Quantum-Selected Configuration Interaction: classical diagonalization of Hamiltonians in subspaces selected by quantum computers

Keita Kanno; Masaya Kohda; Ryosuke Imai; Sho Koh; Kosuke Mitarai; Wataru Mizukami; Yuya O. Nakagawa

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Quantum-Selected Configuration Interaction: classical diagonalization of Hamiltonians in subspaces selected by quantum computers

Keita Kanno, Masaya Kohda, Ryosuke Imai, Sho Koh, Kosuke Mitarai, Wataru Mizukami, Yuya O. Nakagawa

arXiv:2302.11320v1Feb 22, 2023quant-ph

TL;DR

QSCI addresses the challenge of extracting reliable many-electron energies from noisy quantum devices. It samples an approximate state to select important configurations and classically diagonalizes the resulting Hamiltonian subspace. The method is noise resilient, preserves a variational energy bound, and was numerically verified and demonstrated on an 8-qubit molecular Hamiltonian.

Problem
Noisy quantum devices make direct energy estimation vulnerable to statistical and physical errors, while full configuration-interaction calculations face combinatorial growth in Fock-space dimension.
Method
QSCI samples an approximate eigenstate in the computational basis to select important electron configurations, then classically diagonalizes the Hamiltonian in the selected subspace.
Results
QSCI is robust against noise because quantum computation only defines the subspace, and its ground-state energy remains a variational upper bound even with statistical and physical errors.
Takeaways & Limitations
QSCI can refine noisy VQE results, estimate observables without additional quantum cost, and potentially address challenging molecules with quantum devices of several tens of qubits plus classical diagonalization.
Takeaways & Limitations
QSCI performance depends highly on the quality of the input state.

Abstract

from arXiv · show

We propose quantum-selected configuration interaction (QSCI), a class of hybrid quantum-classical algorithms for calculating the ground- and excited-state energies of many-electron Hamiltonians on noisy quantum devices. Suppose that an approximate ground state can be prepared on a quantum computer either by variational quantum eigensolver or by some other method. Then, by sampling the state in the computational basis, which is hard for classical computation in general, one can identify the electron configurations that are important for reproducing the ground state. The Hamiltonian in the subspace spanned by those important configurations is diagonalized on classical computers to output the ground-state energy and the corresponding eigenvector. The excited-state energies can be obtained similarly. The result is robust against statistical and physical errors because the noisy quantum devices are used only to define the subspace, and the resulting ground-state energy strictly satisfies the variational principle even in the presence of such errors. The expectation values of various other operators can also be estimated for obtained eigenstates with no additional quantum cost, since the explicit eigenvectors in the subspaces are known. We verified our proposal by numerical simulations, and demonstrated it on a quantum device for an 8-qubit molecular Hamiltonian. The proposed algorithms are potentially feasible to tackle some challenging molecules by exploiting quantum devices with several tens of qubits, assisted by high-performance classical computing resources for diagonalization.

I. INTRODUCTION

QSCI addresses noisy quantum-device limitations by sampling important electron configurations from an approximate state and classically diagonalizing the Hamiltonian in their subspace. This preserves a variational upper bound under statistical and physical errors while extending to eigenstate estimation and related classical-computational benefits.

QSCI overview: QSCI samples an approximate quantum state to select important electron configurations, then classically diagonalizes the Hamiltonian in their subspace to obtain eigenstates and energies.The selected configurations define a truncated Hamiltonian whose smallest eigenvalue and eigenvector approximate the ground state; excited states can be obtained similarly.
Scope: The approach targets noisy quantum chemistry calculations, while its stated scope also includes other many-body Hamiltonians such as condensed-matter electron and spin problems.The paper focuses its applications on quantum chemistry and notes broader applicability in principle.
Noise resilience: The resulting ground-state energy remains a rigorous upper bound on the exact energy despite statistical fluctuation and physical noise.Noise can degrade the selected subspace, but exact classical diagonalization preserves the variational inequality within that subspace.
Additional outputs: QSCI can estimate many observables without additional quantum cost because the obtained eigenstates have explicit classical representations.This provides an eigenstate-tomography use beyond energies.
Selected-configuration advantage: Sampling can identify configurations missed by fixed-basis conventional methods, while requiring at most one diagonalization per eigenstate rather than iterative adaptive searches.The authors report substantially less classical computational time than adaptive methods under this design.
Relation to prior methods: QSCI avoids quantum estimation of subspace Hamiltonian matrix elements, unlike quantum subspace methods whose matrix elements remain exposed to quantum errors.All QSCI matrix elements are computed classically, trading more complex basis states for robustness against statistical and physical errors.

C. QSCI for multiple energy eigenstates

QSCI extends its ground-state procedure to multiple low-lying eigenstates, using either one common subspace or distinct subspaces constructed for each state.

C. QSCI for multiple energy eigenstates: QSCI offers single and sequential diagonalization schemes for finding multiple low-lying eigenstates.The ground-state algorithm is a special case of both extensions.
C. QSCI for multiple energy eigenstates: The single scheme simultaneously diagonalizes one common subspace, whereas the sequential scheme diagonalizes multiple state-specific subspaces in sequence.
C. QSCI for multiple energy eigenstates: A phase-flip error at the circuit end does not affect computational-basis probabilities or the resulting sampling outcome.

1. Single diagonalization scheme

The single diagonalization scheme combines important configurations from multiple input states into one common subspace, then obtains several eigenstates through one classical diagonalization.

1. Single diagonalization scheme: Multiple input states are sampled to form configuration sets whose union defines a common subspace for the desired low-lying eigenstates.Each input state may use its own retained-configuration count Ri, while the common subspace contains R elements and requires R ≥ Ns.
1. Single diagonalization scheme: The allocation of Ri controls the trade-off between prioritizing one eigenstate and treating all input states more evenly.
1. Single diagonalization scheme: The common subspace is classically diagonalized once to produce the Ns lowest eigenvalues and corresponding eigenvectors.The scheme is illustrated alongside the sequential alternative in Fig. 2.
1. Single diagonalization scheme: In the ideal setting, cycling through input states ensures that at least R′ frequent bit strings from each input are included in the common subspace.
1. Single diagonalization scheme: For single diagonalization, the variational inequality holds for every obtained eigenstate by Cauchy’s interlace theorem.QSE and MCVQE instead require measured subspace-Hamiltonian matrix elements, whereas QSCI calculates them classically.

2. Sequential diagonalization scheme

Sequential diagonalization finds excited states one at a time by sampling a state-specific subspace and adding penalties that enforce orthogonality to previously obtained states.

2. Sequential diagonalization scheme: Sequential diagonalization constructs distinct subspaces and processes the ground state and subsequent excited states through successive Hamiltonian diagonalizations.Its structure is similar to variational quantum deflation.
2. Sequential diagonalization scheme: For the k-th excited state, the method samples an input state, restricts the solution to its important configurations, and enforces orthogonality to earlier output states.
2. Sequential diagonalization scheme: An effective Hamiltonian adds overlap penalties with coefficients βi before the smallest eigenvalue is selected as the k-th-state estimate.The coefficients must be sufficiently large; one stated sufficient condition is βi > E_exact^(k).
2. Sequential diagonalization scheme: The procedure does not require extra circuits to calculate overlap terms, because the effective-Hamiltonian matrix elements are computed classically.
2. Sequential diagonalization scheme: Sequential diagonalization lacks a guaranteed variational inequality because its effective Hamiltonian uses approximate previously obtained eigenstates.

III. BENCHMARK OF QSCI WITH NOISELESS SIMULATIONS

Noiseless simulations test QSCI for ground and excited states, scalability, and statistical effects using molecular Hamiltonians; the ground-state results show substantial refinement of VQE.

III. BENCHMARK OF QSCI WITH NOISELESS SIMULATIONS: The simulations use noiseless VQE or VQD input preparation and molecular Hamiltonians built with the Born–Oppenheimer approximation, Hartree–Fock orbitals, STO-3G unless otherwise stated, and Jordan–Wigner mapping.
A. QSCI for ground state: During intermediate VQE iterations 70–200, QSCI with R ≳16 reaches chemical accuracy even when VQE itself does not.This indicates that partially optimized VQE states can already serve as useful QSCI inputs.
B. QSCI for excited states: Excited-state tests compare sequential and single diagonalization for H2O states T1 and S1 under Sz = 0, using VQD-prepared lower-state inputs and R = 16 settings.
B. QSCI for excited states: Sequential diagonalization performs best except during early iterations with very poor input states and outperforms VQD at moderate Ri = 16.Single diagonalization is expected to improve with larger R, but sequential diagonalization does not guarantee the excited-state variational inequality.

C. Scaling of computational costs

QSCI’s costs trade quantum sampling against classical subspace selection and diagonalization. Across studied systems, it generally reduces sampling cost, while classical feasibility depends strongly on molecular structure and target accuracy.

Classical scalability: For Cr2, the selected-configuration count remains manageable for 0.001 Hartree accuracy beyond 50 qubits, whereas hydrogen chains show clearer exponential growth.The differing trends suggest QSCI is more suited to localized systems with many electrons than spatially extended systems.
Shot estimation: QSCI estimates required shots as 1/|cR|2, where cR is the R-th largest CI coefficient and R is chosen for the target energy error.Sampling this many shots gives an O(1) probability of observing the R-th most significant configuration, making it a rough shot estimator.
Multiple observables: Evaluating additional observables favors QSCI because its classical output eigenvector supports expectation-value calculations without extra quantum computation.Conventional methods require additional measurements, such as Pauli strings introduced by extra operators.
Sampling cost: QSCI generally outperforms QWC in sampling cost within the studied system-size range, although its scaling can be worse for hydrogen chains.The comparison indicates advantages for moderately smaller but classically challenging systems even when considering shot reduction alone.
Sampling accuracy: The 1/|cR|2 estimate tracks the shot count producing an average error near the target tolerance, supporting fair QSCI–QWC cost comparisons.In the H6 sampling simulation, QSCI’s absolute energy errors were much smaller than those from conventional QWC sampling.

IV. BENCHMARK OF QSCI WITH NOISY SIMULATION AND EXPERIMENT

The benchmark evaluates QSCI after VQE state preparation on an 8-qubit H4 Hamiltonian using noisy simulation and an IonQ device. QSCI improves noisy results, and selected runs with 27 configurations achieve chemical accuracy and outperform CISD.

Experimental setup: QSCI uses four VQE states from distinct iterations as inputs, with the selected configurations defining subspaces for multiple R values.The benchmark compares QSCI against conventional QWC energy estimation and CISD under the same H4 setup.
Noise and post-selection: Post-selection is effective in both noisy simulation and experiment, while the quantum-device results agree reasonably with simulation but show greater error.Post-selection fixes the sampled sector using electron number and spin constraints.
Experimental benchmark: On the physical device, some QSCI calculations with R = 27 outperform CISD and achieve chemical accuracy for the 8-qubit H4 system.The comparison uses CISD with 27 Slater determinants and exact diagonalization as the CASCI reference.
Hybrid workflow: Classical diagonalization generates and solves the truncated Hamiltonian using sparse-matrix techniques such as Lanczos or Davidson methods.This classical stage is paired with quantum sampling of the input state and supports explicit eigenstate representations.
Benefits: QSCI can refine VQE results and reduce readout-error effects through post-selection without requiring extra measurement gate operations.For Jordan–Wigner mapping, the readout-error rate is reduced from O(p) to O(p2).

C. Use of QSCI with more general input states

QSCI can use broadly prepared and sampled input states, selecting configurations quantum-mechanically and diagonalizing their Hamiltonian subspace classically. The approach was numerically and experimentally verified and can complement selected-CI methods, while its performance depends strongly on input-state quality.

C. Use of QSCI with more general input states: QSCI selects important configurations by sampling any quantum-preparable input state, then classically diagonalizes the Hamiltonian in the selected subspace.Possible inputs include adiabatic preparation, imaginary-time evolution, boosted or shallow VQE, coupled-cluster circuits, and Clifford-optimized parametrized states.
D. QSCI as selected CI: QSCI’s selected subspace is defined by quantum sampling, distinguishing it from selected-CI methods with fixed or adaptively classically chosen configuration spaces.The sampling distribution is p(x) = |⟨x|ψin⟩|2, and the approach can potentially provide quantum speed-up when the input state is classically hard to sample.
D. QSCI as selected CI: QSCI can be combined with adaptive selected-CI methods such as ASCI, and some hyperparameter settings reportedly outperform ASCI.The proposed hybrid would combine configurations suggested by QSCI with those from another adaptive method.
E. Outlook: For geometry optimization or molecular dynamics, the same selected subspace may be reused across iterations to reduce quantum sampling costs.The paper notes that sampling can be skipped for some iterations while retaining the subspace defined by R configurations.
E. Outlook: QSCI performance depends highly on input-state quality, motivating iterative schemes that improve initially modest input states.The method is also applicable to classically sampled input states and can support observable evaluation for systems with exactly known ground states.
C. Use of QSCI with more general input states: QSCI was numerically verified for molecular ground and excited states and experimentally demonstrated using 8-qubit quantum circuits.The studies also examined refinement of VQE results and eigenstate tomography without additional quantum cost.

Appendix A: The effect of post-selection

Post-selection exploits conserved particle number and spin to reject inconsistent computational-basis outcomes and reduce certain readout errors. Sequential excited-state diagonalization has separate variational caveats when previously obtained states are imperfect.

Appendix A: The effect of post-selection: Particle-number post-selection reduces the bit-flip error rate from approximately O(pN) to n0n1p2 when Np ≪ 1.The procedure keeps measurement strings whose number of 1s matches the known electron count; spin conservation can make it more efficient.
Appendix A: The effect of post-selection: The O(p) to O(p2) error reduction is generally not expected for parity or Bravyi–Kitaev mappings, or for symmetry-reduced Hamiltonians.Those encodings can contain undetectable single-bit flips that preserve the electron count.
Appendix A: The effect of post-selection: If lower eigenstates are known exactly and penalty parameters satisfy βi > Ek − Ei, the effective-Hamiltonian construction recovers the desired variational ordering.A stronger condition based on the qubit-Hamiltonian coefficients can be used without prior spectral knowledge.
Appendix A: The effect of post-selection: Sequential diagonalization can lose a variational upper-bound guarantee for excited states when preceding output states do not perfectly overlap with exact lower-energy eigenstates.The effective Hamiltonian constructed from imperfect lower-state outputs need not be bounded by the target excited-state eigenvalue.

2. An optimal shot allocation for evaluating expectation values of multiple observables in conventional method

The conventional multi-observable strategy allocates measurement shots by operator variances while reusing shared operator terms. Its practical estimates involve grouping choices, adaptive variance estimation, and approximations for specific molecular derivatives.

2. An optimal shot allocation for evaluating expectation values of multiple observables in conventional method: Shared fermionic-basis operator expectations can be reused across observables instead of measuring each observable independently.This is relevant for sets including nuclear gradients and Hessians, which contain many related operator terms.
2. An optimal shot allocation for evaluating expectation values of multiple observables in conventional method: The method allocates shots to minimize total measurement cost subject to a target aggregate variance across multiple observables.A Lagrange-multiplier optimization determines the allocation, with operators decomposed into measurable Pauli-string or commuting groups.
2. An optimal shot allocation for evaluating expectation values of multiple observables in conventional method: Variance estimates may be refined iteratively by using early measurements to adjust later shot allocations.A mildly optimized first allocation provides estimates for recalculating operator variances in subsequent iterations.
2. An optimal shot allocation for evaluating expectation values of multiple observables in conventional method: The analytically available allocation controls summed variance rather than the maximum variance, so it is reasonable but not necessarily optimal for every operator.The total shot count is adjusted so each operator meets the desired worst-case precision.
2. An optimal shot allocation for evaluating expectation values of multiple observables in conventional method: Nuclear Hessians generally require derivatives of the state as well as the Hamiltonian, but QSCI can obtain these derivatives by finite-distance diagonalizations within the same selected subspace.The simulation ignored state-derivative contributions; including them properly is stated to increase QSCI’s advantage.
2. An optimal shot allocation for evaluating expectation values of multiple observables in conventional method: For generic molecules, the QSCI–QWC crossover may occur at larger qubit counts because many hydrogen-chain observables vanish through geometric symmetry.The simulations evaluated O(Natom^2) observables, but symmetry makes that benchmark less representative of generic systems.

Appendix C: Details of numerical simulations and experiments

The numerical simulations and hardware experiment use standard quantum-chemistry Hamiltonians generated from Hartree–Fock/STO-3G data and mapped to qubits with Jordan–Wigner transformation.

Appendix C: Details of numerical simulations and experiments: Electronic Hamiltonians were generated with OpenFermion and PySCF under the Born–Oppenheimer approximation, using Hartree–Fock orbitals and the STO-3G minimal basis unless stated otherwise.The study specifies these tools and basis choices for the examined molecules.
Appendix C: Details of numerical simulations and experiments: The molecular Hamiltonians were mapped to qubit Hamiltonians using the Jordan–Wigner transformation.Geometries came from CCCBDB, Ref., PubChem, or specified non-equilibrium hydrogen-chain configurations.

1. Noiseless simulation for ground state

The noiseless ground-state simulations use a real-valued, symmetry-preserving ansatz, with a depth-10 setup for the H2O calculation.

Noiseless simulation for ground state: The ansatz is defined for n qubits and depth d, providing the circuit structure used in the simulation.
Noiseless simulation for ground state: The H2O ground-state calculation uses a real-valued symmetry-preserving ansatz with depth 10.The circuit starts from the Hartree–Fock state and uses randomly chosen initial parameters for optimization.

2. Noiseless simulations for excited states

The supplementary simulations examine QSCI across molecular systems, observables, and circuit setups, including excited-state preparation and scaling behavior.

Noiseless simulations for excited states: The supplementary setup includes excited-state VQD calculations with particle-number, spin, and orthogonality constraints.The penalty and overlap coefficients used for these constraints are specified in the implementation.
Noiseless simulations for excited states: The Ry ansatz used for noisy simulations and experiments has depth 8, while the illustrated 8-qubit circuit assigns independent parameters to rotational gates.
Noiseless simulations for excited states: Across additional molecules, QSCI shows the same sampling features observed for H6, including small standard deviation and shot-count estimation through 1/|cR|2.
Noiseless simulations for excited states: The standard deviation is nearly constant across hydrogen chains, while absolute error depends strongly on the number of atoms.The relationship between standard deviation and absolute error also varies among three 12-qubit systems.
Noiseless simulations for excited states: QSCI’s expectation values for most nuclear-gradient and Hessian components have accuracy similar to the energy.Some components exhibit larger absolute errors than the energy.

4. Bond length dependence

The supplementary analyses test QSCI across bond lengths, molecular systems, and comparisons with ASCI, including observable-error behavior for hydrogen chains.

4. Bond length dependence: The bond-length analysis evaluates a 14-qubit H2O Hamiltonian across multiple geometries using Hartree–Fock orbitals and an STO-3G basis.Potential-energy curves are shown for reference alongside estimates of R and 1/|cR|2.
4. Bond length dependence: Hydrogen chains exhibit the worst cost scaling among tested molecular classes, whereas Cr2 is among the least expensive systems.
4. Bond length dependence: QSCI observable errors for H4, H6, and H8 are evaluated against exact CASCI values for nuclear gradients and Hessians.
4. Bond length dependence: For ASCI search ratio r = 20, QSCI performs better while ASCI requires less computational cost for searching configurations.The comparison uses idealized sampling from the exact ground state for the 20-qubit H10 molecule.