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Chemistry Beyond the Scale of Exact Diagonalization on a Quantum-Centric Supercomputer
Javier Robledo-Moreno, Mario Motta, Holger Haas, Ali Javadi-Abhari, Petar Jurcevic, William Kirby, Simon Martiel, Kunal Sharma, Sandeep Sharma, Tomonori Shirakawa, Iskandar Sitdikov, Rong-Yang Sun, Kevin J. Sung, Maika Takita, Minh C. Tran, Seiji Yunoki, Antonio Mezzacapo
TL;DR
Exact diagonalization and standalone prefault-tolerant quantum processors face scaling, runtime, circuit-depth, and noise challenges in electronic-structure calculations. The paper combines quantum sampling with distributed classical computing in SQD and applies it to molecular and iron-sulfur-cluster problems, finding results beyond exact-diagonalization sizes within the studied scope.
Problem
Exact diagonalization is limited to small electronic-structure systems, while prefault-tolerant quantum implementations face prohibitive measurement runtimes and deep-circuit noise.
Method
SQD combines quantum experiments with a quantum-centric supercomputing workflow that uses quantum samples and configuration recovery to construct sparse ground-state wavefunction approximations.
Results
SQD addresses realistic N2, [2Fe-2S], and [4Fe-4S] problems using 58, 45, and 77 qubits, respectively, with up to 3.5 K two-qubit gates, while N2 energies differ from HCI by tens of mEh.
Takeaways & Limitations
The results suggest that quantum-centric supercomputing can tackle chemistry problems beyond sizes amenable to exact diagonalization at current error rates.
Takeaways & Limitations
For some problem instances, classical configuration proposals remain the bottleneck, and larger, more strongly correlated systems may challenge classical heuristics.
Abstract
from arXiv · showhide
A universal quantum computer can simulate diverse quantum systems, with electronic structure for chemistry offering challenging problems for practical use cases around the hundred-qubit mark. While current quantum processors have reached this size, deep circuits and large number of measurements lead to prohibitive runtimes for quantum computers in isolation. Here, we demonstrate the use of classical distributed computing to offload all but an intrinsically quantum component of a workflow for electronic structure simulations. Using a Heron superconducting processor and the supercomputer Fugaku, we simulate the ground-state dissociation of N$_2$ and the [2Fe-2S] and [4Fe-4S] clusters, with circuits up to 77 qubits and 10,570 gates. The proposed algorithm processes quantum samples to produce upper bounds for the ground-state energy and sparse approximations to the ground-state wavefunctions. Our results suggest that, for current error rates, a quantum-centric supercomputing architecture can tackle challenging chemistry problems beyond sizes amenable to exact diagonalization.
Introduction
Ground-state energies are central targets in quantum chemistry, but exact solutions become combinatorially expensive and are limited to relatively small systems. Quantum chemistry experiments on prefault-tolerant processors also face prohibitive runtimes and deep-circuit errors, motivating the SQD workflow introduced here.
- Motivation: Ground-state energies are computed by solving the Schrödinger equation in the Born-Oppenheimer approximation, but finite-basis exact solutions scale combinatorially with electrons and orbitals.This scaling makes exact numerical treatment difficult as molecular systems grow.
- Motivation: FCI is limited to approximately (22e,22o) and (26e,23o), requiring approximate methods for larger systems.Examples include diagrammatic techniques, wavefunction ansatzes, and Monte Carlo integration.
- Motivation: Prefault-tolerant quantum chemistry implementations remain limited because energy-estimation runtimes are excessive and chemically motivated circuits are very deep.For unitary coupled cluster and a single time-evolution step, circuit-related quantities scale as M^4 with the number of spin-orbitals M.
- Contribution: The paper introduces Sample-Based Quantum Diagonalization (SQD), a quantum-centric workflow for realistic electronic-structure problems beyond exact diagonalization.Experiments study N2, [2Fe-2S], and [4Fe-4S] using 58, 45, and 77 qubits, respectively, with up to 3.5 K two-qubit gates.
Results
The SQD workflow combines quantum sampling with distributed classical processing to study molecular ground states beyond exact-diagonalization scales. Experiments on N2 and iron-sulfur clusters show useful energy and wavefunction information, while highlighting conditions and limitations for quantum advantage.
- Workflow: SQD combines quantum processors with distributed classical nodes to execute limited large circuits and process sampled configurations for ground-state estimation.The workflow maps chemistry problems to qubits, executes circuits, recovers configurations, and projects and diagonalizes subspaces classically.
- Configuration recovery: Configuration recovery reduces the quantum signal required for N2 energy errors below 10 mEh from approximately 20% to approximately 2%.The comparison uses raw noisy samples without recovery versus samples processed by configuration recovery.
- N2 dissociation: The N2 experiments produce energies within tens of mEh of HCI across the dissociation curve, while restricted CCSD fails for the dissociation.The combined quantum runtime for all dissociation-curve points is approximately 45 minutes.
- Iron-sulfur clusters: At approximately 3.5k two-qubit gates, measurement outcomes retain a valuable signal even when quantum solutions are worse than other classical methods.The [4Fe-4S] analysis compares energy and variance from Heron outcomes with configurations sampled uniformly.
- Implications: SQD’s accuracy can be ranked through its upper-bound ground-state energies, while its sparse wavefunctions can be stored, certified, and further processed classically.Improvement over SCI depends on quantum circuits producing better and more efficient subspaces under the sparse-determinant assumption.
Conventions and notation
The paper formulates electronic configurations in a second-quantized molecular-orbital basis and maps them to qubit bitstrings using Jordan–Wigner notation.
- Hamiltonian and basis: The Born–Oppenheimer Hamiltonian uses orthonormal molecular orbitals, with nuclear, one-electron, and two-electron contributions represented in atomic units.Active-space and relativistic calculations modify the Hamiltonian quantities to account for inactive-electron density and scalar relativistic effects.
- Qubit representation: Jordan–Wigner mapping represents each electronic configuration as a computational-basis bitstring over M = 2N_MO qubits.The bitstring separates spin-up and spin-down orbital occupations.
- Particle sectors: The Hamming weight in each spin sector gives the number of spin-σ electrons, and valid configurations satisfy N_xσ = N_σ.The restricted Hartree–Fock bitstring obeys the required particle-number constraints by construction.
- Configuration space: The Fock space contains all electronic configurations across possible orbital filling factors, while determinant and electronic configuration are used interchangeably.The Hilbert-space dimension grows with the number of electrons and orbitals.
Sample-based Quantum Diagonalization
Sample-Based Quantum Diagonalization samples noisy quantum configurations, recovers valid particle sectors, and diagonalizes Hamiltonians in sampled subspaces to estimate ground states.
- Projection and diagonalization: SQD projects and diagonalizes the Hamiltonian separately in sampled configuration subspaces, whose lowest eigenstates approximate the ground state.The resulting eigenvalues provide ground-state energy estimates and the eigenvectors define sparse wavefunction approximations.
- Symmetry constraints: Particle number and spin projection are enforced directly or approximately, while total-spin conservation is implemented as a soft penalty with λ = 0.2.The eigenstate solver penalizes contributions from states with the wrong total-spin eigenvalue.
- Batch construction: SQD forms batches from unique spin-up and spin-down configurations rather than directly collecting independent samples, supporting more complete spin-coupled subspaces.The construction is motivated by avoiding spin contamination or redundant configurations when partner bitstrings are absent.
- Configuration recovery: Configuration recovery filters noisy samples to the correct particle-number sector and probabilistically flips bits using averaged spin-orbital occupancies.The procedure corrects configurations whose measured spin-sector electron counts differ from the target ground-state counts.
- Energy-variance analysis: Energy-variance extrapolation fits the linear relation between energy and Hamiltonian variance when the approximate state is sufficiently close to an eigenstate.The fitted intercept estimates the energy at zero variance and can reveal nearby eigenstates.
Experimental details
The experiments use LUCJ circuits initialized from classical CCSD information and sparsified for heavy-hex hardware, with orbital permutations mitigating connectivity constraints.
- LUCJ ansatz: LUCJ samples electronic configurations using local number-number interactions and hardware-compatible orbital rotations.Its local approximation balances circuit accuracy with heavy-hex connectivity and gate-depth constraints.
- Circuit construction: The truncated two-layer LUCJ circuit adds exp(K_2) to reduce excessive concentration around the Hartree–Fock configuration for dynamically correlated species.The circuit depth remains identical to that of a single-layer LUCJ circuit.
- Hardware mapping: Heavy-hex sparsification retains interactions compatible with adjacent-qubit connectivity and uses orbital permutations to place dominant terms at favorable positions.Orbital rotations implement the required permutations before sparsification.
- Classical initialization: Circuit parameters are derived from a classical restricted closed-shell CCSD calculation by decomposing and retaining dominant components of the t_2 tensor.The resulting parameters are used directly for hardware sampling without further variational optimization.
- Limitations: Discarding large interaction elements can produce a lower-accuracy wavefunction, while the connectivity limitation requires larger processors or a substantially different fermion-to-qubit mapping.The paper identifies this as a fundamental problem for the current mapping and hardware setting.
Supplementary materials
The supplementary materials include additional figures, tables, and references supporting the main text.
- Supplementary contents: The supplementary materials contain Figures S1–S22, Tables S1–S2, and references 7–106.
Supercomputer
The manuscript lists contributors affiliated with IBM Quantum, IBM Research, the University of Colorado, and RIKEN centers in Japan.
- The author list includes Javier Robledo-Moreno, Mario Motta, Holger Haas, and Ali Javadi-Abhari.
- Contributors are affiliated with IBM Quantum and IBM Research Cambridge, alongside the University of Colorado’s Department of Chemistry.
- Several authors are affiliated with RIKEN’s Center for Computational Science and Center for Quantum Computing.
1 Convergence properties of the components of sample-based quantum diagonalization
The analysis links sample-based diagonalization accuracy and scalability to ground-state wavefunction concentration, while identifying noise and diagnostic uncertainty as important boundaries.
- Ground-state configurations are ordered by probability, with concentration characterized through the weight and minimum probability of the leading m configurations.The analysis uses α_m for cumulative probability and β_m for the individual lower bound.
- The probability of missing one of the first m configurations decreases exponentially with the number of ground-state samples.A sufficient sample count is derived by bounding the failure probability through the least probable configuration among the leading m.
- Polynomially many samples suffice for small energy error when configuration probabilities decay exponentially or algebraically with rank.For exponential decay, the sufficient sample count scales polynomially with qubit number; algebraic decay likewise yields polynomial scaling under the stated conditions.
- Partitioning samples into K batches replaces one large N_s × N_s diagonalization with parallel diagonalizations of K smaller d × d matrices.The batch construction preserves the stated polynomial scaling while reducing the matrix dimensions handled by each diagonalization.
- Concentration diagnostics do not definitively establish whether a ground-state wavefunction is concentrated.The limitation applies to tests based on convergence with sampled-basis dimension, batch variance, and occupations near hypercube corners.
- SQD performs better for more concentrated Hubbard-model ground states, while d = 10^3 configurations are insufficient for the metallic U = 1 case.For U = 16, SQD amplitudes agree better with exact amplitudes; for U = 1, the exact state contains many more relevant configurations than the trial space allows.
- Without quantum error correction, ideal-state sampling requires a number of shots that grows exponentially with circuit depth and noise rate.The analysis states that the maximum operation count is inversely proportional to the noise rate, while configuration recovery can also use some noisy components.
- The recovery probability approaches one as the quantum-signal fraction α approaches one and exceeds direct uniform-distribution sampling for configurations near the reference.Hardware fractions are reported as orders of magnitude larger than the analytical lower bound, which is described as loose in practice.
2 Optimization of the circuit parameters
The section develops circuit-parameter and orbital-rotation optimization for SQD, using subspace energies and Monte Carlo estimators to manage combinatorially large configuration spaces. It also identifies support mismatch and controlled-circuit requirements as practical limitations, while quantum-experiment optimization remains future work.
- Circuit-parameter optimization: SQD optimizes circuit parameters using a subspace-energy cost function rather than standard VQE objectives.The HPC quantum estimator diagonalizes multiple subspaces formed from batches of sampled configurations.
- Circuit-parameter optimization: Monte Carlo estimation replaces the double-combinatorial summation over configuration batches, using K much smaller than K_max.K_max is itself double-combinatorial in the numbers of spin-orbitals and electrons, making exhaustive evaluation inefficient.
- Circuit-parameter optimization: The sampled configurations are drawn according to the circuit state's probability distribution, with each configuration probability given by |⟨x|Ψ⟩|2 in the noiseless case.The same sampling framework can also be combined with configuration recovery.
- Limitations: Gradient estimates can be biased when the support of the sampling distribution differs from the support required by the estimator.The reciprocal probability expression becomes ill-defined for configurations outside the sampling distribution's support.
- Limitations: Controlled versions of variational circuits required by Hadamard-test gradient evaluations can exceed the circuit depths reachable on current quantum processors.Gradient-free methods such as COBYLA or simulated annealing are therefore used in the numerical study.
- Limitations: Optimization of circuit parameters during quantum experiments is deferred to future studies, including analysis of noise effects in gradient-free optimizers.The experiments instead show that fixed parameters can already provide good estimator accuracy.
- Orbital optimization: Full orbital optimization alternates gradient-descent updates of circuit and orbital parameters with eigenstate solves that update wavefunction amplitudes.Orbital rotations transform the Hamiltonian's one- and two-body integrals, and their gradients use reduced density matrices and Monte Carlo estimation.
3 Additional information about experimental details
The supplementary experiments describe compiled LUCJ circuits, their hardware and sampling setup, and numerical comparisons of circuit approximations for N2 dissociation. The results emphasize the role of density-density interactions and report hardware runs using substantial sampling and error mitigation.
- Experimental setup: The supplementary experiments compare compiled LUCJ circuits with a classically efficient reduction and document quantum and classical hardware resources.Reported implementation details include processor mappings, circuit depth, gate counts, measurement outcomes, and classical computing resources.
- Compiled circuits: Displayed circuits separate orbital rotations and density-density interactions with barriers, which are removed during final compilation to reduce circuit depth.Circuits are compiled into single-qubit and two-qubit CNOT gates.
- Compiled circuits: The first orbital-rotation portion prepares the restricted Hartree-Fock Slater determinant directly, reducing circuit size without information loss.Subsequent orbital rotations do not admit the same simplification.
- Numerical circuit study: The numerical comparison uses LUCJ(L=1), LUCJ(L=4), and a determinant circuit, with COBYLA optimization and K=1 configuration batch at each optimization step.The study evaluates N2 dissociation in the 6-31G basis without noise or configuration recovery.
- Numerical circuit study: Density-density interactions play a crucial role in generating relevant electronic configurations for SQD during N2 dissociation.SQD has the largest energy error for the determinant circuit for most bond lengths, while adding LUCJ two-body terms reduces the error.
- Sampling and mitigation: 2.4576 · 10^6 measurement outcomes are collected for each [2Fe-2S] and [4Fe-4S] ground-state experiment.The N2 dissociation experiments collect 1 · 10^5 and 9.8304 · 10^4 outcomes per point at 6-31G and cc-pVDZ, respectively.
- Sampling and mitigation: Readout-error mitigation and dynamical decoupling are used to mitigate measurement and gate errors, respectively.The two probabilities for obtaining configurations with the correct particle number are statistically distinguishable in all experiments.
4 Additional experimental results
Additional experiments show that SQD improves electronic-structure estimates through configuration recovery, orbital optimization, and increased classical resources, while quantum samples retain useful signal beyond uniform noise.
- N2 experiments: The ground-state energy error for N2 remains below 10 mEh across the dissociation curve.The estimator agrees well with exact results throughout the curve.
- N2 experiments: Larger wavefunction amplitudes agree exceptionally with exact N2 results, while tail accuracy deteriorates at larger bond lengths.The deterioration coincides with stronger static correlation at larger separations.
- Scaling and runtime: Increasing the determinant count d lowers energies and suppresses unphysical dissociation-curve oscillations.The same improvement with computational resources appears in energy-variance analyses.
- Scaling and runtime: Distributed classical resources reduce diagonalization runtime as node count increases, although speedup eventually saturates from inter-node communication overhead.The scaling analysis identifies node counts needed to maintain a fixed runtime for different subspace dimensions.
- N2 experiments: Orbital optimization has little effect near equilibrium but yields noticeably lower dissociation energies and reduces oscillations for smaller d.It improves the single-particle basis used by the estimator, while circuit-parameter optimization remains future work.
- Quantum signal in the experiments: Quantum measurement outcomes contain signal distinguishable from white noise, and configuration recovery amplifies it into more accurate ground-state representations.Without recovery, energies remain comparable to RHF; with recovery, the dissociation curve becomes qualitatively correct as d increases.