Variational quantum algorithms for discovering Hamiltonian spectra

Abstract

Calculating the energy spectrum of a quantum system is an important task, for example to analyze 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 [O. Higgott, D. Wang, and S. Brierley, arXiv:1805.08138], we employ the low depth swap test to energetically penalize 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 toward 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 optimization 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.

Variational quantum state diagonalization

“…The final step of VQSD is to prepare the eigenvectors associated with the m-largest eigenvalues, i.e., the eigenvalues in the set in Eq. (15). Let Z = {z :λ est z ∈λ est } be the set of bitstrings z associated with the eigenvalues inλ est .…”

Theory of analytical energy derivatives for the variational quantum eigensolver

“…We also present a method to extract the derivatives of excited state energy based on the technique presented in Refs. [3,4]. The analysis on the computational cost shows that the analytical differentiation provided in this work can be more practical than a finite difference approach, in a sense that one does not need to consider the best step size to extract the energy derivative information with an optimal precision.…”

“…The generation of excited states can be a powerful application of the VQE, because the classical computation, despite the recent significant improvement in the theory and the computational power, still suffers in the calculation of them [47]. Among the several proposals [2][3][4][5] to generate excited states with the VQE, we adopt the one proposed in Refs. [3,4] to compute the derivatives of the excited state energy.…”

“…Originally, the method was devised for finding the ground state of a system [1]. It has subsequently been extended for excited states by several proposals [2][3][4][5]. From the generated eigenstates, one can measure its associated physical quantities, such as the particle densities and transition amplitudes between the different eigenstates.…”

Methodology for replacing indirect measurements with direct measurements

“…The destructive swap test is a direct version of the swap test [8] which measures the overlap | ψ|ϕ | 2 between two quantum states |ψ and |ϕ . Initially proposed in [7], this method has been rediscovered by the machine learning approach [9], and it is now utilized in the application of NISQ devices [10][11][12]. Reference [13] has proposed to use the destructive swap test to measure | ψ|U |ψ | 2 for an arbitrary U by substituting |ϕ with U |ψ , and the protocol was extended to measure the quantity | ψ|P|ϕ | 2 , where P is a qubit-permutation operator, which can be employed to estimate nonlinear functionals of a quantum state ρ such as Tr(ρ n ) [14], with a low-depth circuit.…”