Can Many-Body Hamiltonians Be Learned Without Full Quantum Control?

Why full control is the wrong assumption

Most Hamiltonian-learning algorithms with strong provable guarantees assume fast, arbitrary single-qubit gates interleaved with time evolution, plus measurements in arbitrary bases. The paper argues this assumption is beyond near-term analog quantum simulators, where a single laser sets a common Rabi frequency and detuning for every atom, and individually addressable gates are very limited.

The authors describe the experimentalist as "out of control": the individually addressed gates that learning protocols rely on are often not part of the native toolbox. So they ask what can be learned using only the operations these devices natively support, and formalize two restricted access models.

Two access models define what the learner can do

In the uniform control (a-to-b) model, every experiment prepares |0^n>, applies the same single-qubit rotation U^{⊗n} to each qubit, applies time evolution, applies a second uniform rotation V^{⊗n}, and measures every qubit in the computational basis. Equivalently, every qubit is prepared along a common Bloch direction a and measured along a common direction b.

In the Z-basis access (Z-to-Z') model, every experiment prepares a computational basis state |z>, applies time evolution, and measures in the computational basis. The paper also studies an even weaker all-zero (0-to-Z') model where z must be 0^n. Neither model subsumes the other: uniform control allows an arbitrary common basis but no site resolution, while Z-basis access allows site-resolved bit flips at preparation. In both models no control is applied during evolution, evolution times are short, and the data collected are bitstrings.

Uniform control recovers every parameter

For a general nearest-neighbor Hamiltonian on a known simple graph G=(V,E), the Hamiltonian has 3|V|+9|E| real parameters: three single-site field components per vertex and a 3×3 interaction matrix per edge. The paper shows that for generic instances, uniform control suffices to reconstruct all of them with no gauge ambiguity, for any interaction graph, including graphs of unbounded degree with no geometric locality.

The protocol exploits second- and third-order response in short-time evolutions. The authors note that the standard linear-response approach directly fails here: first-order data determine H only up to a linear subspace of dimension at least |E|, which is insufficient to recover all 3|V|+9|E| real numbers. The sample complexity, total evolution time, and classical post-processing time scale as Õ(ε^{-2} poly(n,R,1/δ,1/ς)), and the protocol uses only marginals of output bitstrings on at most four sites.

Z-basis access hits unavoidable sign symmetries

Under Z-basis access, the paper focuses on nearest-neighbor Hamiltonians with only Pauli X/Z interactions, a class that captures contemporary Rydberg atom platforms. For these, conjugating H by any diagonal unitary, or replacing H with −H*, leaves every transition probability unchanged. The authors show these symmetries are the only obstruction: over 1D and 2D rectangular lattices, all parameters can be learned modulo a discrete group of 2^{n+1} sign flips.

For translation-invariant families, the cost is independent of n altogether, scaling as Õ(ε^{-2}). For spatially varying nearest-neighbor X/Z Hamiltonians, the protocol uses constant-size local experiments, and its sample complexity, total evolution time, and classical post-processing complexity scale as Õ(ε^{-2} poly(n,R,ς^{-1},δ^{-1})).

Polynomial systems replace linear response

At a technical level, the protocols reduce the learning problem to highly structured systems of polynomial equations in the unknown coefficients. The authors prove these systems identify the parameters modulo the unavoidable symmetries, and that they are well conditioned for generic (smoothed) instances, by exhibiting explicit algebraic certificates.

For the spatially varying case, the proof first recovers the off-diagonal coefficients throughout the lattice and aligns their signs. It then uses the same representative of K in every local subtraction and diagonal solve: four-site windows on a chain, and a vertex with its neighbors on a square lattice. On a four-site chain window, eliminating two boundary sums leaves ten polynomial equations in five diagonal variables; the certificate proves these five variables are uniquely determined, after which the two eliminated sums are recovered by division.

The 1/ε² cost is unavoidable

Both protocols use a number of experiments and total evolution time scaling as Õ(poly(n)/ε²). The paper shows this 1/ε² dependence is unavoidable in the absence of interleaved control: a lower bound (Theorem 3) rules out Heisenberg scaling for a 1D nearest-neighbor X/Z family when each experiment consists of a single uninterrupted time evolution under H.

This is a meaningful boundary. The lower bound applies to the specific setting of one uninterrupted evolution per experiment; the paper's outlook asks whether allowing uniform control during the time evolution could change the scaling. That question is left open, so the 1/ε² result should be read as a statement about the no-interleaved-control regime, not about all possible restricted-access protocols.

Where the guarantees stop

The Z-basis reconstruction results rely on finite-range lattice interactions. Rydberg and trapped-ion interactions generally have power-law tails, and extending these reconstructions to sufficiently rapidly decaying long-range interactions remains an open problem. The uniform-control results, by contrast, already apply to generic two-body Hamiltonians on arbitrary known interaction graphs.

The paper also assumes known local depolarizing noise for the Z-basis theorems, but does not address more general preparation and measurement errors. Realistic dynamics may include noise during evolution itself; for time-homogeneous Markovian dynamics the unknown generator would be a Lindbladian rather than a Hamiltonian, and whether the same approach can identify a local Lindbladian using only computational-basis access or uniform control is left open. The results are also stated within a smoothed-analysis framework, where coefficients are perturbed by a small Gaussian of scale ς, so they are guarantees for generic rather than adversarially chosen instances.

What to take from the restricted-access result

The most decision-relevant comparison in this paper is between the two access models. Uniform control gives a stronger guarantee: no gauge ambiguity at all, for generic 2-local Hamiltonians on any known interaction graph, at a cost that scales polynomially in n. Z-basis access gives a weaker but still broad guarantee: reconstruction up to 2^{n+1} sign flips, for the narrower X/Z nearest-neighbor family on chains and rectangular lattices, but with a cost that is independent of n for translation-invariant families.

For a reader deciding what is achievable on a given analog platform, the practical threshold is whether the device can prepare arbitrary computational basis states with site-resolved bit flips. If it can, the Z-basis results apply to the X/Z lattice families studied here. If it cannot, and only global rotations are available, the uniform-control result is the relevant one, and it covers generic 2-local Hamiltonians on any known graph. The paper's own framing is that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.

Sources used in this answer