Adiabatic state preparation from general initial states

TL;DR

The paper tackles the challenge of improving ground-state preparation by enabling adiabatic state preparation (ASP) from a general, efficiently preparable trial state. It builds a parent Hamiltonian for the trial state via a measurement-based, covariance-driven kernel to generate a proto-parent and then applies spectrum folding to produce a valid HP whose ground state includes the trial state, followed by an adiabatic path to the target Hamiltonian. The approach is demonstrated on water dissociation and singlet methylene, showing that Krylov-based trial states can reduce the required adiabatic time relative to standard RHF-based ASP, and that the method is robust to certain practical issues like shot noise when the covariance matrix is well conditioned. While promising, the framework relies on careful operator selection and faces scaling and convergence challenges due to spectrum folding and nonconvex optimization, highlighting directions for improved cost functions, symmetry handling, and applicability to strongly correlated systems.

Abstract

A variety of quantum computing algorithms exist for the preparation of approximate Hamiltonian ground states. A natural and important question is how these ground-state approximations can be further improved using adiabatic state preparation. Here, we present a heuristic method to carry out adiabatic state preparation starting from a generic initial wavefunction. Given a quantum circuit that prepares the initial wavefunction, and a target Hamiltonian for which one wishes to prepare the ground state, we present an algorithm to construct an adiabatic path between these two states. This method works by approximating a parent Hamiltonian for the initial wavefunction, and the quality of this approximation can be can be checked prior to running the ASP algorithm. We apply this technique to simulate the ground state of water and the lowest-lying multireference singlet state of methylene, using various initial wavefunctions as the starting point of the adiabatic path.

Figures (3)

• Figure 1: (a) Estimate of the adiabatic time $T_{\mathrm{est}}$ along dissociation of both OH bonds in the H$_2$O molecule, using a (4e,4o) active space. $T_{\mathrm{est}}$ is estimated for standard adiabatic state preparation (i.e. with RHF trial and the Fock operator as its parent, red circles connected by dashed lines) and for different trial wavefunctions (red, green, blue symbols for RHF, MP2, Krylov-space wavefunctions respectively) in combination with the parent Hamiltonian construction proposed in this work. (b) Spectral gap $\gamma$ of the parent Hamiltonians obtained by applying our method to various trial wavefunctions, compared against the spectral gap of the Fock operator used in standard adiabatic state preparation. For both subplots, $R$ is a dimensionless quantity which scales the length of all molecular bond lengths relative to the equillibrium geometry.
• Figure 2: (a) Estimate of the adiabatic time $T_{\mathrm{est}}$ along dissociation of both OH bonds in the H$_2$O molecule, using a (8e,6o) active space. $T_{\mathrm{est}}$ is estimated for standard adiabatic state preparation (i.e. with RHF trial and the Fock operator as its parent, red circles connected by dashed lines) and for different trial wavefunctions (red, green, blue symbols for RHF, MP2, Krylov-space wavefunctions respectively) in combination with the parent Hamiltonian construction proposed in this work. (b) Spectral gap $\gamma$ of the parent Hamiltonians obtained by applying our method to various trial wavefunctions, compared against the spectral gap of the Fock operator used in standard adiabatic state preparation. For both subplots, $R$ is a dimensionless quantity which scales the length of all molecular bond lengths relative to the equillibrium geometry
• Figure 3: Estimate of the adiabatic time $T_{\mathrm{est}}$ for singlet methylene at equilibrium geometry, using a (8e,6o) active space. $T_{\mathrm{est}}$ is estimated for standard adiabatic state preparation (i.e. with RHF trial and the Fock operator as its parent, red solid line), for an RHF trial with the parent Hamiltonian construction proposed in this work (red dashed line), and for Krylov-space wavefunctions of increasing dimension (blue circles, with darker colors corresponding to higher $d$).