Out-of-control Hamiltonian Learning

Weiyuan Gong; Muzhou Ma; Sitan Chen; Jordan Cotler; Hsin-Yuan Huang

quant-phOct 5, 2026

Out-of-control Hamiltonian Learning

Authors: Weiyuan Gong, Muzhou Ma, Sitan Chen, Jordan Cotler, Hsin-Yuan Huang

Organizations: Harvard University. · Caltech. · Oratomic.

Abstract

Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control--fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases--that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models. Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments. Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli $X/Z$ interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges. Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.

Figures & tables

Access model	Hamiltonian model	Gauges	Evolution time	Classical post-proc.
Uniform control ( $\hat{a}$ -to- $\hat{b}$ ) Theorem 1 , 5	General graphs	No gauge	$\widetilde{O}(\poly(n)\varepsilon^{-2})$	$\widetilde{O}(\poly(n)\varepsilon^{-2})$
$0$ -to- $Z^{\prime}$ Theorem 6	Isotropic TI NN IXZ	$2$ signs ( $Z^{\otimes n}$ , time reversal)	$\widetilde{O}(\varepsilon^{-2})$	$\widetilde{O}(\varepsilon^{-2})$
$Z$ -to- $Z^{\prime}$ Theorem 7	TI NN IXZ, 1D/2D/3D	$2$ signs ( $Z^{\otimes n}$ , time reversal)	$\widetilde{O}(\varepsilon^{-2})$	$\widetilde{O}(\varepsilon^{-2})$
$Z$ -to- $Z^{\prime}$ Theorem 8	1D TI $3$ -g.l. IXZ	$2$ signs ( $Z^{\otimes n}$ , time reversal)	$\widetilde{O}(\varepsilon^{-2})$	$\widetilde{O}(\varepsilon^{-2})$
$Z$ -to- $Z^{\prime}$ Theorem 9	1D/2D NN IXZ	$2^{n+1}$ signs ( $Z^{S}$ , time reversal)	$\widetilde{O}(\poly(n)\varepsilon^{-2})$	$\widetilde{O}(\poly(n)\varepsilon^{-2})$
Lower bound Theorem 3	1D NN IXZ	—	$\Omega(\varepsilon^{-2})$	—

Table 1 : Summary of our results. Upper bounds hold in smoothed analysis framework and suppress polynomial factors in

R,1/\varsigma,1/\delta

, where

R

bounds magnitude of Hamiltonian coefficients and

\varsigma

is smoothing scale (see Section C.1 ).

Figure 1 : (a) Uniform control (

\hat{a}

-to-

\hat{b}

model): all qubits are prepared along one common direction

\hat{a}

|0_{\hat{a}}^{n}\rangle=(U|0\rangle)^{\otimes n}

, and measured along one common direction

\hat{b}

. (b)

Z

-basis access (

Z

-to-

Z^{\prime}

model;

0

-to-

Z^{\prime}

for

|0^{n}\rangle

input): the qubits start in a computational basis state

|z\rangle

and are measured in the computational basis. (c) From the data of (a), all parameters can be learned, for any interaction graph. (d) From the data of (b), a generic NN IXZ Hamiltonian on a 1D chain or square lattice can be learned up to unavoidable Pauli-

Z

conjugation and time-reversal gauges.

Figure 2 : Three-round reconstruction under uniform control. (a) Local data: Taylor coefficients

L_{S}

Q_{S}

and

\mathcal{W}_{S}

(orders one to three) of the Fourier coefficients

m^{\hat{a},\hat{b}}_{S}(t)

on sets

S

of at most four sites, with

J_{e}=T_{e}+A_{e}+\lambda_{e}I

for

A_{e}=[\bm{\mathbf{d}}_{e}]_{\times}

. (b)–(d): checks denote what is learned in that round, crimson and question marks denote the remaining ambiguities. (b) Round 1: first-order data identifies

T_{e}

\bm{\mathbf{h}}_{v}

, as well as

A_{e}

up to a global curl gauge. (c) Round 2: Second-order terms

R_{uvw}

of three-site cumulants determine the skew-symmetric components

A_{e}

for edges on induced paths. For the remaining edges, we use observables adapted to the local graph structure (see Figure 6 ). (d) Round 3: Third-order terms

F_{\{u,v,w,x\}}

of connected four-site cumulants determine the edge traces

\lambda_{e}

for edges with an endpoint of degree at least three. Second-order single-site responses recover the remaining traces (see Figure 7 ).

Figure 3 : Local reconstruction of a generic NN IXZ Hamiltonian from

Z

-to-

Z^{\prime}

data. (a) For transitions that flip the marked site (ring), vary the neighboring input bits (double arrows) and extract the second- and fourth-order coefficients of the local transition probabilities

p_{z\to z^{\prime}}(t)

. (b) Stage 1: Second-order data recover each transverse field and its incident mixed couplings up to a common site sign (

\pm

). Fourth-order tests on nonadjacent pairs resolve the chain-endpoint ambiguities and determine relative

J_{XX}

signs along the shaded paths, leaving one common sign. (c) Stage 2: Subtracting the fourth-order contribution from evolution under the recovered off-diagonal Hamiltonian

K

gives local equations for

h_{Z}

and

J_{ZZ}

. These are solved on windows, with separate equations for lattice sides and corners. The boundary sums

\Sigma

combine fields with exterior

ZZ

couplings. (d) Overlapping windows recover every field and coupling. Once a common representative of

K

is fixed, each local diagonal system has a unique solution. The remaining gauges are site-dependent Pauli-

Z

conjugation, with signs

s_{j}=\pm 1

, and time reversal, with sign

\tau=\pm 1

, as in Figure 1 (d). In particular,

J_{XX}^{j}\mapsto\tau s_{j}s_{j+1}J_{XX}^{j}

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4 : Experimental access to the unknown Hamiltonian

H

. (a) Uniform control (

\hat{a}

-to-

\hat{b}

): Prepare every qubit along a common direction

\hat{a}

, evolve once under

H

, and measure along a common direction

\hat{b}

. Uniform rotations implement the preparation and readout, and the two directions may vary between experiments. (b) Computational-basis access (

Z

-to-

Z^{\prime}

): Prepare

\ket{z}

in the computational basis, evolve once under

H

, and measure in the computational basis, obtaining

z^{\prime}

with probability

p_{z\to z^{\prime}}(t)

. Arbitrary inputs require site-resolved bit flips, and the all-zero-input model fixes

z=0^{n}

. Our protocols vary input bits and measure output probabilities on constant-size regions ( Remark 1 ). Neither (a) nor (b) allows control during evolution. (c) General coherent control: An arbitrary joint state

\rho

of the system and ancillas undergoes evolution segments

U(t_{j})=e^{-iHt_{j}}

on the system, alternating with joint control unitaries

V_{j}

, followed by a collective measurement with outcome

m

. Restricting each experiment to one uninterrupted evolution gives the setting of our lower bound ( Theorem 3 ), even with arbitrary inputs, passive ancillas, and measurements.

Figure 5 : First-order reconstruction under uniform control. (a) Single-site and two-site responses determine the local fields

\bm{\mathbf{h}}_{v}

and the symmetric traceless parts

T_{e}

of the couplings. The skew-symmetric parts

A_{e}

enter only through their divergence at each vertex. (b) The trace terms

\lambda_{e}I

are invisible at first order, leaving one undetermined parameter per edge. (c) Adding a skew-symmetric matrix around a cycle, with signs following the edge orientations, preserves these divergences: the incoming and outgoing contributions cancel at every vertex. The illustrated graph has two independent cycles, supporting the circulations

\Gamma

and

\Gamma^{\prime}

. Together, the trace and circulation ambiguities account for

|E|+3(|E|-|V|+c)

undetermined parameters on a graph with

c

connected components. For the edge couplings on the plot, darker colors indicating stronger interactions, and blue arrows show the local fields.

Figure 6 : Local reconstruction of the skew-symmetric edge vectors. (a) For an induced path

u

v

w

, six fixed settings (basis pairs) give a

6\times 6

system in

(\bm{\mathbf{d}}_{uv},\bm{\mathbf{d}}_{vw})

. Its determinant is a nonzero polynomial of degree six, so independent Gaussian smoothing gives the simultaneous lower bound in Eq. ( 21 ). Each selected path is solved independently. In particular, every edge of a triangle-free component with at least three vertices has a path. (b) For general graphs, first solve every edge lying in an induced path, then recover each remaining edge in a non-complete connected component from a triangle whose other two edges are already known. Complete connected components use the divergence equation for

K_{2}

, the six-variable affine vertex system for

K_{3}

, and four-vertex clique systems for

K_{r}

r\geq 4

Figure 7 : Recovering the edge traces from local responses. (a) The trace term (crimson) exchanges two spins between pair flips generated by the known couplings (amber), making it visible in the four-site third-order response. (b) A vertex and three of its neighbors give a linear system for at most six edge traces. Subtracting the known traceless contribution from the connected third-order response isolates these traces and exterior neighbors do not contribute. (c) At vertices with one or two neighbors, single-site second-order responses recover the incident traces after the known contribution is subtracted. Blue halos mark the edges whose traces are recovered, and the blue rim marks the measured vertex. Crimson dots indicate representative trace terms. See Lemma 6 and Eq. ( 18 ).

Index	$h_{Z}$	$h_{X}$	$J_{ZZ}$	$J_{XZ}$	$J_{XX}$	Gauge Type
1	$\tilde{h}_{Z}$	$\tilde{h}_{X}$	$\tilde{J}_{ZZ}$	$\tilde{J}_{XZ}$	$\tilde{J}_{XX}$	—
2	$-\tilde{h}_{Z}$	$-\tilde{h}_{X}$	$-\tilde{J}_{ZZ}$	$-\tilde{J}_{XZ}$	$-\tilde{J}_{XX}$	(1)
3	$\tilde{h}_{Z}$	$-\tilde{h}_{X}$	$\tilde{J}_{ZZ}$	$-\tilde{J}_{XZ}$	$\tilde{J}_{XX}$	(2)
4	$-\tilde{h}_{Z}$	$\tilde{h}_{X}$	$-\tilde{J}_{ZZ}$	$\tilde{J}_{XZ}$	$-\tilde{J}_{XX}$	(1)(2)

Appendix

Table 2 : The four generic gauge-transformed parameter configurations for 1D, 2D, and 3D isotropic translationally invariant nearest-neighbor IXZ Hamiltonians. The tilded coefficients denote one fixed representative. The gauges are labeled by: (1) time-reversal gauge as stated in Proposition 2 , and (2) conjugation by

Z^{\otimes n}

, a special case of Proposition 2 .

\displaystyle[t^{2}]p_{H}^{B}(t)=\frac{(p_{H}^{B})^{\prime\prime}(0)}{2!},\qquad[t^{4}]p_{H}^{B}(t)=\frac{(p_{H}^{B})^{(4)}(0)}{4!}.

Appendix

Algorithm 2 Reconstructing a 1D TI NN IXZ Hamiltonian with

Z

-to-

Z^{\prime}

access

\displaystyle[t^{2}]p_{H}^{B}(t)=\frac{(p_{H}^{B})^{\prime\prime}(0)}{2!},\qquad[t^{4}]p_{H}^{B}(t)=\frac{(p_{H}^{B})^{(4)}(0)}{4!}.

Appendix

Algorithm 3 Reconstructing a 2D/3D TI NN IXZ Hamiltonian with

Z

-to-

Z^{\prime}

access

\displaystyle[t^{2}]p_{H}^{B}(t)=\frac{(p_{H}^{B})^{\prime\prime}(0)}{2!},\qquad[t^{4}]p_{H}^{B}(t)=\frac{(p_{H}^{B})^{(4)}(0)}{4!}.

Appendix

Algorithm 4 Reconstructing a 1D TI

3

-g.l. IXZ Hamiltonian with

Z

-to-

Z^{\prime}

access

	1D				2D				3D
Monomial	1	2	3	4	1	2	3	4	1	2	3	4
$J_{XX}^{4}$	128	128	336	336	640	1536	3200	128	1536	4480	336	2816
$J_{XZ}^{2}J_{XX}^{2}$	512	768	1136	1136	4992	12800	29184	512	18048	54272	1968	43360
$h_{X}J_{XZ}J_{XX}^{2}$	512	1024	1728	1344	2560	7424	15872	512	6144	19968	1344	22528
$J_{ZZ}^{2}J_{XX}^{2}$	128	128	144	144	2304	3456	4608	288	9600	16000	1200	6464
$h_{Z}J_{ZZ}J_{XX}^{2}$	256	256	320	320	1536	2304	3072	384	3840	6400	960	3584

Appendix

Table 3 : Monomial coefficients of the fourth derivatives

F_{1},\ldots,F_{4}

for the isotropic TI IXZ models; the observables are ordered as in Appendix E.1 .

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