Variational Quantum Computation of Excited States

TL;DR

This work extends the variational quantum eigensolver to efficiently compute excited-state energies by introducing variational quantum deflation (VQD). By augmenting the objective with overlap penalties against previously found states, VQD enforces orthogonality and yields the $k$-th excited energy using the same qubit count as VQE, with at most roughly double the circuit depth. The authors analyze sampling costs, provide low-depth overlap estimation strategies (including a destructive SWAP variant), and demonstrate robust, accurate results on H$_2$ with favorable error accumulation properties. They also discuss alternative deflation schemes, symmetry constraints, and error-mitigation approaches, arguing that VQD is well-suited for near-term quantum devices and practical excited-state calculations. Overall, VQD offers a resource-efficient, modular pathway to excited-state quantum chemistry and related eigenvalue problems on noisy intermediate-scale quantum platforms.

Abstract

The calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum computers.

Figures (4)

• Figure 1: A schematic of our variational quantum deflation method for finding the $k$-th excited state of a Hamiltonian $H$.
• Figure 2: All ground and excited state energy levels of H$_2$ in the STO-3G basis, calculated using exact diagonalisation (blue dotted line) and our variational quantum deflation (VQD) method (red filled circles) over a range of internuclear separations.
• Figure 3: Median error using VQD to determine each energy level $k$ in the spectrum of $H_2$ at bond distance (0.7414 Å) in the STO-3G basis. Red, blue and green lines show results using $10^6$, $10^7$ and $10^8$ samples (per Hamiltonian subterm and overlap term) respectively. For the solid lines, the standard VQD algorithm was used, whereas for dashed lines states $i<k$ in $H_k$ were computed exactly for comparison. Chemical accuracy ($1.6\times 10^{-3}$ Ha) is also shown for reference (black solid line). Error bars show 1$\sigma$ standard errors for the median estimates (calculated using bootstrap resampling efron1992bootstrap).
• Figure 4: The $N$-qubit generalisation of the destructive SWAP test as applied to two ansatz states $\ket{\psi({\lambda}_i)}$ and $\ket{\psi({\lambda}_k)}$, prepared using state preparation circuits $R({\lambda}_i)$ and $R({\lambda}_k)$ respectively.