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Subspace-search variational quantum eigensolver for excited states

Ken M Nakanishi, Kosuke Mitarai, Keisuke Fujii

arXiv:1810.09434v2quant-ph

TL;DR

Excited states are important for molecular and physical processes but remain difficult for classical computation and lack suitable NISQ algorithms. The paper introduces SSVQE, which uses orthogonal inputs and unitary ansätze to search excited-state subspaces without ancillas, and reports numerical validation while extending the approach to transition amplitudes and multiple excited states.

  • Problem

    Excited states matter for chemical and physical processes, yet classical computation has increasing cost and relatively poor results, while NISQ-compatible methods were lacking.

  • Method

    SSVQE maps orthogonal input states through a parameterized unitary into a low-energy subspace, then identifies excited states while avoiding swap-test ancilla requirements.

  • Results

    Numerical simulations validate the proposed algorithms, including third-excited-state agreement for helium hydride and accurate eigenvalues through the third excited state.

  • Takeaways & Limitations

    The framework extends VQE to excited states, multiple states through the k-th, and transition-amplitude measurements for properties including permittivity and spontaneous-emission rates.

Abstract

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The variational quantum eigensolver (VQE), a variational algorithm to obtain an approximated ground state of a given Hamiltonian, is an appealing application of near-term quantum computers. The original work [A. Peruzzo et al.; \textit{Nat. Commun.}; \textbf{5}, 4213 (2014)] focused only on finding a ground state, whereas the excited states can also induce interesting phenomena in molecules and materials. Calculating excited states is, in general, a more difficult task than finding ground states for classical computers. To extend the framework to excited states, we here propose an algorithm, the subspace-search variational quantum eigensolver (SSVQE). This algorithm searches a low energy subspace by supplying orthogonal input states to the variational ansatz and relies on the unitarity of transformations to ensure the orthogonality of output states. The $k$-th excited state is obtained as the highest energy state in the low energy subspace. The proposed algorithm consists only of two parameter optimization procedures and does not employ any ancilla qubits. The disuse of the ancilla qubits is a great improvement from the existing proposals for excited states, which have utilized the swap test, making our proposal a truly near-term quantum algorithm. We further generalize the SSVQE to obtain all excited states up to the $k$-th by only a single optimization procedure. From numerical simulations, we verify the proposed algorithms. This work greatly extends the applicable domain of the VQE to excited states and their related properties like a transition amplitude without sacrificing any feasibility of it.

I. INTRODUCTION

VQE uses a parameterized circuit to approximate a Hamiltonian’s ground state, but excited states are important and harder to compute classically. The paper proposes SSVQE to find excited states on NISQ devices while preserving near-term feasibility.

  • VQE background: VQE minimizes a Hamiltonian expectation value over parameterized circuit states to approximate the ground state.The ansatz circuit U(θ) generates |ψ(θ)⟩, whose energy expectation is optimized iteratively.
  • Motivation: Excited states contribute to chemical reactions, physical processes, and phenomena such as luminescence.Intermediate reaction states are generally not ground states, making their properties relevant for analysis.
  • Motivation: Classical computation faces increasing cost and relatively poor results for excited states, while NISQ-ready algorithms were still lacking.This motivates using quantum computers for finding and analyzing excited states.
  • Contribution: SSVQE supplies orthogonal input states to a parameterized circuit and minimizes energy in their spanned subspace.Unitary evolution preserves output orthogonality, enabling excited-state searches without enforcing output overlap constraints directly.
  • Contribution: The method can find the k-th excited state with two optimization procedures and generalize to all states through the k-th with one.The paper also describes transition-amplitude measurement and numerical simulations for random transverse Ising models and helium hydride.

A. Subspace-search variational quantum eigensolver

SSVQE first maps mutually orthogonal, easily prepared inputs into a low-energy subspace, then searches that subspace for the k-th excited state. Unitarity preserves orthogonality throughout the mapping.

  • Algorithm: SSVQE begins with mutually orthogonal input states and an ansatz circuit acting on an n-qubit quantum computer.The inputs can be chosen from easily preparable states such as computational-basis states.
  • Subspace search: The first cost function minimizes the summed energies of the circuit outputs to identify a subspace containing the target state.At the optimized parameters, the mapped subspace is arranged so the k-th excited state is its highest-energy state.
  • Target-state search: A second circuit acts only within the mapped subspace, and maximizing its energy expectation selects the k-th excited state.An arbitrary input index s is used for the second optimization.
  • Optimization variant: For k ≥ 2^n−1, choosing fewer orthogonal inputs and reversing the optimization directions can be faster.The stated alternative uses 2^n−k input states.

B. Weighted SSVQE for finding the k-th excited state

Weighted SSVQE targets the k-th excited state with a single optimization by assigning unequal weights to orthogonal input states. The method may require longer optimization because its cost landscape is more complicated.

  • Method: Weighted SSVQE extends the subspace method to find the k-th excited state using a single optimization procedure.The circuit uses mutually orthogonal input states.
  • Cost function: The weighted cost function assigns a weight w in (0, 1) to the target input while summing the remaining output energies.This weighting determines the mapping of the target input relative to the other inputs.
  • Outcome: At the global optimum, the target input maps to the k-th excited state and the other inputs map into the lower-energy subspace.Minimizing the weighted cost therefore identifies the target state directly.
  • Caveat: The optimization may take longer than the simple SSVQE because the weighted cost function can have a more complicated landscape.A numerical example reports convergence requiring more iterations than the overall simple-SSVQE procedure.

C. Weighted SSVQE for finding up to the k-th excited states

A generalized weighted SSVQE uses one optimization to obtain every excited state through the k-th. An ordered weight vector determines which orthogonal input maps to each energy eigenstate.

  • Generalization: The generalized method finds all excited states up to the k-th with one optimization procedure.It optimizes a weighted sum over orthogonal input states.
  • Weight design: Weights satisfying w_i > w_j when i < j determine the assignment of input states to excited states.The weight ordering controls which input is converted to which eigenstate.
  • Outcome: At the global optimum, each input |ϕ_j⟩ maps to the j-th excited state |E_j⟩ for j from 0 through k.The authors note that optimization time may increase because of the cost landscape.

III. RELATED WORKS

The paper reviews excited-state VQE methods and contrasts them with SSVQE, emphasizing orthogonality preservation, reduced optimization, and the absence of swap tests and hyperparameters.

  • Existing approaches: Prior approaches include quantum phase estimation hybridization, quantum subspace expansion, constrained VQE, and penalty-based inductive methods.These methods differ in their use of controlled time evolution, excitation-operator subspaces, constraints, penalty terms, and swap tests.
  • Advantages of the proposed methods: When the ansatz can represent the exact mapping from input states to eigenstates, SSVQE and weighted SSVQE produce the exact energy spectrum.This condition concerns the representational ability of the unitaries used by the algorithms.
  • Advantages of the proposed methods: SSVQE has no hyperparameters, while weighted SSVQE gives unique results when its hyperparameters satisfy the required conditions.The stated uniqueness applies to the weighted method under those conditions.
  • Advantages of the proposed methods: Optimization runs only twice for SSVQE and only once for weighted SSVQE.The weighted version therefore reduces the number of optimization procedures required to obtain the target excited states.

IV. CALCULATION OF TRANSITION MATRIX ELEMENTS

The optimized SSVQE unitary enables transition amplitudes between Hamiltonian eigenstates to be measured by transforming the operator into the input-state basis and evaluating its terms separately.

  • Transition amplitudes: The optimized unitary relates an eigenstate transition amplitude ⟨E_i|A|E_j⟩ to the input-state matrix element ⟨ϕ_i|U†(θ*)AU(θ*)|ϕ_j⟩.This connection uses the mapping established by the optimized SSVQE circuit.
  • Measurement procedure: The transformed matrix element is expanded into separately measurable expectation values involving prepared superposition states.The input states are assumed to be simple states, so superpositions such as |+x_ij⟩ can be prepared.
  • Measurement procedure: Each term is measured independently on the NISQ device and the results are summed on a classical computer.

V. NUMERICAL SIMULATION

The numerical simulations use a four-qubit variational ansatz with specified circuit depths, computational-basis inputs, and repeated randomized optimizations; the circuit parameters minimize the cost L.

  • Simulation setup: The simulations use 4-qubit Hamiltonians with input states {|0000⟩, |0001⟩, |0010⟩, |0011⟩}.The input states are computational-basis states.
  • Simulation setup: The ansatz uses D1 = 2, with D2 = 6 for target-state searches and D2 = 8 for finding all states up to the k-th.The circuit depth differs between the two simulation goals.
  • Simulation setup: The initial parameters are uniformly sampled from [0, 2π), and each simulation is optimized 10 times from different starts.
  • Simulation setup: The parameters are optimized to minimize L, and the circuit repetitions are controlled by D1 and D2.

A. Transverse Ising model

The paper demonstrates the proposed algorithms on a fully connected transverse Ising model with four sites, randomly sampled couplings, and target excitation index k = 3.

  • Model and parameters: The transverse Ising demonstration uses a fully connected model with N = 4.
  • Model and parameters: The coefficients a_i and J_ij are sampled randomly from a uniform distribution on [0, 1).
  • Model and parameters: All experiments in this subsection target the k = 3 excited state.

1. SSVQE

The SSVQE uses two optimization stages to identify the k-th excited state through a low-energy subspace and then isolate its highest-energy state. Transverse-Ising simulations show the cost functions approach their targets while fidelity approaches unity.

  • Step 1: The first optimization minimizes L1(θ) to identify a low-energy subspace containing the desired excited state.For the third excited state, the optimal L1/4 equals the average of the first four energy levels.
  • Step 1: As L1/4 approaches its global optimum, the fidelity to the target subspace approaches unity.
  • Step 2: The second optimization minimizes L2(φ) while searching within the identified subspace for the third excited state.Its fidelity is defined by the overlap between the third excited state and the final transformed input state.
  • Step 2: The transverse-Ising simulation indicates that the subspace-search approach successfully finds the third excited state.

2. Weighted SSVQE for finding the k-th excited state

The weighted SSVQE incorporates energy weights to find a selected excited state in one optimization sequence and all states through the k-th in one procedure. Simulations on a transverse Ising model and HeH show successful recovery or approximation of the targeted energy levels.

  • Weighted SSVQE for finding the k-th excited state: Weighted SSVQE finds the third excited state in one optimization sequence, although convergence requires more iterations than the two-stage SSVQE.The longer convergence is attributed to a more complicated cost-function landscape.
  • Weighted SSVQE: The weighted SSVQE finds the ground and excited states through the third simultaneously using a weight vector w = (4, 3, 2, 1).The number of convergence iterations is almost equivalent to that of the preceding single-state procedure.
  • B. Helium hydride: The HeH calculation uses a 4-qubit Hamiltonian derived with the STO-3G minimal basis set.The ansatz does not conserve electron number, so calculated excited states can have different electron counts.
  • B. Helium hydride: In HeH, both SSVQE and weighted SSVQE agree with the exact third-excited-state energies at each bond length.
  • B. Helium hydride: Weighted SSVQE approximates the HeH energy eigenvalues for all states up to the third excited state.The results are shown separately for the ground, first-, second-, and third-excited states.

VI. CONCLUSION

The work extends VQE to excited-state finding by mapping orthogonal input states onto energy eigenstates through a designed cost function. The algorithms require, in principle, only one or two optimization procedures and can find one or more arbitrary excited states.

  • The proposed algorithms extend VQE to finding excited states of a given Hamiltonian.
  • Orthogonal input states and a carefully designed cost function map each state onto an energy eigenstate.
  • Only one or two optimization procedures are required in principle to find one or more arbitrary excited states.
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