Explicitly correlated wavefunctions as classical seeds and property references for sample-based quantum diagonalization beyond the Born–Oppenheimer approximation
This work evaluates how well hybrid classical–quantum simulation methods capture not only the energy of small atomic and molecular systems, but also their other properties: contact densities, molecular structure, and state identity — the spin sector a computed state actually lives in. It employs explicitly correlated Gaussian (ECG) wavefunctions both as references for evaluating these properties and as classical seeds for sample-based quantum diagonalization (SQD), on the simulator and on IBM quantum hardware. Our testbeds — positronium hydride, the muonic heliumlike ions, the HD+ molecular ion, and the positronium anion and molecule — realize three typical failure regimes: boxes that lack the property no matter how configurations are selected, boxes that hold it exactly with no selection needed, and the middle regime where the selection itself decides. In every regime the property error dwarfs the energy error: a subspace can converge the energy to microhartrees while misstating an annihilation rate, a bond length, or the spin by whole percent. Property fidelity, not energy, is the discriminating test of a multicomponent simulation.
This work evaluates how well hybrid classical–quantum simulation methods capture not only the energy of small atomic and molecular systems, but also their other properties: contact densities, molecular structure, and state identity — the spin sector a computed state actually lives in. It employs explicitly correlated Gaussian (ECG) wavefunctions both as references for evaluating these properties and as classical seeds for sample-based quantum diagonalization (SQD), on the simulator and on IBM quantum hardware. Our testbeds — positronium hydride, the muonic heliumlike ions, the HD+ molecular ion, and the positronium anion and molecule — realize three typical failure regimes: boxes that lack the property no matter how configurations are selected, boxes that hold it exactly with no selection needed, and the middle regime where the selection itself decides. In every regime the property error dwarfs the energy error: a subspace can converge the energy to microhartrees while misstating an annihilation rate, a bond length, or the spin by whole percent. Property fidelity, not energy, is the discriminating test of a multicomponent simulation.
The accurate treatment of electron correlation in extended molecular systems remains computationally challenging using classical electronic structure methods. Hybrid quantum-classical algorithms offer a potential route to overcome these limitations; however, their practical deployment on existing quantum computers requires strategies that both reduce problem size and mitigate hardware noise. In this work, we investigate ground-state energy calculations of ligand-like molecules using Sample-based Quantum Diagonalization (SQD) within the Density Matrix Embedding Theory (DMET) framework, focusing on low-symmetry systems with diverse bonding motifs that exhibit subsystem-dependent variations in fragment-environment entanglement. These entanglement-based variations directly influence bath orbital construction, impurity sizes, and the structure of the embedded Hamiltonians, posing nontrivial challenges for both embedding and quantum sampling. By combining DMET fragmentation with SQD-based construction of reduced configuration spaces through quantum sampling and iterative configuration recovery, we perform quantum simulations on IBM's Eagle R3 (IBM Sherbrooke) and IBM's Heron R3 (IBM Boston) superconducting quantum hardware thereby, showing that the entanglement structure across embedding subsystems plays a central role in determining the efficiency and accuracy of the simulations. Despite these complexities, we show that the DMET-SQD framework yields ground-state energies in strong agreement with DMET-FCI benchmarks, achieving chemical accuracy (1 kcal/mol) across all systems studied. These results demonstrate that SQD-based quantum simulations can be robustly extended to low-symmetry, chemically realistic, industry relevant molecules, and highlight the importance of entanglement-aware embedding strategies for scalable quantum electronic structure calculations.
We present a large-scale experimental study of quantum-computing-based molecular simulation carried out on IQM's Sirius 24-qubit superconducting processor, utilizing up to 16 operational qubits. The work employs Sample-based Quantum Diagonalization (SQD) together with the Local Unitary Cluster Jastrow (LUCJ) ansatz to estimate ground-state energies for a set of benchmark molecules, including H2, LiH, BeH2, H2O, and NH3. In addition, we introduce a Linear-CNOT variant of the Unitary Coupled-Cluster Singles and Doubles (LCNot-UCCSD) ansatz within the SQD workflow, trading higher circuit depth for reduced classical preprocessing. A comparison between these ansätze is provided, clarifying their respective strengths, limitations, and suitability for near-term quantum hardware. We further explore potential energy landscapes through 1D scans for H2 and HeH+ using both STO-3G and 6-31G basis sets, and for LiH and BeH2 in STO-3G. Extending beyond this, we demonstrate the experimental construction of a full 2D potential energy surface for the water molecule on quantum hardware, mapped over a 32 × 32 grid in bond length and bond angle. To move beyond small benchmark systems, we combine SQD(LUCJ) with Density Matrix Embedding Theory (DMET) to compute active-space energies for a set of ligand-like molecules, as well as the pharmacologically relevant amantadine system. Across all studies, the majority of quantum-computed energies agree with reference FCI results, as well as with DMET-CASCI energies for embedded systems, to within chemical accuracy for the chosen basis sets. These results demonstrate the reliability of sample-based diagonalization approaches and underscore the potential of hybrid embedding strategies for extending quantum simulations to increasingly complex molecular systems, while also highlighting their practicality on current IQM quantum hardware.
The energy is the last thing to fail. Sample-based quantum diagonalization (SQD), the method behind the largest chemistry calculations on quantum processors, grades its results by the energy, the quantity least sensitive to errors in the state. This work asks what else the state gets right, on archived hardware data from two sources: the author's published N₂ run on IBM hardware, and IBM's archive for the 77-qubit iron–sulfur demonstration of 2025, the largest SQD calculation whose raw samples are public. Three density properties are followed alongside the energy against converged references: two stop improving while the energy still does; the third, set by the diffuse tail, oscillates while the energy descends smoothly. Four controls show whose work the energy is: random bitstrings seeded with five valid strings lag the hardware data by one iteration and match them at equal subspace size, a coupled-cluster prior matches or beats them there, and the device beats only the published control, uniform in-sector strings, whose loop has nothing to repair. The ordering repeats on the iron–sulfur cluster, where the spin is read alongside the energy and a penalty confirms the classical seed's lead as a singlet energy. Exact samples of the one-layer LUCJ circuit stall on a few hundred configurations: the noise feeds the repair. The recommendation is cheap: report a density property and the spin alongside the energy, at fixed subspace size, against a converged reference. Looking forward, exact samples of a time-evolved state broaden the support ten- to twentyfold over the LUCJ circuit, take the static loop to half a millihartree where LUCJ stalls, and, repaired from a noise channel, lead the hardware shots at equal subspace size at equilibrium and at the stretched bond; generative circuit compression would bring such a sampler inside the Nighthawk gate budget. v2 (2026-09-11): Table 10 gains the one-layer LUCJ circuit with optimized parameters, all-to-all and hardware pattern, as the rung between the coupled-cluster-parametrized circuit and the time-evolved state; §3.6.3 and §3.7.6 state that the configuration prior is coupled cluster's output, not the circuit's input. Changes in v3 (2026-09-13): the optimized-circuit rows of Table 10 restated from four converged optimizations (hardware pattern reproducible to chemical accuracy; three all-to-all optima, a perturbed start returning to one of them); a new §3.6.4 measuring the classical solver's envelope to fifty million determinants on a laptop (Table 12). Changes in v4: Table 10 gains the full-CI weight missing from each sample's string product space, 1 − W(U×U), and two full-CI yardstick rows; §3.6.3 reads it: the missing weight orders the static-loop error where configuration and string counts do not, and a millihartree needs less than about 3 × 10⁻⁴ missing. Changes in v5: a new §3.6.5, "Which root is the state", summarizes in one paragraph a measurement reported in full in a companion paper in preparation: at the stretched geometries the lowest root of a subspace closed under spin inversion is a spin mixture, which a DMRG-X-type overlap rule cannot repair; the abstract and §4 follow it. Table 10's hardware pattern read from the CCSD amplitudes is given as a range over ten builds, because round-off-level differences between coupled-cluster runs change that circuit; three optimizations "from the same starting state".
Quantum and hybrid quantum-classical algorithms have the potential to improve the accuracy and scale of quantum chemistry and impact a broad range of applications such as drug discovery and catalysis. In particular, subspace methods such as sample-based quantum diagonalization (SQD) can already tackle classically challenging systems. However, practical algorithms that go beyond the computation of electronic ground and excited states are still scarce. Here, we introduce the variable-geometry sample-based quantum diagonalization (VGSQD) algorithm, which extends SQD to variable molecular geometries by reusing quantum samples from a reference geometry at displaced ones, thus significantly reducing quantum computational cost. Using circuits of up to 124 qubits on ibm-quebec, an IBM Heron r2 quantum processor, we determine the equilibrium geometry and energy of the weakly bound, strongly correlated beryllium dimer, as well as the water molecule. Finally, we extend VGSQD to chemical reactions and compute the minimum energy path and energy barrier of a hydrogen exchange reaction. The results are comparable with state-of-the-art classical methods such as CCSD(T) and DMRG. This work broadens the scope of quantum algorithms for chemistry, from energies to geometry optimization and reaction pathways, both essential for practical applications.
We provide a systematic evaluation of the sample-based quantum diagonalization (SQD) method for electronic structure based on the W4-11 thermochemistry dataset, comprising 124 total atomization, 83 bond dissociation, 20 isomerization, 505 heavy-atom transfer, and 13 nucleophilic substitution processes, covering diverse bonding situations and reaction mechanisms. This is the largest study assessing the accuracy and precision of a quantum-hybrid algorithm on a digital quantum device across a variety of molecular systems and chemical reactions, using 16.85 hours on the superconducting quantum processor ibm_rensselaer and 724.22 node hours on the supercomputer AiMOS. To ensure a fair comparison, our study employs commensurate resource allocation for both classical and quantum simulations. Although SQD exhibits large statistical deviations from ground-state reference energies, energy extrapolations yield CCSD-level accuracy. While bond-breaking reactions show a systematic improvement as computational resources increase, nucleophilic substitution or heavy atom transfer reactions do not. The limitations quantified in this manuscript indicate opportunities for improvement in SQD-based algorithms. This work provides a benchmark and community resource for exploring new quantum algorithms and devices, supported by an online benchmark challenge and an open-source Python library for direct comparison.
Ground-State Energy Estimation of HeH+, ArH+, and H2O via Sample-Based Quantum Diagonalization
Accurate ground-state energies are essential for understanding molecular structure, chemical bonding, and reaction energetics in quantum chemistry. In this work, we investigate the ground-state properties of the molecular systems HeH+, ArH+, and H2O using Sample-Based Quantum Diagonalization (SQD), a hybrid quantum-classical framework designed for near-term quantum devices. Unlike variational approaches such as VQE, which require deep parameterized circuits and repeated expectation-value measurements, SQD reconstructs a low-energy determinant subspace directly from measured bitstrings. For the present calculations, bitstrings were generated on IBM quantum hardware using shallow local unitary cluster Jastrow (LUCJ) circuits whose parameters were constructed from the t1 and t2 amplitudes of coupled-cluster singles and doubles (CCSD) calculations based on restricted Hartree--Fock (RHF) references. From these samples, we compute ground-state potential-energy curves of HeH+, ArH+, and H2O with the 6-31G and cc-pVDZ basis sets. For all three systems, the SQD results obtained with the cc-pVDZ basis closely follow the corresponding same-basis CCSD energies and reproduce the equilibrium-region trends of the potential-energy curves. HeH+ and ArH+ were chosen as simple yet astrophysically important molecular-ion benchmarks, while H2O was included as a representative polyatomic molecule to assess the applicability of SQD beyond diatomic ionic systems. At the adopted equilibrium geometries, the deviations from the same-active-space CASCI references are 0.00, 2.51, and 6.34 mHa for HeH+, ArH+, and H2O, respectively. These results demonstrate the feasibility of hardware-assisted SQD for the present benchmark systems and motivate further studies of its accuracy and computational scaling for larger molecular active spaces.