In the problem of ground-state energy estimation, one aims to estimate the smallest eigenvalue of a Hamiltonian, often given a guiding state, with some promised overlap $γ$ with the ground space. The main approach to this problem is to simulate its evolution, and estimate the smallest (or equivalently, largest) eigenphase of the resulting unitary $U$. We give a quantum algorithm that estimates the largest eigenphase of a unitary $U$ in this guided setting using a factor of $\log\frac{1}γ$ fewer queries to $U$ than the previous best approach. The result matches an existing lower bound, and answers an open question from Mande and de Wolf. The algorithm is based on transducers, which often allow composition of quantum algorithms without overhead from error reduction.
In the problem of ground-state energy estimation, one aims to estimate the smallest eigenvalue of a Hamiltonian, often given a guiding state, with some promised overlap γ with the ground space. The main approach to this problem is to simulate its evolution, and estimate the smallest (or equivalently, largest) eigenphase of the resulting unitary U. We give a quantum algorithm that estimates the largest eigenphase of a unitary U in this guided setting using a factor of logγ1 fewer queries to U than the previous best approach. The result matches an existing lower bound, and answers an open question from Mande and de Wolf. The algorithm is based on transducers, which often allow composition of quantum algorithms without overhead from error reduction.
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.
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.
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.
We construct classical algorithms computing an approximation of the ground state energy of an arbitrary k-local Hamiltonian acting on n qubits. We first consider the setting where a good ``guiding state'' is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation (i.e., an approximation with constant relative accuracy) of the ground state energy can be computed classically in poly(1/χ,n) time and poly(n) space, where χ denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up a to polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in 2O(n) time and poly(n) space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results.
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.
We construct classical algorithms computing an approximation of the ground state energy of an arbitrary k-local Hamiltonian acting on n qubits. We first consider the setting where a good "guiding state" is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation (i.e., an approximation with constant relative accuracy) of the ground state energy can be computed classically in poly (1/χ,n) time and poly(n) space, where χ denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up to a polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in 2^O(n) time and poly(n) space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results.