Real time evolution for ultracompact Hamiltonian eigenstates on quantum hardware

TL;DR

This work develops VQPE, a real-time evolution–based variational method for extracting ground and excited-state energies on near-term quantum hardware. It introduces a unitary formulation that reduces measurements, ties the overlap structure to Toeplitz matrices, and uses phase cancellation to recover eigenstates with a compact set of time-evolved states, aided by SVD for noise robustness. The approach is validated through classical simulations of weakly to strongly correlated molecules (including Cr$_2$) achieving chemical accuracy with ~50 timesteps, and through quantum hardware/simulator demonstrations of the transverse field Ising model, highlighting its NISQ suitability. The results indicate VQPE offers a highly efficient, scalable path for simulating complex many-body systems with significantly fewer variational parameters than conventional classical methods.

Abstract

In this work we present a detailed analysis of variational quantum phase estimation (VQPE), a method based on real-time evolution for ground and excited state estimation on near-term hardware. We derive the theoretical ground on which the approach stands, and demonstrate that it provides one of the most compact variational expansions to date for solving strongly correlated Hamiltonians. At the center of VQPE lies a set of equations, with a simple geometrical interpretation, which provides conditions for the time evolution grid in order to decouple eigenstates out of the set of time evolved expansion states, and connects the method to the classical filter diagonalization algorithm. Further, we introduce what we call the unitary formulation of VQPE, in which the number of matrix elements that need to be measured scales linearly with the number of expansion states, and we provide an analysis of the effects of noise which substantially improves previous considerations. The unitary formulation allows for a direct comparison to iterative phase estimation. Our results mark VQPE as both a natural and highly efficient quantum algorithm for ground and excited state calculations of general many-body systems. We demonstrate a hardware implementation of VQPE for the transverse field Ising model. Further, we illustrate its power on a paradigmatic example of strong correlation (Cr2 in the SVP basis set), and show that it is possible to reach chemical accuracy with as few as ~50 timesteps.

Figures (11)

• Figure 1: Relative error of the first four eigenvalues from the real time NOVQE secular equation for a Hamiltonian of linear spectrum $E_N = N\Delta E$ ($\Delta E = 0.75$ here) and different time steps $\Delta t$ as a funciton of the number of time steps. The reference state follows $\ket{\Psi_0} \propto \sum_N e^{-E_N}\ket{N}$, such that only the 16 Hamiltonian eigenstates of lowest energy are part of the support space, choosing a coefficient threshold of $10^{-12}$. When solving the secular equation, we choose the same threshold for the SVD decomposition of the overlap matrix. The vertical dashed line marks the 15-th time step, after which we have as many expansion states as vectors in the support space. The large subplot corresponds to the smallest time step which fulfills the phase cancellation condition Eq. \ref{['eq:PCC']}, which reduces to a single condition for this Hamiltonian. The insets in each subfigure correspond to a geometric representation of the phase cancellation condition, with each phase $e^{-it_j\Delta E}$ a point in the unit circle on the complex plane. See text for details.
• Figure 2: Relative error and corresponding singular values of the overlap matrix for Hamiltonian of linear spectrum with different noise values. Upper panels: Relative error of the first four eigenvalues from the VQPE secular equation for a Hamiltonian of linear spectrum $E_N = N\Delta E$ ($\Delta E = 0.75$ here) and perfect time step $\Delta t_P$, including Gaussian noise $\mathcal{N}(0,\epsilon)$ to the Hamiltonian and overlap matrix elements. We choose the singular value truncation threshold $s_{SV}$ to be at least two orders of magnitude larger than the noise standard deviation $\epsilon$. The reference state follows $\ket{\Psi_0} \propto \sum_N e^{-E_N}\ket{N}$, the effective support space size determined using the singular value truncation threshold $N^{max}_{SVD} = -\frac{1}{2\Delta E}\ln(s_{SV})-1$. Lower panels: Corresponding singular values of the overlap matrix as a function of time. The horizontal dashed line represents the threshold $s_{SV}$. Note that, until a given singular value is larger than $s_{SV}$, we cannot extract the corresponding eigenvalue from the generalized eigenvalue problem. This is represented by the horizontal lines in the upper panels. See text for details.
• Figure 3: Relative error and corresponding singular values of the overlap matrix for Hamiltonian of linear spectrum for a large enough number of time steps to extract eigenstates below the error threshold. Upper Panel: Relative error of the first five eigenstates from the VQPE secular equation for a Hamiltonian of linear spectrum $E_N = N\Delta E$ ($\Delta E = 0.75$ here) and perfect time step $\Delta t_P$, including Gaussian noise $\mathcal{N}(0,10^{-2})$ to the Hamiltonian and overlap matrix elements. We choose the singular value truncation threshold $s_{SV} = 9\cdot10^{-1}$. The reference state follows $\ket{\Psi_0} \propto \sum_N e^{-E_N}\ket{N}$. Lower Panel: Corresponding singular values of the overlap matrix as a function of time. The horizontal dashed line represents the singular value threshold $s_{SV}$. The time step at which each singular value becomes larger than $s_{SV}$ is marked with a vertical dashed line, connecting upper and lower panels. See text for details.
• Figure 4: Four qubit version of asymptotically optimal phase estimation circuit.
• Figure 5: Accuracy comparison between VQPE and the Heisenberg limit for the ground state energy of LiH. Varying numbers of samples per expectation value are shown for VQPE alongside the idealization of VQPE without finite-sampling errors. We use SVD cutoffs of 2, 0.9, and 0.3 respectively corresponding to $10^3$, $10^4$, and $10^5$ samples and $10^{-6}$ for the ideal case. All methods use $\Delta t = 0.5$. The top and bottom plots show the the maximal evolution time (i.e. how many time steps need to be included for energy convergence), related to the circuit depth, and the total evolution time, i.e. the sum of all time segments needed.