Subspace-search variational quantum eigensolver for excited states

TL;DR

The paper addresses the challenge of computing excited states with near-term quantum devices by extending VQE through the subspace-search VQE (SSVQE), which preserves orthogonality via orthogonal input states and unitary evolution. It introduces two main variants: the original SSVQE requiring two optimization steps and a family of weighted SSVQE methods that can find the k-th excited state or all excited states up to k with a single optimization, all without ancilla qubits or swap tests. The authors derive a practical transition-matrix-element calculation within this framework and validate the methods with numerical simulations on a 4-qubit fully connected transverse Ising model and the Helium hydride molecule, demonstrating accurate excited-state energies and transition properties. Overall, the work broadens the applicability of VQE to excited-state physics and chemistry on noisy quantum hardware, offering efficient procedures for accessing excited states and their transitions.

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.; \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.

Figures (11)

• Figure 1: Variational quantum circuit used in the simulations of \ref{['sec:numerical_simulation']}. These parameters $\bm{\phi}, \bm{\theta}$ are optimized to to minimize $\mathcal{L}$. $D_1$ and $D_2$ denote the number of repetition of a circuit in each bracket. Note that, in the explanation of weighted SSVQE, $\bm{\theta}$ denotes $\{\bm{\phi}, \bm{\theta}\}$ in this figure.
• Figure 2: Step 1 of the SSVQE to find third excited state of a transverse Ising model. (black dashed line) $\mathrm{Avg.}(E_{0,1,2,3})$ = $\frac{1}{4}\sum_{k=0}^3 E_k$ , which is the globally optimal value of $\mathcal{L}_1/4$ in this case. (red solid lines) The evolution of $\mathcal{L}_1/4$ and the fidelity (see the main text for the definition) during the optimization process.
• Figure 3: Step 2 of the SSVQE to find the third excited state of a transverse Ising model. (red solid lines) The evolution of $\mathcal{L}_2$ and the fidelity (see the main text for the definition) during the optimization process.
• Figure 4: The weighted SSVQE to find the third excited state of a transverse Ising model. In the energy diagram, SSVQE($E_3$) (red solid line) is $\braket{\varphi_3|U^\dagger(\bm{\theta})HU(\bm{\theta})|\varphi_3}$ at each iteration.
• Figure 5: The weighted SSVQE to find excited states of a transverse Ising model up to the third. In the energy diagram, SSVQE($E_k$) (solid lines) are $\braket{\varphi_k|U^\dagger(\bm{\theta})HU(\bm{\theta})|\varphi_k}$ for each $k$ at each iteration.