Warm-Start Quantum Computation for Energy Readout and State Preparation

Abstract

Estimating a ground-state energy and preparing a ground state both split into a front-end that supplies an approximate state $|\chi ⟩$ and a back-end that reads or refines it. We argue, and measure at every step, that the only thing crossing that split is the front-end's spectral weight distribution $p_n=|⟨E_n|\chi ⟩{|}^2$ over the eigenstates of the problem Hamiltonian $H_P$. Everything downstream is priced in it, so the front-end is a swappable module, and we swap it seven times without altering anything to its right. We call the arrangement WARP. Four back-ends spend that distribution. A readout (A) samples $g(t)=⟨\chi |e^{-i{H}_{P}t}|\chi ⟩$ on the device and extracts true eigenvalues by classical signal processing, returning an energy below the variational bound of the state it was handed, from a signal $50\times$ shorter than a Fourier transform needs. Three routes spend the same account on a state instead. An adiabatic schedule (B) has a minimum gap set by the overlap; a discrete amplification (C) is cheaper in depth but must halt at an index the overlap fixes, so it needs a measured value and not a bound; a recursion (D) of depth $\mathcal{O}(3^m)$ raises the question of whether it can be run at all. The interactions are the substance. On stretched N${}_2$ a sample-based diagonalisation reports a variational bound 79 mHa high, fifty times outside chemical accuracy; reading the state it has already prepared returns $E_0$ to $1.4\times 10^{-6}$ Ha, three orders of magnitude inside it, and needs no new front-end and no new circuit. And because the weights price everything, they must be measurable when scarce, so we beat a weak reference against a strong one and recover overlaps four orders of magnitude below the shot-noise floor. We map the operating envelope throughout, and state the limitations.

59 pages, 47 figures, and 9 tables