The low-temperature properties of interacting quantum systems are believed to require exponential resources to compute in the general case. Quantifying the extent to which such properties can be approximated using efficient algorithms remains a significant open challenge. Here, we consider the task of approximating the ground state energy of two-local quantum Hamiltonians with bounded-degree interaction graphs. Most existing algorithms optimize the energy over the set of product states. We propose and analyze a family of shallow quantum circuits that can be used to improve the approximation ratio achieved by a given product state. The algorithm takes as input an n-qubit product state with variance Var and improves its energy by an amount proportional to Var^{2}/n. In a typical case, this results in an extensive improvement in the estimated energy. We extend our results to k-local Hamiltonians and entangled initial states.
The low-temperature properties of interacting quantum systems are believed to require exponential resources to compute in the general case. Quantifying the extent to which such properties can be approximated using efficient algorithms remains a significant open challenge. Here, we consider the task of approximating the ground state energy of two-local quantum Hamiltonians with bounded-degree interaction graphs. Most existing algorithms optimize the energy over the set of product states. We propose and analyze a family of shallow quantum circuits that can be used to improve the approximation ratio achieved by a given product state. The algorithm takes as input an n-qubit product state with variance Var and improves its energy by an amount proportional to Var^{2}/n. In a typical case, this results in an extensive improvement in the estimated energy. We extend our results to k-local Hamiltonians and entangled initial states.
We consider a computational problem where the goal is to approximate the maximum eigenvalue of a two-local Hamiltonian that describes Heisenberg interactions between qubits located at the vertices of a graph. Previous work has shed light on this problem's approximability by product states. For any instance of this problem the maximum energy attained by a product state is lower bounded by the Max Cut of the graph and upper bounded by the standard Goemans-Williamson semidefinite programming relaxation of it. Gharibian and Parekh described an efficient classical approximation algorithm for this problem which outputs a product state with energy at least 0.498 times the maximum eigenvalue in the worst case, and observe that there exist instances where the best product state has energy 1/2 of optimal. We investigate approximation algorithms with performance exceeding this limitation which are based on optimizing over tensor products of few-qubit states and shallow quantum circuits. We provide an efficient classical algorithm which achieves an approximation ratio of at least 0.53 in the worst case. We also show that for any instance defined by a 3- or 4-regular graph, there is an efficiently computable shallow quantum circuit that prepares a state with energy larger than the best product state (larger even than its semidefinite programming relaxation).
The ground state energy and the free energy of Quantum Local Hamiltonians are fundamental quantities in quantum many-body physics, however, it is QMA-Hard to estimate them in general. In this paper, we develop new techniques to find classical, additive error product-state approximations for these quantities on certain families of Quantum k-Local Hamiltonians. Namely, those which are either dense, have low threshold rank, or are defined on a sparse graph that excludes a fixed minor, building on the methods and the systems studied by Brandão and Harrow, Gharibian and Kempe, and Bansal, Bravyi and Terhal. We present two main technical contributions. First, we discuss a connection between product-state approximations of local Hamiltonians and combinatorial graph property testing. We develop a series of weak Szemerédi regularity lemmas for k-local Hamiltonians, built on those of Frieze and Kannan and others. We use them to develop constant time sampling algorithms, and to characterize the `vertex sample complexity' of the Local Hamiltonian problem, in an analog to a classical result by Alon, de la Vega, Kannan and Karpinski. Second, we build on the information-theoretic product-state approximation techniques by Brandão and Harrow, extending their results to the free energy and to an asymmetric graph setting. We leverage this structure to define families of algorithms for the free energy at low temperatures, and new algorithms for certain sparse graph families.
Near-Optimal Bounds on the Density of Low-Energy States of k-Local Hamiltonians and Faster Quantum Algorithms
Low-energy estimation and state preparation for general k-local Hamiltonians are fundamental challenges in quantum complexity theory. Buhrman et al.~ [BGLGST, PRL 2025] recently broke the natural Grover bound O∗(2n/2) for both problems, with the improvement depending on the relative accuracy ε and the locality k. In this work, we present faster exponential quantum algorithms for these problems, where the binary entropy function governs the runtime exponent. For sufficiently small ε/k, our algorithms improve the exponent by a factor of log(k/ε) over [BGLGST, PRL 2025]. Our main technical result is an entropy-governed lower bound on the dimension of the Hamiltonian's low-energy subspace, obtained by depolarizing its ground state. For fixed k, this bound is optimal up to constant factors in the exponent. The same framework yields tighter bounds for Heisenberg, XY, and Ising models on arbitrary interaction graphs.
Computing many-body ground state energies and resolving electronic structure calculations are fundamental problems for fields such as quantum chemistry or condensed matter. Several quantum computing algorithms that address these problems exist, although it is often challenging to establish rigorous bounds on their performances. Here we detail three algorithms that produce approximate ground states for many-body and electronic structure problems, generalizing some previously known results for 2-local Hamiltonians. Each method comes with asymptotic bounds on the energies produced. The first one produces a separable state which improves on random product states. We test it on a spinless Hubbard model, validating numerically the theoretical result. The other two algorithms produce entangled states via shallow or deep circuits, improving on the energies of given initial states. We demonstrate their performance via numerical experiments on a 2-dimensional Hubbard model, starting from a checkerboard product state, as well as on some chemistry Hamiltonians, using the Hartree-Fock state as reference. In both cases, we show that the approximate energies produced are close to the exact ones. These algorithms provide a way to systematically improve the estimation of ground state energies and can be used stand-alone or in conjunction with existing quantum algorithms for ground states.
Estimating ground state energies of many-body Hamiltonians is a central task in many areas of quantum physics. In this Letter, we give quantum algorithms which, given any k-body Hamiltonian H, compute an estimate for the ground state energy and prepare a quantum state achieving said energy, respectively. Specifically, for any ϵ>0, our algorithms return, with high probability, an estimate of the ground state energy of H within additive error ϵM, or a quantum state with the corresponding energy. Here, M is the total strength of all interaction terms, which in general is extensive in the system size. Our approach makes no assumptions about the geometry or spatial locality of interaction terms of the input Hamiltonian and thus handles even long-range or all-to-all interactions, such as in quantum chemistry, where lattice-based techniques break down. In this fully general setting, the run-time of our algorithms scales as 2^{cn/2} for c<1, yielding the first quantum algorithms for low-energy estimation breaking a standard square root Grover speedup for unstructured search. The core of our approach is remarkably simple, and relies on showing that an extensive fraction of the interactions can be neglected with a controlled error. What this ultimately implies is that even arbitrary k-local Hamiltonians have structure in their low energy space, in the form of an exponential-dimensional low energy subspace.
Low-energy estimation and state preparation for general k-local Hamiltonians are fundamental challenges in quantum complexity theory. For constant relative accuracy, Buhrman et al. (PRL 2025) recently broke the natural Grover bound O(2n/2), where n denotes the number of qubits, for both problems. In this paper, for any sufficiently small parameter d≥0, we present an even faster quantum algorithm that outputs a quantum state with energy bounded by the minimum energy over all depth-d states (i.e., states obtained by applying a depth-d circuit to the all-zero state), together with an estimate of this energy. For the class of Hamiltonians with depth-d ground states, our algorithm furthermore achieves exactly the same energy guarantees as Buhrman et al. Our results also provide insight into the distinction between strongly entangled states and those admitting efficient classical descriptions.