Molecular Properties from Quantum Krylov Subspace Diagonalization

TL;DR

This work addresses the challenge of obtaining nuclear energy gradients within quantum Krylov subspace diagonalization for molecular systems. It derives analytical first-order derivatives and shows how to obtain relaxed one- and two-particle density matrices of Krylov eigenstates, then reduces the otherwise quadratic measurement cost by directly preparing Krylov eigenstates with quantum signal processing, using a coherent measurement scheme to lower estimator variance. The approach employs block-encoding and qubitization to realize polynomial transforms of the Hamiltonian via quantum signal processing, with Chebyshev polynomials serving as a convenient Krylov basis. Numerical benchmarks on H$_2$O demonstrate that the coherent gradient-measurement strategy lowers gradient-variance by about a factor of two, while the regularization threshold and Krylov dimension balance energy accuracy against measurement overhead. Overall, the results provide a practical route to analytic gradients in Krylov-based quantum chemistry, with potential extensions to excited states and density-m matrix properties.

Abstract

Quantum Krylov subspace diagonalization is a prominent candidate for early fault tolerant quantum simulation of many-body and molecular systems, but so far the focus has been mainly on computing ground-state energies. We go beyond this by deriving analytical first-order derivatives for quantum Krylov methods and show how to obtain relaxed one and two particle reduced density matrices of the Krylov eigenstates. The direct approach to measuring these matrices requires a number of distinct measurement that scales quadratically with the Krylov dimension $D$. Here, we show how to reduce this scaling to a constant. This is done by leveraging quantum signal processing to prepare Krylov eigenstates, including exited states, in depth linear in $D$. We also compare several measurement schemes for efficiently obtaining the expectation value of an operator with states prepared using quantum signal processing. We validate our approach by computing the nuclear gradient of a small molecule and estimating its variance.

Figures (4)

• Figure 1: Quantum circuit for block-encoding a polynomial $P(x)$ of fixed parity using quantum signal processing as defined in Eq. \ref{['eq:qsp']}.
• Figure 2: Quantum circuit template for block-encoding the real part of $P(x)$ presented in Eq. \ref{['eq:qsp']}. where $U_\Phi$ the circuit depicted in Figure \ref{['fig:circiut_1']} for block-encoding $P$ and $P^*$ respectively.
• Figure 3: Total variance, i.e. sum of the variance of all the nuclear gradient $dE_0/dx$ vector components, as a function of the Krylov dimension. The blue and orange plot represent the variance of the coherent approach, that measures $\hat{P_\nu}$ and $\hat{R_{0}}\hat{P_\nu}$, and the post-selection approach respectively for $s=10^{-3}$.
• Figure 4: FIG \ref{['fig:eta_norm_vs_degree']} infinity norm $\eta$ and FIG. \ref{['fig:delta_vs_degree']} the absolute difference of the Krylov minimal energy and the ground-state energy.