Rapid initial state preparation for the quantum simulation of strongly correlated molecules

TL;DR

This work tackles the expensive resource cost of quantum phase estimation for ground-state energy calculations when the input state has imperfect overlap with the true ground state. It delivers two complementary advances: (1) a significantly more efficient matrix-product-state (MPS) preparation via a new unitary-synthesis scheme that reduces Toffoli count by about a factor of 7, enabling rapid generation of high-overlap initial states; (2) two filtering strategies, sampling and binary search, improved by window-function phase estimation (Kaiser and prolate spheroidal windows) to minimize error tails and overall runtime. The binary-search approach offers a square-root speedup in the overlap $p$ at the cost of larger constants, becoming advantageous for very small $p\lesssim 0.003$, while sampling excels at moderate to large overlaps. Applying these methods to Fe-S cluster problems, including FeMo-cofactor, the authors demonstrate feasible resource estimates (e.g., $7.3\times 10^{10}$ Toffoli gates for FeMoco under plausible overlaps) and present an overlap-extrapolation protocol to inform initial-state choices, illustrating a practical pathway to high-overlap state preparation in difficult chemical systems.

Abstract

Studies on quantum algorithms for ground state energy estimation often assume perfect ground state preparation; however, in reality the initial state will have imperfect overlap with the true ground state. Here we address that problem in two ways: by faster preparation of matrix product state (MPS) approximations, and more efficient filtering of the prepared state to find the ground state energy. We show how to achieve unitary synthesis with a Toffoli complexity about $7 \times$ lower than that in prior work, and use that to derive a more efficient MPS preparation method. For filtering we present two different approaches: sampling and binary search. For both we use the theory of window functions to avoid large phase errors and minimise the complexity. We find that the binary search approach provides better scaling with the overlap at the cost of a larger constant factor, such that it will be preferred for overlaps less than about $0.003$. Finally, we estimate the total resources to perform ground state energy estimation of Fe-S cluster systems, including the FeMo cofactor by estimating the overlap of different MPS initial states with potential ground-states of the FeMo cofactor using an extrapolation procedure. {With a modest MPS bond dimension of 4000, our procedure produces an estimate of $\sim 0.9$ overlap squared with a candidate ground-state of the FeMo cofactor, producing a total resource estimate of $7.3 \times 10^{10}$ Toffoli gates; neglecting the search over candidates and assuming the accuracy of the extrapolation, this validates prior estimates that used perfect ground state overlap. This presents an example of a practical path to prepare states of high overlap in a challenging-to-compute chemical system.

Figures (20)

• Figure 1: The sequence of operations used to prepare a MPS. The input state $\ket{0}$ second from the top is of dimension $\chi$, as is the final output state at the bottom $\ket{0}$.
• Figure 2: The windows (a) and error probability distribution (b) for phase measurements. In red is the flat window, and in green is the cosine window, which provides the minimum phase variance. In blue is the Kaiser window with $\alpha\approx 1.5$, and the black crosses are the Slepian window with $c=5$.
• Figure 3: The decomposition of a multiport interferometer for 8 modes, equivalent to a transformation on 3 qubits. The triangular Reck-Zeilinger Reck decomposition is in (a), and (b) is the rectangular decomposition from Ref. Clements.
• Figure 4: The crossings depicted in Fig. \ref{['fig:interferometer']} are general beam splitters with arbitrary phases and reflectivities (left). They can be implemented with two 50/50 beam splitters with phase shifts (right).
• Figure 5: Two layers of phases controlled by the first $n-1$ qubits and Hadamards on the last qubit. The boxes labelled $R(\phi)$ and $R(\theta)$ are phase rotations controlled by the first $n-1$ qubits.