TIMES-ADAPT: A Quantum algorithm for real-time evolution in low-energy subspaces using fixed-depth circuits

TL;DR

A new variational quantum algorithm that prepares time-evolved states in a low-energy or symmetric subspace of a time-independent Hamiltonian on a quantum computer, and builds fixed-depth circuits for real-time evolution in the subspace, where time only enters as a circuit parameter.

Abstract

We propose a new variational quantum algorithm, which we refer to as TIMES-ADAPT, that prepares time-evolved states in a low-energy or symmetric subspace of a time-independent Hamiltonian on a quantum computer. Using a specially trained unitary that diagonalizes the Hamiltonian in a subspace, we construct fixed-depth circuits for real-time evolution in the subspace, where time only enters as a circuit parameter. We present two versions of the algorithm depending on whether the initial state is specified in the energy eigenbasis or computational basis. We consider two important applications of our methods: wave packet evolution and energy transport in spin systems. We benchmark our algorithms using variants of the Heisenberg XXZ model.

Figures (6)

• Figure 1: General block diagram for the variational ansatz of TEPID-ADAPT. $U_m(\vec{\mu})$ prepares $\rho_m$ on the system register (top), as indicated by the red line. $V_A(\vec{\theta})$ is an adaptively generated unitary on the system register that approximately evolves $\rho_m$ to the target Gibbs state.
• Figure 2: A diagrammatic workflow for TIMES-ADAPT. We first use TEPID-ADAPT to train an adaptively generated circuit $V_A$ to diagonalize the Hamiltonian in a low-energy subspace. We also readily obtain the energy differences from this procedure. We propose two related versions of our algorithm to perform time-evolution of a state in the low-energy subspace with a fixed-depth circuit. In TIMES-ADAPT-I, we create a particular linear combination of computational basis states, and use $V_A$ to obtain the time-evolved state $\ket{\psi(t)}$. In TIMES-ADAPT-II, we compile an effective unitary that is equivalent to the evolution operator in the low-energy subspace, and then apply it to the initial state to obtain the time-evolved state $\ket{\psi(t)}$.
• Figure 3: The fidelity of various time-evolved states using TIMES-ADAPT-I (dashed) and TIMES-ADAPT-II (solid) with exact diagonalization. The initial state in (a), (b), (c) are assumed to be in the span of the first five eigenstates of $H_{\text{XXZ}}$. In (d), the initial state is a linear combination of all the eigenstates with their Boltzmann weights as coefficients. The various colors corresponds to which portion of the subspace is known via TEPID-ADAPT, as indicated. Panels (a), (b) show results for random initial states, whereas (c) and (d) show results for a uniform and exponential distribution of coefficients, respectively.
• Figure 4: Time evolution of a single-magnon Gaussian wave packet in the longitudinal-field XXZ model. (left) Evolution of the on-site magnetization of the indicated sites using TIMES-ADAPT-I (circles), first-order Trotter (triangles), and exact evolution (solid lines). (right) The infidelity relative to the exact evolution for TIMES-ADAPT-I (circles), and first-order Trotter (triangles). The step size for the Trotter results was chosen to yield the same two-qubit gate depth as the unitary TIMES-ADAPT-I protocol. For clarity, only the even Trotter steps are shown in the plots.
• Figure 5: Time evolution of an inhomogeneous low-energy state in the staggered longitudinal-field XXZ model. (left) Evolution of the energy density on the indicated sites using TIMES-ADAPT-II (dashed lines), first-order Trotter (dotted lines), and exact evolution (solid lines). (right) The infidelity relative to the exact evolution for TIMES-ADAPT-II (dashed line), and first-order Trotter (dotted line). The step size for the Trotter results was chosen to yield the same two-qubit gate depth as the TIMES-ADAPT-II protocol.