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arXiv:2610.12485

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

Tien D. Kieu

quant-phphysics.chem-phSubmitted Sep 17, 2026• Announced Sep 18, 2026• 0 citations• Revised Oct 13, 2026 (v2)
v1 · Sep 17, 2026v2 · Oct 13, 2026
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Abstract

Estimating a ground-state energy and preparing a ground state both split into a front-end that supplies an approximate state $|χ\rangle$ 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 $p_n=|\langle E_n|χ\rangle|^2$ 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 $g(t)=\langleχ|e^{-iH_Pt}|χ\rangle$ 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 $1.4\times10^{-6}$ 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.

Comments
60 pages, 47 figures, and 9 tables

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Estimating a ground-state energy and preparing a ground state both split into a front-end that supplies an approximate state ∣χ⟩|χ\rangle∣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 pn=∣⟨En∣χ⟩∣2p_n=|\langle E_n|χ\rangle|^2pn​=∣⟨En​∣χ⟩∣2 over the eigenstates of the problem Hamiltonian HPH_PHP​. 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)=⟨χ∣e−iHPt∣χ⟩g(t)=\langleχ|e^{-iH_Pt}|χ\rangleg(t)=⟨χ∣e−iHP​t∣χ⟩ 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 50×50\times50× 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(3m)\mathcal{O}(3^m)O(3m) raises the question of whether it can be run at all.
The interactions are the substance. On stretched N2_22​ a sample-based diagonalisation reports a variational bound 79 mHa high, fifty times outside chemical accuracy; reading the state it has already prepared returns E0E_0E0​ to 1.4×10−61.4\times10^{-6}1.4×10−6 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.
Computing the ground state properties of quantum systems is an important potential application of quantum computing. The success probability of quantum phase estimation approaches to ground state problems is proportional to the overlap of the input state with the ground state. Local ansatz state approaches suffer from the orthogonality catastrophe, whereas adiabatic state preparation (ASP) can prepare good approximations with a cost growing with the inverse square of the minimum spectral gap along the adiabatic path. When the initial and target Hamiltonians lie in quantum phases separated by a first order phase transition, the minimum spectral gap along the adiabatic path becomes exponentially small as a function of system size. We pursue a solution to this problem based on choosing an initial Hamiltonian for adiabatic state preparation whose ground state lies in the same quantum phase as the target ground state. We develop a protocol for universal adiabatic warm starts with universal parent Hamiltonians (UPHAWS) that can initialize ASP in any state whose preparation circuit is known. We use the Feynman--Kitaev clock Hamiltonian as a universal parent Hamiltonian, for preparation circuits with and without mid circuit measurement. We benchmark the framework on a Z2\mathbb{Z}_2Z2​-symmetric matrix product state (MPS) family interpolating to a target GHZ Hamiltonian, and on the linear H6H_6H6​ chain under symmetric bond stretching. For the H6H_6H6​ system a bond-dimension-444 matrix product state warm-start increases the minimum gap on the adiabatic path by a factor of two relative to the Hartree-Fock initialization. To perform these classical benchmark simulations we develop a momentum-space truncation of the adiabatic Hamiltonian that may be of independent interest.
Suppose a Hamiltonian HHH has a unique ground state ∣ψ0⟩|ψ_0\rangle∣ψ0​⟩ with an eigenvalue E0E_0E0​, and we have an estimate E~0\tilde{E}_0E~0​ such that ∣E~0−E0∣≤δ|\tilde{E}_0-E_0|\leqδ∣E~0​−E0​∣≤δ, and there is a gap of at least 3δ3δ3δ between E0E_0E0​ and all other eigenvalues. Suppose we have a unitary AAA available that can produce a "guiding state'" A∣0⟩A|0\rangleA∣0⟩ that has overlap at least γγγ with ∣ψ0⟩|ψ_0\rangle∣ψ0​⟩. We show how to obtain an ηηη-approximation of ∣ψ0⟩|ψ_0\rangle∣ψ0​⟩ with probability at least 1−ε1-\varepsilon1−ε using O(log⁡(1/ε)/γδ+log⁡(1/η)/δ)O(\log(1/\varepsilon)/γδ+ \log(1/η)/δ)O(log(1/ε)/γδ+log(1/η)/δ) applications of U=eiHU=e^{iH}U=eiH its inverse, and O(log⁡(1/ε)/γ)O(\log(1/\varepsilon)/γ)O(log(1/ε)/γ) applications of AAA and its inverse. We give two different algorithms, one based on interleaving amplitude amplification and error-reduction in the style of [HMdW03], and one using the composition of transducers. This paper is the state-preparation follow-up to our two recent ground-state-energy estimation papers [JW26,SdW26]. Combined with the optimal energy-estimation of [JW26], our results show the claimed upper bound for ground-state preparation. The bound is optimal up to a constant factor in terms of the number of applications of UUU and U†U^\daggerU† if ε=η\varepsilon=ηε=η, as follows from [SdW26]. In one of our approaches to ground state preparation, we construct transducers for amplitude amplification and LCU that might be of independent interest.
Quantum algorithms for ground-state energy estimation of chemical systems require a high-quality initial state. However, initial state preparation is commonly either neglected entirely, or assumed to be solved by a simple product state like Hartree-Fock. Even if a nontrivial state is prepared, strong correlations render ground-state overlap inadequate for quality assessment. In this work, we address the initial state preparation problem with an end-to-end algorithm that and the quality of initial states, accomplishing the latter with a new metric—the energy distribution. To be able to prepare more complicated initial states, we introduce an implementation technique for states in the form of a sum of Slater determinants that exhibits significantly better scaling than all prior approaches. We also propose low-precision quantum phase estimation (QPE) for further state quality refinement. The complete algorithm is capable of generating high-quality states for energy estimation, and is shown in select cases to lower the overall estimation cost by several orders of magnitude when compared with the best single product state ansatz. More broadly, the energy distribution picture suggests that the goal of QPE should be reinterpreted as generating improvements compared to the energy of the initial state and other classical estimates: such an improvement can still be achieved even if QPE does not project directly onto the ground state. Finally, we show how the energy distribution can help in identifying potential quantum advantage. Published by the American Physical Society 2024
We derive eigenstate comparisons from ordered Schrödinger operators, extending energy bounds to overlaps, spatial distributions and nodes. For three form-ordered operators with discrete comparison ground states, suppose the upper ground-state energy lies in the lower operator's ground-state gap. Three comparison spectral values then bound all three squared ground-state overlaps and guarantee a simple discrete intermediate ground state. Related gap estimates control low-energy subspaces and individual excited-state overlaps under additional gap conditions. In one dimension and radially, a single crossing of the potential difference relative to the energy shift orders ground-state likelihood ratios and monotone observables; a nondecreasing difference suffices. The results cover asymmetric wells and shortened right Dirichlet walls, with coordinatewise comparisons for submodular reference potentials and coordinatewise nondecreasing perturbations. A two-sided Sturm argument orders corresponding internal nodes for nondecreasing potential differences, even between orthogonal one-dimensional excited states. For Yukawa ground states, Hulthén comparisons bound overlaps, radial moments and the density at the nucleus; the latter two-sided bounds hold for screening 0<δ<2/60<δ<2/\sqrt60<δ<2/6​. Optimizing scaled Hulthén majorants gives a lower bound of approximately 1.136111.136111.13611 for n2δc(n,0)n^2δ_c(n,0)n2δc​(n,0), where δc(n,0)δ_c(n,0)δc​(n,0) is the critical screening for the nnnth sss-wave bound state. This bound is optimal within that two-parameter comparison family.
We develop a phase-estimation method with a distinct feature: its maximal run time (which determines the circuit depth) is δ/ , where is the target precision, and the preconstant δ can be arbitrarily close to 0 as the initial state approaches the target eigenstate.The total cost of the algorithm satisfies the Heisenberg-limited scaling O( -1 ).As a result, our algorithm may significantly reduce the circuit depth for performing phase-estimation tasks on early fault-tolerant quantum computers.The key technique is a simple subroutine called quantum complex exponential least squares (QCELS).Our algorithm can be readily applied to reduce the circuit depth for estimating the ground-state energy of a quantum Hamiltonian, when the overlap between the initial state and the ground state is large.If this initial overlap is small, we can combine our method with the Fourier-filtering method developed in [Lin and Tong, PRX Quantum 3, 010318, 2022], and the resulting algorithm provably reduces the circuit depth in the presence of a large relative overlap compared to .The relative-overlap condition is similar to a spectral-gap assumption but it is aware of the information in the initial state and is therefore applicable to certain Hamiltonians with small spectral gaps.We observe that the circuit depth can be reduced by around 2 orders of magnitude in numerical experiments under various settings.
Developing efficient quantum computing algorithms is essential for tackling computationally challenging problems across various fields. This paper presents a novel quantum algorithm, XZ24, for efficiently computing the eigen-energy spectra of arbitrary quantum systems. Given a Hamiltonian H^\hat{H}H^ and an initial reference state ∣ψref⟩|ψ_{\text{ref}} \rangle∣ψref​⟩, the algorithm extracts information about ⟨ψref∣cos⁡(H^t)∣ψref⟩\langle ψ_{\text{ref}} | \cos(\hat{H} t) | ψ_{\text{ref}} \rangle⟨ψref​∣cos(H^t)∣ψref​⟩ from an auxiliary qubit's state. By applying a Fourier transform, the algorithm resolves the energies of eigenstates of the Hamiltonian with significant overlap with the reference wavefunction. We provide a theoretical analysis and numerical simulations, showing XZ24's superior efficiency and accuracy compared to existing algorithms. XZ24 has three key advantages: 1. It removes the need for eigenstate preparation, requiring only a reference state with non-negligible overlap, improving upon methods like the Variational Quantum Eigensolver. 2. It reduces measurement overhead, measuring only one auxiliary qubit. For a system of size LLL with precision εεε, the sampling complexity scales as O(L⋅ε−1)O(L \cdot ε^{-1})O(L⋅ε−1). When relative precision εεε is sufficient, the complexity scales as O(ε−1)O(ε^{-1})O(ε−1), making measurements independent of system size. 3. It enables simultaneous computation of multiple eigen-energies, depending on the reference state. We anticipate that XZ24 will advance quantum system simulations and enhance applications in quantum computing.