-dominated, suggesting that further improvements in Hamiltonian simulation algorithms, combined with scalable mitigation, will make quantum utility in NMR spectroscopy and related fields a near-term reality.

Reference: "Large circuit execution for NMR spectroscopy simulation on NISQ quantum hardware" (2512.14513)