Warm-Start Quantum Computation: Spectral-Weight Resources for Energy Readout and State Preparation

Estimating a ground-state energy and preparing a ground state both split into a front-end that supplies an approximate state χ 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 nχ squared 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)=χe-iH Ptχ 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 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 times10-6 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.