In this work, we study the problems of certifying and learning quantum Ising Hamiltonians. Our main contributions are as follows: Certification of Ising Hamiltonians. We show that certifying an Ising Hamiltonian in normalized Frobenius norm via access to its time-evolution operator requires only $\widetilde O(1/\varepsilon)$ time evolution. This matches the Heisenberg-scaling lower bound of $Ω(1/\varepsilon)$ up to logarithmic factors. To our knowledge, this is the first nearly-optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Lemma from Fourier analysis. Learning Ising Gibbs states. We design an algorithm for learning Ising Gibbs states in trace norm that is sample-efficient in all parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state) but suffered from exponential sample complexity in the inverse temperature. Certification of Ising Gibbs states. We give an algorithm for certifying Ising Gibbs states in trace norm that is both sample and time-efficient, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022). Finally, we extend our results on learning and certification of Gibbs states to general $k$-local Hamiltonians for any constant $k$.
In this work, we study the problems of certifying and learning quantum Ising Hamiltonians. Our main contributions are as follows: Certification of Ising Hamiltonians. We show that certifying an Ising Hamiltonian in normalized Frobenius norm via access to its time-evolution operator requires only O(1/ε) time evolution. This matches the Heisenberg-scaling lower bound of Ω(1/ε) up to logarithmic factors. To our knowledge, this is the first nearly-optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Lemma from Fourier analysis. Learning Ising Gibbs states. We design an algorithm for learning Ising Gibbs states in trace norm that is sample-efficient in all parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state) but suffered from exponential sample complexity in the inverse temperature. Certification of Ising Gibbs states. We give an algorithm for certifying Ising Gibbs states in trace norm that is both sample and time-efficient, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022). Finally, we extend our results on learning and certification of Gibbs states to general k-local Hamiltonians for any constant k.
In this work, we study the problems of certifying and learning quantum k-local Hamiltonians, for a constant k. Our main contributions are as follows:
- Certification of Hamiltonians. We show that certifying a local Hamiltonian in normalized Frobenius norm via access to its time-evolution operator can be achieved with only O(1/ε) evolution time. This is optimal, as it matches the Heisenberg-scaling lower bound of Ω(1/ε). To our knowledge, this is the first optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Hypercontractivity Lemma from Fourier analysis.
- Learning Gibbs states. We design an algorithm for learning Gibbs states of local Hamiltonians in trace norm that is sample-efficient in all relevant parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state), and thus inevitably suffered from exponential sample complexity scaling in the inverse temperature.
- Certification of Gibbs states. We give an algorithm for certifying Gibbs states of local Hamiltonians in trace norm that is both sample and time-efficient in all relevant parameters, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022).
We study the problem of certifying local Hamiltonians from real-time access to their dynamics. Given oracle access to e−itH for an unknown k-local Hamiltonian H and a fully specified target Hamiltonian H0, the goal is to decide whether H is exactly equal to H0 or differs from H0 by at least ε in normalized Frobenius norm, while minimizing the total evolution time. We introduce the first intolerant Hamiltonian certification protocol that achieves optimal performance for all constant-locality Hamiltonians. For general n-qubit, k-local, traceless Hamiltonians, our procedure uses O(ck/ε) total evolution time for a universal constant c, and succeeds with high probability. In particular, for O(1)-local Hamiltonians, the total evolution time becomes Θ(1/ε), matching the known Ω(1/ε) lower bounds and achieving the gold-standard Heisenberg-limit scaling. Prior certification methods either relied on implementing inverse evolution of H, required controlled access to e−itH, or achieved near-optimal guarantees only in restricted settings such as the Ising case (k=2). In contrast, our algorithm requires neither inverse evolution nor controlled operations: it uses only forward real-time dynamics and achieves optimal intolerant certification for all constant-locality Hamiltonians.
Abstract. We study the problem of learning a local quantum Hamiltonian [Formula: see text] given copies of its Gibbs state [Formula: see text] at a known inverse temperature [Formula: see text]. Anshu et al. [2020 IEEE 61st Annual Symposium on Foundations of Computer Science, pp. 685–691] gave an algorithm to learn a Hamiltonian on [Formula: see text] qubits to precision [Formula: see text] with only polynomially many copies of the Gibbs state, but which takes exponential time. Obtaining a computationally efficient algorithm has been a major open problem [ Alhambra, PRX Quantum, 4 (2023), 040201 ; Anshu and Arunachalam, Nature Rev. Phys., 6 (2023), pp. 59–69 ], with prior work only resolving this in the limited cases of high temperature [ Haah, Kothari, and Tang, Markov field on finite graphs and lattices, 1971 ] or commuting terms [ Anshu et al., Efficient learning of commuting Hamiltonians on lattices, 2021 ]. We fully resolve this problem, giving a polynomial time algorithm for learning [Formula: see text] to precision [Formula: see text] from polynomially many copies of the Gibbs state at any constant [Formula: see text]. Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multivariate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system. We then show that solving a low-degree sum-of-squares relaxation of this polynomial system suffices to accurately learn the Hamiltonian.
We study the problem of learning a Hamiltonian H to precision ε, supposing we are given copies of its Gibbs state ρ=exp(−βH)/Tr(exp(−βH)) at a known inverse temperature β. Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] recently studied the sample complexity (number of copies of ρ needed) of this problem for geometrically local N-qubit Hamiltonians. In the high-temperature (low β) regime, their algorithm has sample complexity poly (N, 1/β, 1/ε) and can be implemented with polynomial, but suboptimal, time complexity. In this paper, we study the same question for a more general class of Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error ε with sample complexity S=O(logN/(βε)2) and time complexity linear in the sample size, O(SN). Furthermore, we prove a matching lower bound showing that our algorithm’s sample complexity is optimal, and hence our time complexity is also optimal. In the appendix, we show that virtually the same algorithm can be used to learn H from a real-time evolution unitary e−itH in a small t regime with similar sample and time complexity.
We consider the problems of testing and learning an unknown n-qubit Hamiltonian H from queries to its evolution operator e−iHt under the normalized Frobenius norm. We prove: 1. Local Hamiltonians: We give a tolerant testing protocol to decide if H is ε1-close to k-local or ε2-far from k-local, with O(1/(ε2−ε1)4) queries, solving open questions posed in a recent work by Bluhm et al. For learning a k-local H up to error ε, we give a protocol with query complexity exp(O(k2+klog(1/ε))) independent of n, by leveraging the non-commutative Bohnenblust-Hille inequality. 2. Sparse Hamiltonians: We give a protocol to test if H is ε1-close to being s-sparse (in the Pauli basis) or ε2-far from being s-sparse, with O(s6/(ε22−ε12)6) queries. For learning up to error ε, we show that O(s4/ε8) queries suffice. 3. Learning without memory: The learning results stated above have no dependence on n, but require n-qubit quantum memory. We give subroutines that allow us to learn without memory; increasing the query complexity by a (logn)-factor in the local case and an n-factor in the sparse case. 4. Testing without memory: We give a new subroutine called Pauli hashing, which allows one to tolerantly test s-sparse Hamiltonians with O(s14/(ε22−ε12)18) queries. A key ingredient is showing that s-sparse Pauli channels can be tolerantly tested under the diamond norm with O(s2/(ε2−ε1)6) queries. Along the way, we prove new structural theorems for local and sparse Hamiltonians. We complement our learning results with polynomially weaker lower bounds. Furthermore, our algorithms use short time evolutions and do not assume prior knowledge of the terms in the support of the Pauli spectrum.
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.