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 $0<δ<2/\sqrt6$. Optimizing scaled Hulthén majorants gives a lower bound of approximately $1.13611$ for $n^2δ_c(n,0)$, where $δ_c(n,0)$ is the critical screening for the $n$th $s$-wave bound state. This bound is optimal within that two-parameter comparison family.
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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<2/6. Optimizing scaled Hulthén majorants gives a lower bound of approximately 1.13611 for n2δc(n,0), where δc(n,0) is the critical screening for the nth s-wave bound state. This bound is optimal within that two-parameter comparison family.
Suppose we can apply the unitary U=eiH for some Hamiltonian H, and are given access to a unitary that prepares a guiding state promised to have overlap at least γ>0 with the ground space of H. Our goal is to estimate the ground-state energy of H within additive error δ>0 and success probability at least 1−ε, ε>0. How many applications of U and its inverse U−1 are necessary and sufficient? This quantity corresponds to the total Hamiltonian-simulation time needed. An upper bound O(log(1/ε)log(1/γ)/γδ) was known, and was improved to O(log(1/ε)/γδ) very recently [JW26]. A matching lower bound was known whenever one of the three parameters δ,γ,ε was held constant [MdW26]. In this paper we prove the joint lower bound Ω(log(1/ε)/γδ) with the tight ε-dependence provided the dimension of H is at least log(1/ε)/γ2. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and H has a gap of δ between its first and second eigenvalue; and for ground-state preparation, where δ denotes the spectral gap and ε now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of U are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when H is nonnegative and presented as a sum of squares; and our results imply the lower bound Ω(log(1/ε)/γδ) for this case.
Suppose a Hamiltonian H has a unique ground state ∣ψ0⟩ with an eigenvalue E0, and we have an estimate E~0 such that ∣E~0−E0∣≤δ, and there is a gap of at least 3δ between E0 and all other eigenvalues. Suppose we have a unitary A available that can produce a "guiding state'" A∣0⟩ that has overlap at least γ with ∣ψ0⟩. We show how to obtain an η-approximation of ∣ψ0⟩ with probability at least 1−ε using O(log(1/ε)/γδ+log(1/η)/δ) applications of U=eiH its inverse, and O(log(1/ε)/γ) applications of A 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 U and U† if ε=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.
We determine the optimal query complexity of ground-state preparation to trace-distance error ε when an energy threshold in the spectral gap is known. Let UH be an α-block-encoding of a Hamiltonian with unique ground state ∣ψ0⟩, and suppose ∣⟨ψ0∣UI∣0⟩∣≥γ for a state-preparation oracle UI. The threshold lies at least Δ/2 above the ground-state energy and at least Δ/2 below every excited-state energy. We give two algorithms that prepare a state within trace distance ε of the ground state. One uses O((α/Δ)(γ−1+log(1/ε))) calls to UH in expectation; the other uses O((α/(γΔ))log(1/ε)) calls to UH in the worst case. We prove a lower bound matching the expected query count; the corresponding worst-case lower bound follows from Somma and de Wolf [SdW26]. The respective bounds on calls to UI are O(1/γ) in expectation and O(γ−1log(1/ε)) in the worst case. On (N+1)-dimensional systems, these UI bounds are also optimal when the expected or worst-case count of UH calls, respectively, is o((α/Δ)N). Both algorithms use a constant-accuracy spectral filter to construct a purifier, which we then sequentially compose during amplitude amplification to prepare a state with constant overlap with the ground state. The expected-query algorithm repeats the preparation followed by one high-accuracy spectral filter until success. The worst-case algorithm uses filters of increasing accuracy and limits the total number of queries.
Ground-state preparation, energy estimation, and property estimation are fundamental tasks in quantum simulation. Existing ground-state preparation algorithms achieve near-optimal complexity using block encodings or controlled Hamiltonian evolution, raising a natural question: is controlled Hamiltonian evolution necessary, or can the same scaling be attained in the weaker access model that provides access to only uncontrolled Hamiltonian evolution? We resolve this question by giving two algorithms that access the input Hamiltonian H only through queries to UH and its inverse UH†, where ∥H∥≤αH and τ=Θ(1/αH). Given an input state with ground-state weight at least η and a spectral gap of H of at least γ, our first algorithm uses fresh copies of the input state and requires O(αH/(γη)) queries to UH and UH†, while our second algorithm uses coherent access to the input-state preparation unitary and its inverse to improve this complexity to O(αH/(γη)). The latter bound matches the known lower bound up to logarithmic factors, showing that algorithms with additional access to controlled Hamiltonian evolution cannot improve the query complexity for ground-state preparation beyond logarithmic factors. Both algorithms are based on a simple mechanism that we call the spectral descent principle, in which the algorithm starts from the input state ∣ψ⟩ and progressively descends through the energy spectrum, moving closer to the ground state until the desired fidelity is reached. We also extend this framework to ground-state property and energy estimation and give explicit complexity guarantees for both tasks.
Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.
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 pn=∣⟨En∣χ⟩∣2 over the eigenstates of the problem Hamiltonian HP. 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⟩ 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× 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) raises the question of whether it can be run at all.
The interactions are the substance. On stretched N2 a sample-based diagonalisation reports a variational bound 79 mHa high, fifty times outside chemical accuracy; reading the state it has already prepared returns E0 to 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.