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Constrained variational quantum eigensolver: Quantum computer search engine in the Fock space

Ilya G. Ryabinkin, Scott N. Genin, Artur F. Izmaylov

arXiv:1806.00461v1physics.chem-phphysics.comp-phquant-ph

TL;DR

Qubit-space diagonalization can hide electron-number and spin labels, so states from different sectors may intersect in the spectrum. The paper proposes constrained VQE to target a chosen electronic property; its constrained curves retain target spin without kinks, while the mean-field singlet has an incorrect dissociation limit.

  • Problem

    Qubit-space diagonalization hides electron-number and spin labels, bringing states from different Hilbert spaces into a shared spectrum where they can intersect.

    Source: A. Operators in the Fock and qubit… · p. 3

  • Method

    Constrained VQE adds constraints on conserved properties to target an electronic state with a specified spin, charge, or other property.

    Source: D. Discussion · p. 5

  • Results

    Constrained PESs retain their target spin at all R and show no kinks; the mean-field singlet still has an incorrect dissociation limit.

    Source: C. Constrained Variational Quantum… · p. 4

  • Takeaways & Limitations

    The approach targets specific states and removes root-switching kinks with virtually no additional quantum costs.

    Source: D. Discussion · p. 5

    The constrained mean-field singlet has an incorrect dissociation limit, and correcting it requires an entangler.

    Source: C. Constrained Variational Quantum… · p. 4

Abstract

from arXiv · show

Variational quantum eigensolver (VQE) is an efficient computational method promising chemical accuracy in electronic structure calculations on a universal-gate quantum computer. However, such a simple task as computing the electronic energy of a hydrogen molecular cation, H$_2^+$, is not possible for a general VQE protocol because the calculation will invariably collapse to a lower energy of the corresponding neutral form, H$_2$. The origin of the problem is that VQE effectively performs an unconstrained energy optimization in the Fock space of the original electronic problem. We show how this can be avoided by introducing necessary constraints directing VQE to the electronic state of interest. The proposed constrained VQE can find an electronic state with a certain number of electrons, spin, or any other property. The new algorithm does not require any additional quantum resources. We demonstrate performance of the constrained VQE by simulating various states of H$_2$ and H$_2$O on Rigetti Computing Inc's 19Q-Acorn quantum processor.

A. Operators in the Fock and qubit spaces

Mapping the electronic Hamiltonian into qubit space preserves its spectrum but hides electron-number and spin labels. As a result, states from different sectors can intersect and produce kinks in potential energy surfaces.

A. Operators in the Fock and qubit spaces

  • For H2 in STO-3G, the four-qubit Hamiltonian represents 16 electronic states, and its spectrum matches the fermionic Hamiltonian while hiding N and S2 labels.Classical quantum chemistry projects onto the electronic state of interest using Slater determinants, which fix particle number and spin.Source: A. Operators in the Fock and qubit… · pp. 2–3
  • For R ≤ 0.7 Å, the first excited state is H2+; at larger R it is neutral H2 triplet, and the cation ground state intersects that triplet.The triplet’s change in spectral position is one source of the kink.Source: A. Operators in the Fock and qubit… · p. 3
  • Intersections between states from different Hilbert spaces can create kinks when the Fock-space Hamiltonian brings them into the same qubit space.Figure 2 compares the two lowest Hamiltonian eigenstates with the QMF curve.Source: Figure 2 · p. 3, A. Operators in the Fock and qubit… · p. 3

B. Variational quantum eigensolver

VQE estimates energy from a parameterized qubit state and minimizes it classically. The mean-field solution can switch spin character as molecular geometry changes, producing a kink in its energy curve.

B. Variational quantum eigensolver

  • VQE measures Pauli-term averages for a parameterized state, contracts them with Hamiltonian coefficients, then minimizes the energy on a classical computer.The wavefunction is built from an initialized qubit state followed by mean-field rotations and an entangler.Source: B. Variational quantum eigensolver · p. 3
  • The ansatz entangler is intended to add electron-correlation effects beyond the mean-field rotations.Its parameters enter an exponent of multi-qubit anti-Hermitian operators.Source: B. Variational quantum eigensolver · p. 3
  • The qubit mean-field state is a product of spin-coherent states whose orientations on the Bloch sphere are varied to minimize the energy functional.Each orientation is specified by φ and θ; one Euler angle contributes only a global phase and does not affect energy.Source: B. Variational quantum eigensolver · p. 4
  • Near R ≈ 1.6 Å, the mean-field minimum switches from a singlet to a triplet solution, producing a second type of kink.The exact curve is smooth in this case, but approximate entanglers do not generally guarantee convergence to the exact answer and may leave such kinks.Source: B. Variational quantum eigensolver · p. 4

C. Constrained Variational Quantum Eigensolver

Constrained minimization preserves target electron number and spin while producing smooth potential-energy curves for the studied H2 and H2+ states. In H2O, the spin constraint avoids convergence to a closely spaced triplet, and measurement post-processing can reduce environmental noise.

C. Constrained Variational Quantum Eigensolver

  • The constrained minimization adds a penalty functional for deviations from desired values of operators such as electron number and total spin.The constraints reuse the Hamiltonian's Pauli terms, so their expectation values can be formed at no additional quantum cost.Source: C. Constrained Variational Quantum… · p. 4
  • The constrained H2 singlet and triplet PESs retain their target spins without kinks; the triplet matches the exact PES, while the singlet has an incorrect dissociation limit in the minimal basis.The singlet limit lies between the purely covalent and ionic solutions; correcting it requires an entangler.Source: C. Constrained Variational Quantum… · p. 4
  • The constrained H2+ curve is smooth and coincides with the exact PES because electron–electron correlation is absent.Source: C. Constrained Variational Quantum… · p. 4
  • For H2O, imposing both S = 0 and Ne = 10 avoids convergence to the closely spaced triplet solution.The PES varies one OH-bond length from the symmetric equilibrium configuration.Source: C. Constrained Variational Quantum… · p. 5
  • Reweighting measurements by overlap with eigenfunctions of target-property operators reduces noise from uncontrolled environmental interactions without changing intrinsic quantum uncertainty.Source: C. Constrained Variational Quantum… · p. 5

D. Discussion

The discussion presents constrained VQE as a low-cost way to target electronic states with specified properties and remove root-switching kinks. It also reports measurement-noise reduction and specifies the constraints used in this study.

D. Discussion

  • Constrained VQE targets states with specified spin, charge, or other properties and removes PES kinks associated with root switching, with virtually no additional quantum cost.Source: D. Discussion · p. 5
  • The study imposed constraints on electron number and total spin; restricting Sz was not beneficial in the cases considered.Source: D. Discussion · p. 5
  • Post-processing with conserved-quantity operators reduces noise from qubit–environment interactions without affecting statistics from ideal projective measurements.Source: D. Discussion · p. 5

A. Spin coherent states

A spin coherent state is constructed from a maximal-projection eigenfunction using the spin-lowering operator. The resulting states form an overcomplete, non-orthogonal set on the Bloch sphere.

A. Spin coherent states

  • A spin coherent state is generated by applying a scaled exponential of the lowering operator to a normalized eigenfunction of Sz with maximal projection M = J.Source: A. Spin coherent states · p. 5
  • The states form an overcomplete, non-orthogonal set on the unit Bloch sphere, parameterized by spherical angles φ and θ.Source: A. Spin coherent states · p. 5

B. Classical minimization

Classical constrained and unconstrained minimizations use sequential quadratic programming. Mean-field solutions are optimized over eight Bloch angles.

B. Classical minimization

  • Classical constrained and unconstrained minimizations use sequential quadratic programming through MATLAB's fmincon routine.Source: B. Classical minimization · p. 5
  • Each mean-field solution is obtained by minimizing its energy over all eight Bloch angles.The eight angles arise from four qubits, each with two Bloch angles.Source: B. Classical minimization · p. 6

C. Quantum computer simulation details

The simulations used a non-entangled, single-qubit rotation ansatz on Rigetti’s 19Q-Acorn processor, with grouped measurements and post-processing. Sampling was limited, and Nelder–Mead was chosen for its robustness to noise.

Simulation setup

  • The wavefunction ansatz applied RZ and RX rotations to individual qubits without entanglement, using qubits with one-gate fidelity above 0.99.The simulations ran on Rigetti’s 19Q-Acorn QPU using pyQuil and Forest API.Source: C. Quantum computer simulation details · p. 6

Measurement

Sampling

  • Measurements used 1,000 shots for H2 and 10,000 for H2O, with 20 and 4 Hamiltonian estimates, respectively; limited system time prevented deeper sampling.The four experiments were completed in 24 hours over five sessions.Source: C. Quantum computer simulation details · p. 6

Optimization

D. Post-processing procedure

Post-processing removes measurement outcomes that violate the target electron number or other known good quantum number. The procedure is equivalent to projecting into the desired electron-number subspace, preserving the expectation value there while removing some noise contributions.

Procedure

  • Repeated measurements of non-commuting Hamiltonian parts can collapse the wavefunction into different eigenfunctions, so post-processing removes eigenvalues whose readout states violate the target quantum number.For H = A + B, the averages of the two parts are measured separately; their eigenfunctions need not be shared.Source: D. Post-processing procedure · p. 6

Projector interpretation

  • The electron-number filter is equivalent to applying a projector onto the subspace with the correct electron number.The projector is defined as P_N = |N⟩⟨N|, where N is the eigenvalue of the electron-number operator.Source: D. Post-processing procedure · p. 6

Invariance

  • For a wavefunction already in the target subspace, post-processing leaves the Hamiltonian expectation value unchanged.Outcomes associated with eigenfunctions orthogonal to that subspace do not contribute.Source: D. Post-processing procedure · p. 6

Noise reduction

  • Filtering removes noise components that violate the target electron number, producing a more accurate Hamiltonian average from the remaining density matrix.The paper distinguishes removable noise from the projected, irreducible component.Source: D. Post-processing procedure · p. 6
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