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Variational quantum algorithms for discovering Hamiltonian spectra

Suguru Endo, Tyson Jones, Sam McArdle, Xiao Yuan, Simon Benjamin

arXiv:1806.05707v2quant-ph

TL;DR

Excited-state energies remain less developed than ground-state energies, despite their importance for molecular applications and the classical intractability of large Hamiltonian diagonalisation. The paper introduces a low-depth variational algorithm that sequentially penalises discovered states using swap-test overlaps and imaginary-time evolution. Simulations on 3SAT Hamiltonians and LiH successfully estimate excitation spectra, while ansatz limitations reduce accuracy for some highly entangled states.

  • Problem

    Excited-state calculation has received comparatively less development, although energy spectra inform molecular dynamics and classical diagonalisation becomes prohibitively expensive for large systems.

  • Method

    The algorithm sequentially discovers Hamiltonian eigenstates by using low-depth swap tests to penalise known states and variational imaginary-time evolution to target the modified Hamiltonians’ ground states.

  • Results

    The method successfully obtained estimates of excitation spectra for SAT and LiH Hamiltonians in simulations using two different ansätze.

  • Takeaways & Limitations

    The low-depth, variational design provides a route to estimating Hamiltonian spectra on hybrid quantum-classical machines.

  • Takeaways & Limitations

    Accuracy decreases when the ansatz lacks sufficient power to generate highly entangled eigenstates, including around LiH bond length l ≈2.5˚A.

Abstract

from arXiv · show

Calculating the energy spectrum of a quantum system is an important task, for example to analyse reaction rates in drug discovery and catalysis. There has been significant progress in developing algorithms to calculate the ground state energy of molecules on near-term quantum computers. However, calculating excited state energies has attracted comparatively less attention, and it is currently unclear what the optimal method is. We introduce a low depth, variational quantum algorithm to sequentially calculate the excited states of general Hamiltonians. Incorporating a recently proposed technique, we employ the low depth swap test to energetically penalise the ground state, and transform excited states into ground states of modified Hamiltonians. We use variational imaginary time evolution as a subroutine, which deterministically propagates towards the target eigenstate. We discuss how symmetry measurements can mitigate errors in the swap test step. We numerically test our algorithm on Hamiltonians which encode 3SAT optimisation problems of up to 18 qubits, and the electronic structure of the Lithium Hydride molecule. As our algorithm uses only low depth circuits and variational algorithms, it is suitable for use on near-term quantum hardware.

I. INTRODUCTION

Classical Hamiltonian diagonalisation becomes intractable for large quantum systems, while excited-state calculation remains less developed despite its importance for chemical reactions. The paper proposes a shallow, variational method that sequentially discovers Hamiltonian energy levels using imaginary-time evolution and swap-test penalties.

  • Motivation: Classical Hamiltonian diagonalisation becomes prohibitively expensive as quantum-system size grows, limiting studies of complicated compounds.The cost of storing and operating on the quantum state grows exponentially.
  • Research gap: Excited-state calculation has received less development than ground-state calculation despite its importance for analysing chemical reactions and drug discovery.
  • Research gap: Existing VQE-based excited-state proposals require either many measurements or deep quantum circuits.Examples include quantum subspace expansion and von-Neumann entropy methods.
  • Approach: The proposed algorithm uses imaginary-time evolution and a shallow swap test to penalise discovered states and sequentially find additional eigenstates.It trades deeper circuits for additional measurements.
  • Approach: Variational imaginary-time evolution tends to converge to Hamiltonian eigenstates from arbitrary initial states within a restricted submanifold.The paper identifies this convergence as the mechanism enabling reliable spectrum discovery.

II. IMAGINARY TIME EVOLUTION

Imaginary-time evolution suppresses higher-energy components and approaches the ground state when the initial state overlaps it. The paper simulates this non-unitary process with a parametrised circuit whose parameters are updated by a hybrid quantum-classical procedure, while noting an ansatz limitation.

  • Imaginary-time dynamics: Imaginary-time evolution makes energy-eigenstate amplitudes decay exponentially according to their energies and approaches the ground state when initial overlap is nonzero.
  • Variational simulation: The non-unitary evolution is approximated by a parametrised trial state produced by an ansatz quantum circuit.
  • Variational simulation: Quantum processors obtain the evolution elements, while a classical processor updates circuit parameters using the Euler rule.
  • Convergence: Repeatedly solving the parameter-update equation evolves a sufficiently expressive ansatz toward a state close to the ground state.Convergence is monitored through changes in the parameters and the converged energy is evaluated with Pauli operators.
  • Limitation: With a less powerful ansatz, imaginary-time evolution may miss the ground state and instead converge to a higher excited eigenstate; this behavior lacks a proof but appears consistently in simulations.

III. EVALUATION OF THE ENERGY SPECTRUM OF THE HAMILTONIAN USING IMAGINARY TIME EVOLUTION

After approximating a discovered eigenstate, the algorithm adds a projector penalty so that the next state becomes the ground state of a modified Hamiltonian. Repeating this process can recover the spectrum without directly modifying the Hamiltonian or requiring state tomography.

  • State penalisation: The modified Hamiltonian H′ = H + α|˜g⟩⟨˜g| raises the approximate ground-state energy by α.
  • State penalisation: For sufficiently large α, the original first excited state becomes the modified Hamiltonian’s ground state while orthogonal states remain unaffected.
  • Sequential discovery: Repeating projector penalties for discovered eigenstates yields successive excited states and can in principle recover the complete spectrum, including degeneracies.The penalty α must exceed the gap between the ground state and the highest state sought.
  • Implementation: The algorithm modifies imaginary-time evolution equations rather than the Hamiltonian itself because full state-vector tomography would be exponentially costly.
  • Implementation: Low-depth swap tests estimate overlaps between the trial state and previously discovered states for the additional evolution terms.

IV. ERROR MITIGATION

The paper proposes symmetry measurements as an error-detection and mitigation step for the conventional swap test. Register measurements can identify symmetry-breaking errors, although additional mitigation strategies are left untested.

  • Conventional swap test: The conventional swap test has depth growing linearly with qubit number and measures the input-state overlap through an ancilla.
  • Symmetry mitigation: Measuring a shared symmetry on swap-test register qubits detects errors that break that symmetry, allowing erroneous results to be discarded.
  • Symmetry mitigation: For molecular Hamiltonians, electron-number measurements can detect errors when the ansatz conserves electron number.The method can mitigate single-qubit bit-flip errors and certain combinations of two-qubit errors.
  • Symmetry mitigation: Figure 1 illustrates using symmetry measurements to detect and mitigate errors in the conventional swap test.
  • Scope: The algorithm is compatible with other proposed error-mitigation techniques, but the paper does not test them.

V. NUMERICAL SIMULATIONS

The simulations evaluate the algorithm on 3SAT and LiH Hamiltonians, comparing discovered spectra with exact diagonalisation. Variational imaginary time evolution generally identifies eigenstates, while accuracy depends on the ansatz and molecular setting.

  • Simulation setup: The simulations use random initial parameters, re-randomisation after Hamiltonian excitation, and Tikhonov regularisation for smooth parameter updates.The study explores both low depth and compact ansätze.
  • Test systems: The benchmark Hamiltonians encode 3SAT problems of up to 18 qubits and reduced LiH electronic-structure models with 10 and 6 qubits.The 3SAT spectra are equally spaced and highly degenerate, while LiH is represented using physical approximations.
  • 3SAT simulations: Imaginary time evolution discovers low-lying 3SAT states by monotonically lowering expected energy until convergence, followed by excitation of the discovered state.The ground state need not be discovered first; in one 16-qubit simulation it was the third state found.
  • LiH simulations: For reduced 6-qubit LiH, imaginary time evolution successfully discovers eigenstates, whereas gradient descent converges to non-eigenstates and produces errors after excitation.The discovered spectra are compared with direct numerical diagonalisation.
  • LiH simulations: For physical 10-qubit LiH, the 145-parameter low depth ansatz improves accuracy and degeneracy discovery over the 70-parameter compact ansatz, but both lose accuracy near l ≈2.5˚A.The compact ansatz shows smooth ground-state deviations, potentially because it cannot generate the ground state.

VI. DISCUSSION

The paper reports successful spectrum estimation with variational imaginary time and identifies unresolved questions about ansatz design, simulation fidelity, and the hybrid algorithm’s classical component.

  • VI. DISCUSSION: The algorithm successfully obtained excitation-spectrum estimates for SAT and LiH Hamiltonians using two ansätze.The simulations rarely produced non-eigenstates, whereas gradient descent often became trapped in local minima that distorted excited-state energies.
  • VI. DISCUSSION: Variational imaginary time was rarely observed to converge to non-eigenstates, unlike gradient descent, which could produce erroneous reported spectra after excitation.
  • VI. DISCUSSION: The required ansatz design for accurately generating energy eigenstates remains an open direction for future research.
  • VI. DISCUSSION: How faithfully variational evolution must reproduce true imaginary-time evolution to converge to the lowest-lying states remains unresolved.
  • VI. DISCUSSION: The relationship between the hybrid algorithm’s classical components also requires further investigation.

Appendix A: Quantum circuits to obtain the elements of M and V

Appendix A explains how inserted ansatz gates and Pauli-decomposed Hamiltonian terms are evaluated through circuits measuring real parts of unitary expectation values.

  • Appendix A: Quantum circuits to obtain the elements of M and V: U_k,i is the full ansatz unitary with a new gate G_k,i inserted before the ith parameterised gate.
  • Appendix A: Quantum circuits to obtain the elements of M and V: Hamiltonian terms are expanded as sums of Pauli operators and reduced to real parts of expectation values of unitary operators.
  • Appendix A: Quantum circuits to obtain the elements of M and V: The circuit in Fig. 7 evaluates ℜ(eiφ ⟨¯0| V |¯0⟩) using a Hadamard-based measurement of the second qubit’s Pauli-Z expectation.
  • Appendix A: Quantum circuits to obtain the elements of M and V: The implementation can use circuits simpler than a controlled-V circuit.

Appendix B: Overlap calculation by using shallow quantum circuit

Appendix B describes a shallow, constant-depth procedure for estimating the overlap of two L-qubit input states through paired-qubit gates, measurements, and classical post-processing.

  • Appendix B: Overlap calculation by using shallow quantum circuit: The shallow overlap algorithm pairs corresponding qubits of ρ and σ, applies controlled-NOT gates, and then applies transversal Hadamard gates.
  • Appendix B: Overlap calculation by using shallow quantum circuit: Fig. 8 represents a shallow swap-test circuit that evaluates the overlap of two input density matrices, ρ and σ.
  • Appendix B: Overlap calculation by using shallow quantum circuit: Measurements of the paired-qubit outcomes are combined through post-processing to evaluate Tr(ρσ).

1. 3SAT

The paper uses 3SAT instances to construct diagonal Hamiltonians with known spectral structure, testing the method on problems up to 18 qubits and on LiH representations.

  • 3SAT: Each 3SAT variable maps to one qubit, with computational-basis states representing Boolean assignments.The qubit states 1 and 0 correspond to true and false, respectively.
  • 3SAT: The constructed diagonal Hamiltonian assigns each basis state an energy equal to the number of clauses its assignment fails to satisfy.The unique satisfying assignment is therefore the zero-energy ground state, while excited energies are integer-valued and highly degenerate.
  • 3SAT: The simulations test 3SAT Hamiltonians containing up to 18 qubits.The paper uses these instances to probe spectrum discovery rather than to present an efficient 3SAT solver.
  • LiH: LiH is studied in both reduced and full spin-orbital bases, with the full 12-orbital representation mapped to 10 qubits.A further six-qubit reduction has a qualitatively different, non-physical spectrum but remains useful for testing.
  • LiH: Active-orbital reduction uses natural molecular orbitals to remove orbitals identified as nearly filled or nearly empty.These orbitals diagonalise the single-particle reduced density matrix before corresponding fermionic operators are removed.

Appendix D: Implementation of numerical simulations

The numerical simulations initialize a highly excited variational state and evolve it through regularized imaginary-time updates, recording and penalizing converged eigenstates sequentially.

  • Initialization and evolution: Random parameters in [0, 2π) initialize an ansatz state expected to have high energy, after which matrices M and V drive variational imaginary-time evolution.The initial state is generated in QuEST and the matrices are passed to the numerical update procedure.
  • Regularisation: Tikhonov regularisation stabilizes the ill-posed parameter update by balancing accuracy against small, smooth parameter changes.The simulations select λ from a 3-point L-curve while restricting it to [10^-4, 10^-2].
  • Regularisation: λ = 0 allows residuals in M^-1 to disrupt monotonic energy decrease, whereas excessively large λ can converge to non-eigenstates.The regularized update therefore controls both numerical stability and the quality of convergence.
  • Time stepping: The simulations typically use an imaginary-time step of δτ = 10^-1, which Tikhonov regularisation permits to be larger than tested alternatives.The alternatives include LU decomposition, least-squares minimisation, SVD, and truncated SVD.
  • Sequential discovery: After convergence, detected by ||Δθ⃗|| < 10^-2 for three iterations, the state is recorded and later excited through modified V updates.Resetting the parameters restores the initial or higher-energy state before evolution resumes.
  • Simulation shortcuts: The simulations approximate parameter derivatives with fourth-order central finite differences using Δθ_i = 10^-5 instead of explicitly simulating the corresponding subcircuits.M and V are then populated from inner products, and recorded eigenstates introduce excitations through inner products rather than simulated swap-test circuits.
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