Learning a Quantum System’s Rules at Any Temperature
A flat polynomial approximation enables learning a quantum Hamiltonian from thermal states at any constant temperature.
12 May 2026, Yanjiang
Imagine trying to guess the rules of a game by watching only the final board position after many moves have been made. The pieces are scattered, the path that led here is hidden — but from the static arrangement alone, you must infer the allowed moves, the scoring, the very logic of the game. This is, in a sense, the problem physicists have been wrestling with when it comes to learning the “rules” — the Hamiltonian — of a quantum system from its thermal equilibrium state.
In a new preprint on arXiv (arXiv:2310.02243), a team led by Ewin Tang — spanning MIT and UC Berkeley — has now fully solved this problem, giving an algorithm that can learn a local quantum Hamiltonian from copies of its thermal (Gibbs) state in polynomial time, at any constant temperature. It is the culmination of a research arc that began with a landmark paper in 2020, which showed that the sample complexity — the number of copies needed — could be polynomially bounded, but left open the question of whether a computationally efficient algorithm existed at all. The answer, as it turns out, is yes.
The Hamiltonian Learning Problem
To understand why this is a big deal, let us first be precise about the problem. A local quantum Hamiltonian is an operator that describes the energy of a system, written as a sum of terms that each involve only a few particles (typically nearby neighbours on a lattice). Knowing H is essentially knowing everything about the system’s dynamics — its ground states, its excitations, its response to probes. The trouble is that we rarely have direct access to H; what we can measure are copies of the system’s thermal state at a given inverse temperature beta = 1/(kBT). The Gibbs state rho = e^{–betaH} / Z is the system’s equilibrium probability distribution over all possible quantum states, weighted by their Boltzmann factor.
The problem is: given many copies of rho (but not beta, which is assumed known), reconstruct H to a desired precision ϵ. This is a kind of quantum analogue of classical graphical model learning, where one tries to infer the interactions between variables from observed samples. But in the quantum case, the state space is exponentially large, and the Gibbs state itself is a highly non-trivial function of the Hamiltonian. The stakes are high: success would open the door to automatically discovering the effective Hamiltonians of quantum materials, quantum simulators, and even possibly biological systems that exhibit coherent quantum effects.
Why It Was So Hard
In 2020, Anshu, Arunachalam, Kuwahara, and Soleimanifar gave the first algorithm that required only polynomially many copies — exponentially fewer than a naive approach would demand — but their method ran in exponential time. This was like having a key that unlocked the door, but only after months of labour. For a system of n qubits, the total number of parameters in a local Hamiltonian grows as O(n), so one might hope for an algorithm that scales as some polynomial in n. The exponential runtime came from the need to search over all possible Hamiltonians of a given form, a brute-force approach that could not be improved with any known tricks, except in special cases.
Subsequent work made progress in limited settings: at very high temperature (large beta ≈ 0), where the Gibbs state is nearly maximally mixed and simpler structure emerges, polynomial-time algorithms were found. Also, when all terms in the Hamiltonian commute — like the classical Ising model — the problem becomes essentially classical and similarly tractable. But the general case, where quantum effects such as superposition and entanglement are fully operative at arbitrary constant temperature, remained stubbornly open.
The Flat Polynomial Insight
What the MIT—Berkeley team accomplished is to completely close this gap. Their main technical contribution is a new flat polynomial approximation to the exponential function. Here is the core idea: the Gibbs state rho is proportional to e^{–betaH}. If we could somehow write e^{–betaH} as a polynomial in H — or more precisely, as a sum of terms that are low-degree polynomials in the coupling constants defining H — then the measurement statistics from the Gibbs state would become polynomial constraints on those unknown constants. Solving the constraint system would yield the Hamiltonian.
But there is a catch: e^{–betaH} is not a polynomial in H; it is an analytic function with an infinite series expansion. However, over the domain of possible energy values (which is bounded because H is local on a finite system), e^{–betaH} can be approximated by a polynomial. The team discovered a construction — using Chebyshev polynomials — that gives a uniform approximation of exponential decay over an interval, with a remarkable property: the polynomial’s coefficients are all positive. This “flat” polynomial allows them to replace the exponential with a polynomial without introducing spurious signs that would break the convexity of the optimization problem.
The second key step is to translate between multivariate scalar polynomials (which are easy to work with algebraically) and nested commutators — algebraic expressions that appear when you write the measurements from the Gibbs state. This translation allows them to encode the learning problem as a system of polynomial equations, where the unknowns are the coupling constants of the Hamiltonian. Finally, they show that this system can be solved by a low-degree sum-of-squares (SoS) relaxation, a standard algorithmic technique in convex optimization. The degree of the SoS relaxation is chosen to be large enough to capture all the constraints, yet small enough that the overall runtime is polynomial in n.
A New Toolbox for Quantum Many-Body Physics
The result is not just a solution to a long-standing open question; it is a demonstration of a new way to think about Hamiltonian learning. Instead of directly inverting the relationship between parameters and measurements — which is often ill-posed — the team reduces the problem to a well-behaved algebraic system. This is analogous to how, in classical statistics, the method of moments allows one to infer parameters of a distribution from sample moments. Here, the raw measurements play the role of “quantum moments,” and the flat polynomial connects them to the Hamiltonian’s parameters.
This is not a metaphor; the correspondence between polynomial approximation and Hamiltonian reconstruction is mathematically exact. But it is worth pausing to appreciate the conceptual leap. The exponential function, for all its beauty, is a stubbornly non-algebraic object. By approximating it with a carefully designed polynomial, the team has built a bridge between the continuous world of thermal physics and the discrete world of polynomial equation solving. Unlike dinner guests who can occupy only one seat at a time, quantum measurement outcomes can be combined into algebraic constraints that jointly determine the underlying Hamiltonian.
What This Means for Experimental Science
The practical implications are significant. In current quantum computing experiments — with superconducting qubits, trapped ions, or neutral atoms — the actual Hamiltonian describing the device is never perfectly known. There are always stray couplings, residual interactions, and inhomogeneities. Researchers calibrate these devices by running simple experiments and fitting models, but the process is ad hoc and rarely scales well. A polynomial-time algorithm that can learn a Hamiltonian from Gibbs state samples offers a principled way to do this calibration: prepare the system at a given temperature (or, equivalently, at a given time after a quench), take many copies, and run the algorithm. The output is an accurate description of the device’s dynamics.
Of course, the current theoretical work assumes access to many identical copies of the exact Gibbs state — a tall order in practice, as preparing thermal states in a quantum computer is itself a hard problem. But The road ahead is clear, even if the exact timeline for experimental implementation remains uncertain. The method is polynomial, not linear, but the constants are not astronomically large — the team provides explicit bounds on the runtime, which suggest that for systems of a few dozen qubits, the algorithm could be feasible on a classical computer with reasonable resources.
A Window into the Architecture of Nature
Beyond calibration, there is something more philosophical at play. The Hamiltonian of a quantum system is the fundamental law that governs its behaviour in time. To “learn” it from data is to recover one of nature’s deepest patterns from its surface traces. The fact that this is now possible in polynomial time — given sufficient data — means that the inverse problem is not as intractable as one might have feared. The structure of locality and the special form of the Gibbs state provide just enough constraints to make the inference computationally efficient.
This work does not solve all problems in Hamiltonian learning: it assumes knowledge of the inverse temperature beta (which in an experiment would need to be estimated too), and it requires that the Hamiltonian be exactly local with a known interaction graph. But it resolves the central theoretical question that had been open for years. And it offers a template for approaching other inverse problems in quantum science: approximate the hard function with a simple one, translate the problem into an algebraic system, and solve it by convex relaxation.
The team’s preprint closes an important chapter in the story of quantum learning theory. The next ones will be written in the lab, where the algorithms must contend with noise, limited data, and the messy reality of physical devices. For now, however, we have a clean, elegant proof that the rules of the quantum game can be inferred — at any temperature — and that the search need not take exponential time.
Yanjiang is an online editor of LoomSci
References
- Ainesh Bakshi et al., Learning quantum Hamiltonians at any temperature in polynomial time, arXiv:2310.02243