[Paper Review] Subspace-search variational quantum eigensolver for excited states

Ken Nakanishi, Kosuke Mitarai+1 more|arXiv (Cornell University)|Oct 22, 2018

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes the Subspace-Search Variational Quantum Eigensolver (SSVQE), a near-term quantum algorithm that efficiently finds excited states of molecular and spin Hamiltonians without ancilla qubits. By initializing orthogonal input states into a parameterized quantum circuit and minimizing a cost function that preserves orthogonality via unitary evolution, SSVQE locates the k-th excited state in just one or two optimization steps, validated via simulations on transverse Ising models and HeH molecules with high fidelity.

ABSTRACT

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.; extit{Nat. Commun.}; extbf{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.

Motivation & Objective

• To extend the Variational Quantum Eigensolver (VQE) framework to efficiently compute excited states of quantum systems on near-term quantum computers.
• To overcome the limitations of classical methods, which struggle with computational cost and accuracy for excited states.
• To eliminate the need for ancilla qubits and swap tests used in prior approaches, enhancing feasibility for NISQ devices.
• To develop a single-optimization procedure that simultaneously finds multiple excited states up to the k-th level.
• To enable practical applications such as transition amplitude calculations for material property analysis.

Proposed method

• The SSVQE initializes multiple orthogonal input states into a parameterized quantum circuit (ansatz) to span a subspace of interest.
• It uses a cost function that minimizes the average energy of the subspace while preserving orthogonality through unitary evolution.
• The k-th excited state is identified as the highest-energy state within the low-energy subspace after optimization.
• A generalized weighted SSVQE variant uses a weighted cost function to simultaneously optimize for all states from ground to the k-th excited state in one run.
• The method avoids ancilla qubits and the swap test, relying solely on unitary transformations to maintain state orthogonality.
• The algorithm is implemented via variational optimization of circuit parameters using classical optimizers on quantum hardware.

Experimental results

Research questions

• RQ1 Can a variational quantum algorithm find excited states of a Hamiltonian without requiring ancilla qubits or swap tests?
• RQ2 Can the SSVQE locate the k-th excited state using only one or two optimization procedures?
• RQ3 Can the weighted SSVQE simultaneously approximate multiple excited states (up to the k-th) in a single optimization run?
• RQ4 How accurately can SSVQE compute excited state energies for molecular systems like HeH and spin models?
• RQ5 Can the SSVQE be used to estimate transition amplitudes between eigenstates for material property analysis?

Key findings

• The SSVQE successfully finds the third excited state of a transverse Ising model with high fidelity using only two optimization steps.
• The weighted SSVQE accurately computes all excited states up to the third excited state in a single optimization procedure, matching exact diagonalization results.
• Simulations on the HeH molecule at 24 different bond lengths show excellent agreement between SSVQE-predicted and exact excited state energies.
• The fidelity of the output states in the SSVQE reaches near-unity levels during optimization, indicating convergence to target eigenstates.
• The weighted SSVQE achieves comparable convergence speed to the standard SSVQE while enabling simultaneous computation of multiple excited states.
• The method enables the estimation of transition amplit between eigenstates, which is useful for calculating physical properties like permittivity and spontaneous emission rates.