Sample-Based Quantum Diagonalization for Variable Molecular Geometries

AI summary

Overview

Research area: Quantum computing for quantum chemistry — specifically hybrid quantum-classical subspace methods (sample-based quantum diagonalization, SQD) applied to molecular geometry optimization and reaction pathways.

Technical level: Advanced. The paper assumes familiarity with second quantization, active spaces, Hartree-Fock molecular orbitals, coupled cluster (CCSD/CCSD(T)), tensor-network methods (DMRG), and variational circuit ansätze.

Scope (one sentence): The paper introduces VGSQD, an extension of SQD that reuses quantum-sampled bitstrings from a reference molecular geometry at neighboring geometries, and demonstrates it on IBM hardware for geometry optimization of Be₂ and H₂O and for the minimum energy path of the H₂ + H → H + H₂ reaction.

What This Paper Is About

Subspace methods like SQD can already compute ground and excited state energies of classically difficult molecules, but practical workflows such as geometry relaxation and reaction-path search require a separate round of quantum circuit execution at every molecular geometry, which drives up quantum cost. This paper asks whether the bitstrings sampled at one geometry can be reused, after a careful rotation of the molecular orbital basis, at nearby geometries. The goal is to make structural relaxation and reaction-barrier calculations feasible on near-term quantum hardware without re-sampling at every step.

Key Contributions

1. The VGSQD algorithm. A six-step procedure that (i) samples bitstrings from a truncated LUCJ circuit for a reference geometry A, (ii) diagonalizes the Hamiltonian in the resulting subspace, (iii) updates nuclear coordinates to a new geometry B, (iv) checks maximal orbital misalignment δ_align against a threshold ε_align (typically 0.03), (v) checks the nuclear displacement against ε_pos (typically 0.02 Å), and (vi) projects and diagonalizes the rotated Hamiltonian in the reused subspace — steps (v) and (vi) require no quantum circuit execution.

2. A molecular-orbital alignment scheme. Orbital identities are not unique, since the Hartree-Fock reference is invariant under unitary rotations within the occupied and virtual spaces, so bitstrings cannot be reused directly. The authors rotate the occupied and virtual blocks of geometry B to minimize the summed squared difference from the reference orbitals; the problem reduces to maximizing Tr(S_MO^(AB) U) and is solved as U = U_R U_Lᵀ from a singular value decomposition.

3. A second, many-body reuse criterion. Because orbital alignment can miss changes in the many-body ground state, the authors also resample on the QPU when the CI infidelity δ_CI = 1 − |c(A)†c(B̃)|² exceeds ε_CI = 0.01.

4. Hardware demonstrations and benchmarking. Geometry optimization of Be₂ (cc-pVDZ, 62 qubits; cc-pVTZ, 124 qubits) and H₂O (cc-pVDZ, 53 qubits; cc-pVTZ, 122 qubits), plus a CI-NEB minimum energy path for the hydrogen exchange reaction (85-qubit circuits), all run on the IBM Heron r2 processor ibm-quebec with classical postprocessing on the Harfang cluster, and compared against CCSD(T), DMRG, and FCI.

Main Findings

• SQD validation on Be₂. SQD ground-state energies converged within three iterations for both basis sets, reaching −29.2356 Ha (cc-pVDZ, at a bond length of 2.564 Å) and −29.2486 Ha (cc-pVTZ, at 2.385 Å). The paper reports SQD as on par with CCSD for the larger basis set and outperforming CCSD(T)/DMRG for the smaller one; the cc-pVTZ result is slightly below CCSD(T) (−29.2540 Ha) and DMRG (−29.2546 Ha at M = 1000), which the authors attribute to deeper circuits and memory limits on the subspace size.

• Be₂ geometry optimization beats CCSD(T) on bond length. Starting from the experimental bond length of 2.46 Å, the optimization converged at iteration 5 with VGSQD giving 2.555 Å (cc-pVDZ) versus 2.611 Å for CCSD(T), both compared to the experimental 2.460 Å. With cc-pVTZ, VGSQD gives 2.383 Å and −29.2485 Ha versus CCSD(T) at 2.340 Å and −29.2541 Ha. Hartree-Fock and CCSD both failed to bind the dimer, yielding unbounded monomers.

• Only one QPU resampling was needed for the Be₂ relaxation. The δ_align criterion, not δ_CI, triggered the single resampling between iterations 3 and 4. δ_align grew much faster than δ_CI, which stayed below 10⁻³ throughout, indicating that the rotated orbitals made the reference bitstrings reusable.

• H₂O converges to near-experimental geometry with multiple degrees of freedom. VGSQD gives 0.966 Å / 102.16° (cc-pVDZ, −76.2417 Ha) and 0.956 Å / 104.07° (cc-pVTZ, −76.3021 Ha) against experimental values of 0.958 Å and 104.48°. The Results section reports convergence within 0.002 Å and 0.41° for cc-pVTZ (the abstract and introduction describe this as within 0.5°).

• CCSD(T) remains better on H₂O energies. CCSD(T) gives −76.2434 Ha (cc-pVDZ) and −76.3458 Ha (cc-pVTZ), which the authors attribute to H₂O's correlation being mainly dynamical and thus well captured by CCSD(T).

• Reaction barrier agrees closely with FCI. For H₂ + H → H + H₂ with CI-NEB and cc-pVTZ, VGSQD finds a collinear H₃ saddle point with a barrier of 9.75 kcal/mol and a saddle-point H–H distance of 0.929 Å, versus FCI values of 9.94 kcal/mol and 0.930 Å — within 0.2 kcal/mol and 0.001 Å. No resampling was triggered during the path optimization.

• Circuit and hardware scale. The largest circuits used 124 qubits with depth 2033 (2-qubit depth 477, 2-qubit gate count 21,370). The H₃ image circuit used 85 qubits at depth 2979 (2-qubit depth 637, 19,742 two-qubit gates). Each circuit used 256,000 shots, and median two-qubit gate errors across the reported runs ranged from 0.160% to 0.205%.

Methodology in Plain English

The workflow keeps the heavy lifting classical and asks the quantum processor only for a good starting set of electron configurations.

1. Set up the electronic problem classically. For each geometry, a Hartree-Fock calculation (PySCF) produces molecular orbitals; the electronic Hamiltonian is written in second quantization with one- and two-electron integrals. The authors note that no molecular orbitals were frozen anywhere in this study.

2. Sample configurations on the quantum computer. A truncated local unitary cluster Jastrow (LUCJ) circuit, built with the ffsim library and parameterized from CCSD amplitudes without further optimization, is run on ibm-quebec. Bitstrings are read out in the Jordan-Wigner mapping, with the first N qubits holding spin-β occupations and the last N holding spin-α, so each bitstring is directly a Slater determinant.

3. Recover and diagonalize classically. Configuration recovery and subspace diagonalization (qiskit_addon_sqd, 5 batches of up to 9000 samples per iteration, final subspace sizes up to 9.6E7) produce the ground state; diagonalization runs with DICE on 378 vCPUs across 14 compute nodes with 230 GB per node.

4. Reuse instead of resample. When the geometry moves slightly, the new geometry's orbitals are rotated to line up with the reference orbitals (via SVD), the Hamiltonian is re-expressed in that aligned basis, and the stored determinants are reused for a new subspace diagonalization. Quantum execution is invoked only when the alignment or CI-overlap criteria are violated.

5. Get forces without new quantum work. Nuclear gradients are computed semi-numerically by central finite differences of the energy with the one- and two-electron reduced density matrices held fixed, solving Hartree-Fock and realigning orbitals at each displaced geometry.

6. Drive the workflow. Geometry optimization uses a trust-region BFGS, converged when all force components fall below 0.1 mHa/Bohr and per-iteration energy reduction is below 1 µHa. Reaction paths use climbing-image nudged elastic band with a 1 mHa/Bohr force threshold. Error suppression on the hardware was chosen by running a mirrored circuit and comparing all combinations of measurement twirling, Pauli twirling, and dynamical decoupling; only measurement twirling was used.

Why This Matters

Subspace methods had been shown to deliver accurate energies, but not to support the iterative, many-geometry loops that real chemistry workflows depend on. VGSQD turns the quantum processor into an occasional supplier of configurations rather than a per-step requirement, which changes the economics of running these algorithms on today's hardware.

Real-world applications:

• Drug discovery — conformer searches and structural relaxation of candidate molecules, cited in the paper as a motivating application area.
• Catalysis — mapping minimum energy paths and barriers for reactions such as the hydrogen exchange studied here.
• Reaction mechanism analysis — locating saddle points and transition-state geometries, not just equilibrium structures.
• Strongly correlated systems — the Be₂ dimer, whose weak bond defeats Hartree-Fock and CCSD, shows where quantum sampling adds value over standard classical workhorses.

Industry relevance: The work is a collaboration between PINQ2 (Sherbrooke, Canada) and IBM Quantum, and it runs on a commercially available 156-qubit Heron r2 device with a documented quantum-centric supercomputing split between QPU and HPC cluster — a template for production-style hybrid workflows.

Future Directions

• Tracking excited states. The authors caution that the alignment and CI criteria may fail for molecules with near-degenerate ground states and suggest modifying VGSQD to track multiple states, including low-lying excited states, for example with the ExtSQD algorithm.
• Reusing circuits across NEB images. The reaction-path study resampled every image separately; the authors note that QPU cost could be reduced by sharing circuits across images or across iterations when the δ_CI and δ_align thresholds are not exceeded.
• Interchangeable parts. VGSQD is stated to work with ansätze other than LUCJ, and to be combinable with sample-based Krylov quantum diagonalization, TrimSQD, and embedding methods.
• Scaling past current limits. The cc-pVTZ Be₂ result was limited by circuit depth and by the memory available for classical postprocessing, which capped subspace size; extending to larger molecules and more degrees of freedom is the obvious next test.

Target Audience

Researchers and practitioners in quantum chemistry and quantum algorithms — particularly those working on hybrid subspace methods, quantum-centric supercomputing workflows, and near-term applications of gate-based quantum computers. It is also useful for computational chemists who want a concrete comparison of quantum sampling against CCSD(T) and DMRG on bond lengths, angles, and barriers, and for industry teams evaluating where quantum hardware fits into existing HPC chemistry pipelines. Beginners will find the algorithmic and hardware details demanding without prior exposure to second quantization and subspace diagonalization.

Authors’ abstract

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.