Quantum State Preparation with Resolution Refinement

TL;DR

The paper addresses the challenge of preparing ground states in quantum many-body systems on quantum computers and introduces resolution refinement to bootstrap eigenstates from a low-resolution to a high-resolution Hamiltonian. It formalizes an adiabatic path between $H_{low}-\mu$ and $H_{high}-\mu$ using a prolongation operator $P$, and shows the fidelity loss obeys $P_{loss} \le 4B/(T^2 \Delta E_{min}^2) + O(T^{-3})$, yielding $T = O\left(\frac{\sqrt{B}}{\sqrt{\epsilon}\Delta E_{min}}\right)$, with $B = O(N)$ for extensive $V$. Benchmark results cover basis refinement with a Busch model in 1D, lattice refinement with Hartree-Fock nuclear states in a central Woods-Saxon potential in 3D, and a 1D multi-species Hubbard model, demonstrating that overlap probabilities approach 1 as the adiabatic time grows and that energy-flow along the interpolation is smooth. The method leverages low-energy structure preservation to achieve favorable scaling and can be augmented with filtering methods; while broadly applicable, its success may degrade for systems where the low-resolution Hamiltonian fails to capture essential correlations.

Abstract

We introduce a method called resolution refinement that allows one to bootstrap eigenstate preparation on a quantum computer. We first prepare an eigenstate of a low-resolution Hamiltonian using any method of choice. The eigenstate is then lifted to higher resolution and adiabatically evolved to produce the corresponding eigenstate of a higher-fidelity Hamiltonian. We give examples of resolution refinement applied to both single-particle basis states as well as a spatial lattice grid. For basis refinement, we compute few-body ground states of the Busch model for interacting particles in a harmonic trap in one dimension. For lattice refinement, we compute Hartree-Fock nuclear states for a central Woods-Saxon potential in three dimensions, and we compute bound states and continuum states in a multi-species Hubbard model of fermions in one dimension. In all cases, the method is efficient and requires an adiabatic evolution time that scales with the inverse of the energy gap times the square root of the system size. We show that this very favorable scaling arises from the fact that resolution refinement does not make large changes to the structure or energies of the low-energy eigenstates.

Figures (6)

• Figure 1: Plot of the ground state overlap probabilities versus total adiabatic evolution time for $K=N=2$, $K=N=3$, and $K=N=4$ for the Busch model in one dimension.
• Figure 2: Sketch of the fine lattice grid with spacing $a$ and the coarse lattice grid with spacing $2a$.
• Figure 3: The circuit decomposition for the unitary $U$ operator built from single-qubit rotations and two CNOT gates.
• Figure 4: Plot of the overlap probabilities versus total adiabatic evolution time for $^4$He, $^{16}$O, $^{24}$Mg, $^{28}$Si and $^{40}$Ca.
• Figure 5: Single-particle energy levels for one nucleon species for the $^{40}$Ca calculation as a function of interpolation parameter $\lambda$. We have labeled the corresponding irreducible cubic representations.