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Out-of-control Hamiltonian Learning

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

arXiv:2610.06709v1quant-phcs.ITcs.LGmath.ITphysics.atom-ph

TL;DR

Hamiltonian learning matters across quantum science, but many existing protocols assume control and measurement choices unavailable or costly on analog simulators. This paper studies uniform-control and computational-basis access models and shows that Hamiltonian parameters can be reconstructed under both, subject to specified scope and gauge conditions.

  • Problem

    Many Hamiltonian-learning protocols assume the ability to choose each qubit’s preparation and measurement basis, an assumption that can exceed analog simulators’ control capabilities.

    Source: §1 · p. 3

  • Method

    The authors analyze uniform-control and Z-basis protocols and reduce learning to structured polynomial systems whose identifiability and conditioning are certified algebraically.

    Source: §1 · p. 4, §3.2 · p. 14

  • Results

    Generic 2-local Hamiltonians on any known interaction graph are recoverable with uniform control, while nearest-neighbor IXZ Hamiltonians on 1D chains and 2D rectangular lattices are recoverable with Z-basis access up to Pauli-Z-conjugation and time-reversal gauges.

    Source: §1 · p. 4, §2.2 · p. 7

  • Takeaways & Limitations

    These results show that Hamiltonian learning is possible with the restricted preparation and measurement access considered, including on the stated lattice geometries.

    Source: §1 · p. 4, §2 · p. 5, §2.2 · p. 7

    The Z-basis protocol requires arbitrary computational-basis preparation, and the stated SPAM analysis assumes known local depolarizing noise rather than general preparation and measurement errors.

    Source: §5 · p. 16

Abstract

from arXiv · show

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.

1 Introduction

Hamiltonian learning often assumes individually chosen preparation and measurement bases or interleaved control, capabilities that can be unavailable or costly on analog simulators. This section asks what can be learned with native, uniform or computational-basis access, and reports that both models permit substantial reconstruction despite their constraints.

1 Introduction

  • Analog atom and ion simulators often lack fast, individually addressed gates, motivating learning protocols that use operations these devices natively support.Rydberg arrays use common laser settings, while analog trapped-ion simulators use global rotations and interactions.Source: §1 · p. 3
  • Uniform control prepares and measures every qubit along common directions, whereas Z-basis access prepares and measures computational-basis states and can use site-resolved bit flips.Neither model subsumes the other, and neither applies control during evolution.Source: §1 · p. 3
  • Z-basis transition probabilities are invariant under diagonal-unitary conjugation and H → −H*, while uniform-control first-order data leave at least |E| dimensions unresolved.Uniform control uses only two common directions per experiment to recover 3|V| + 9|E| Hamiltonian parameters.Source: §1 · p. 3
  • Generic 2-local Hamiltonians on any known interaction graph are recoverable with uniform control, while nearest-neighbor X/Z Hamiltonians on chains and rectangular lattices are recoverable with Z-basis access up to discrete sign gauges.The uniform-control protocol has no gauge ambiguity; the Z-basis case has 2n+1 sign flips.Source: §1 · p. 4

2 Our results

The results establish reconstruction under two restricted access models: uniform control recovers generic 2-local Hamiltonians on any known graph, while computational-basis access recovers nearest-neighbor IXZ Hamiltonians on chains and rectangular lattices up to unavoidable gauges. Further results cover translation-invariant and all-zero-input settings, and show that the protocols’ 1/ε² evolution-time scaling cannot be improved without interleaved control.

2.1 Uniform control

  • Uniform control reconstructs every parameter of a generic 2-local Hamiltonian on any known interaction graph, without gauge ambiguity.The graph need not have geometric locality.Source: §2.1 · pp. 5–6
  • Connected second- and third-order responses on at most four sites recover parameters that first-order data leave undetermined.The protocol uses short-time evolution and a fixed finite set of common preparation and measurement directions.Source: §2.1 · p. 6

2.2 𝑍-basis access

  • Computational-basis access reconstructs generic nearest-neighbor IXZ Hamiltonians on 1D chains and rectangular lattices up to Pauli-Z conjugation and time-reversal gauges.The theorem covers open or periodic lattices and uses constant-size regions of output bitstrings.Source: §2.2 · p. 7
  • For translation-invariant IXZ Hamiltonians, reconstruction extends to 1D, 2D, and 3D lattices, with only two signs unresolved and cost independent of system size.The extension also covers 3-body IXZ Hamiltonians on a 1D chain.Source: §2.2 · p. 7
  • With all-zero input on open lattices, second- and fourth-order data determine all five isotropic translation-invariant IXZ parameters up to the same two signs.Boundary sites break translation symmetry to provide enough independent equations; cost remains independent of system size.Source: §2.2 · p. 7
  • Without interleaved control, total evolution time scales as 1/ε² rather than 1/ε, and this standard-quantum-limit scaling is unavoidable for the considered access models.The no-go result holds for an explicit 1D nearest-neighbor IXZ family even with entangled inputs, ancillas, arbitrary measurements, and classical adaptivity.Source: §2.2 · p. 7

3 Overview of techniques

This section presents reconstruction techniques from short-time transition-probability data for uniform-control and computational-basis access, using access-specific polynomial constraints. For Z-basis access, it develops polynomial-system methods for approximate recovery modulo unavoidable gauges, including a two-stage strategy and quantitative certificates under smoothed analysis.

3.1 𝑈niform control

  • Uniform-control higher-order constraints become local and linear after combining them with earlier-round estimates, enabling recovery from otherwise extensive nonlinear systems.Three-site cumulant Taylor coefficients localize constraints to connected terms; the second-order terms involve distinct edges connecting the measured sites.Source: §3.1 · pp. 8, 10, 11
  • Under uniform control, reconstruction proceeds in three rounds: first-order data recover symmetric couplings and fields, second-order data recover skew-symmetric couplings, and third-order data recover trace terms.First-order data leave the skew-symmetric couplings ambiguous up to a global circulation; higher-order constraints resolve that ambiguity.Source: §3.1 · pp. 8–10
  • The uniform-control error analysis uses constant-size linear systems and Gaussian-polynomial anti-concentration to ensure bounded conditioning with high probability and benign error propagation.The systems avoid long chains of substitutions, and their coefficients depend polynomially on parameters estimated in earlier rounds.Source: §3.1 · p. 12

3.2 𝑍-basis access

  • With computational-basis access, transition probabilities are even in time, so first-order data reveal no Hamiltonian information; recovery is possible only up to Pauli-Z conjugation and time-reversal gauges.The overview treats general, non-translation-invariant nearest-neighbor IXZ Hamiltonians.Source: §3.2 · p. 12
  • For computational-basis access, second- and fourth-order data recover off-diagonal terms and resolve their signs up to gauge, while localized window systems stitch together the diagonal fields and couplings.On chains, overlapping four-site windows recover individual coefficients; on square lattices, center, side, and corner systems provide the corresponding reconstruction.Source: §3.2 · p. 13
  • Because the computational-basis local systems are nonlinear, the method fixes a globally consistent gauge representative and uses an algebraic identifiability argument, quantifiable through smoothed analysis.This addresses why the polynomial constraints identify the parameters up to gauge despite the lack of the uniform-control error-propagation argument.Source: §3.2 · p. 13
  • A nonsingular coefficient matrix built from products of observables’ polynomials and monomials certifies approximate uniqueness modulo gauges for generic ground-truth parameters.The matrix includes the constant-monomial indicator vector and coefficient vectors of the product polynomials.Source: §3.2 · p. 14
  • The method first solves directly for a subset of unknowns, then applies generic polynomial-system solving to the remaining unknowns.The certification procedure can be adapted to this two-stage approach.Source: §3.2 · p. 14
  • Smoothed-analysis polynomial anti-concentration makes the certificate polynomials bounded away from zero, yielding quantitative guarantees.The certificates establish that certain polynomials do not vanish.Source: §3.2 · p. 14

4 Related works

Prior real-time Hamiltonian-learning methods range from short-time derivative and trace approaches to Heisenberg-scaling protocols, but they generally assume site-resolved basis choices or substantial control. This work’s access models remove site-resolved basis changes, aligning with constraints on analog platforms and extending restricted-access learning work.

Hamiltonian learning from real-time dynamics

  • Earlier real-time approaches infer couplings from short-time derivatives, traces, or correlation matrices, while later protocols achieve Heisenberg scaling using substantial quantum control.The cited work establishes Heisenberg scaling for known local structure [9], unknown interaction structure [10], and sparse ansatz-free Hamiltonians [12] [13] [43].Source: §4 · pp. 14–15
  • Without control, Heisenberg scaling is impossible in general, while the standard quantum limit can be attained by a control-free, ancilla-free protocol using Pauli product-state preparation and measurement.The impossibility result is attributed to Dutkiewicz, O’Brien, and Schuster [11], and the standard-quantum-limit protocol to Zhou and Gong [44].Source: §4 · p. 15
  • Unlike prior methods, including control-free protocols, these models allow only uniform rotations or computational-basis preparation and measurement, with no site-resolved basis changes.Earlier control-free protocols still assume each qubit can be prepared and measured in a chosen basis.Source: §4 · p. 15

Hamiltonian learning on experimental platforms

  • Rydberg and trapped-ion simulators motivate the access models through global operations and computational-basis readout; Guo et al. [45] learned a 300-ion all-to-all Ising Hamiltonian with global manipulations.That experiment uses single-qubit-resolved state detection and is described as closely related to the uniform-control model.Source: §4 · p. 15

Learning under restricted access

  • Restricted-access work includes Heisenberg-limit protocols with static or infrequent control, graph-state certification from uniform measurements, and Hamiltonian learning from time evolution with a single local probe [35].The latter sets polynomial constraints and uses algebraic geometry to certify identifiability up to unavoidable gauge within a smoothed-analysis model.Source: §4 · p. 15

5 Outlook

The outlook recaps reconstruction under restricted access and identifies practical constraints and open extensions, including broader Hamiltonian families, all-zero inputs, adaptivity, and global control.

5 Outlook

  • The outlook recaps reconstruction of generic 2-local Hamiltonians on known graphs and nearest-neighbor IXZ Hamiltonians up to gauges, using one evolution per experiment.It also notes that Heisenberg scaling is ruled out for a 1D nearest-neighbor IXZ family without interleaved control.Source: §5 · pp. 15–16
  • Whether the Z-basis ambiguities remain the only generic ones for nearest-neighbor Hamiltonians with Y terms, or spatially varying IXZ models can be learned from all-zero input, remains open.Source: §5 · p. 16
  • The algorithm adapts estimation precision between stages, while whether adapting inputs, measurements, or evolution times reduces sample complexity remains unanswered.Source: §5 · p. 16
  • The lower bound assumes one uninterrupted evolution; whether interleaved global pulses enable Heisenberg scaling with uniform preparation and measurement remains an open question.Source: §5 · p. 16
  • The implementation analysis requires site-resolved bit flips for arbitrary Z-basis preparation, assumes known local depolarizing noise, and does not cover general preparation and measurement errors.Joint calibration of Hamiltonian and SPAM parameters remains open.Source: §5 · p. 16

B Hamiltonian structures and access models

B.1 Access models

  • Each experiment uses one uninterrupted evolution between preparation and final measurement, and one such cycle counts as an experimental shot.Source: §B.1 · p. 22

B.2 Unobservable gauges

  • Computational-basis probabilities are invariant under diagonal-unitary conjugation and H ↦ −H*, and these gauges can move a Hamiltonian outside its local family.Source: §B.2 · p. 23

B.3 Evolution time lower bound without interleaved control

Without interleaved controls, even broad experimental access cannot avoid a quadratic-in-precision evolution-time lower bound for a 1D nearest-neighbor IXZ family. The proof bounds how distinguishable two hypotheses become under uninterrupted evolution, including in adaptive experiments.

B.3 Evolution time lower bound without interleaved control

  • T ≥ 1/(144ε^2) is required to reconstruct some 1D nearest-neighbor IXZ Hamiltonians to coefficient accuracy ε ≤ π/48 with success probability at least 2/3.The lower bound still applies with adaptive experiments, arbitrary inputs and measurements, and passive ancillary qubits, provided each experiment uses one continuous evolution without interleaved controls.Source: §B.3 · p. 23
  • For the constructed family, the evolution-unitary distance between hypotheses is at most min(t, 1)Δ, limiting the information from each experiment.The estimate follows from the family’s anticommuting Pauli terms and applies to mixed inputs with passive ancillas.Source: §B.3 · p. 23
  • The adaptive proof bounds full-transcript distinguishability by Δ^2T and covers stopping by padding histories with zero-duration rounds.The same history determines the next experiment under both hypotheses.Source: §B.3 · p. 24

C Mathematical tools

  • The reconstruction analysis combines low-order derivative estimates with conditioning bounds and error control through successive reconstruction steps.The cited overview identifies these ingredients as the tools for converting local equations into accurate estimates.Source: C Mathematical tools · p. 25

C.1 Smoothed analysis

The analysis handles unstable local solves by perturbing arbitrary fixed Hamiltonians and bounding the chance that conditioning quantities become small. Derivative estimates and a fixed number of stable reconstruction stages then support the protocols’ resource bounds.

C.1 Smoothed analysis

  • Smoothed analysis perturbs the free coefficients of any fixed base Hamiltonian and bounds failure probability over that perturbation, targeting the perturbed Hamiltonian itself.For nonzero conditioning polynomials, the small-value bound is uniform in the means of the independently smoothed coordinates.Source: §C.1 · pp. 25–26

C.2 Derivative estimation

  • For fixed local parameters and interaction strength, derivatives through order r can be estimated with Õ(η^-2 log(2/δ′)) measurements, independently of n.The method uses measurements after evolution and queries short times whose minimum decreases only inverse-polylogarithmically with precision.Source: §C.2 · pp. 27–28

C.3 Constant-size reconstruction and stability

  • Reconstruction uses constant-size local systems and a fixed number of stages, preventing conditioning losses from accumulating with system size.Under the stated conditioning bounds, finite-stage stability gives output error bounded by a polynomial factor times input error.Source: §C.3 · pp. 28–29

D Learning NN Hamiltonians on graphs with uniform control and all-zero input

  • Uniform-control reconstruction proceeds in three rounds: first-order data recover fields, symmetric traceless couplings, and divergences; second-order responses recover skew parts; third-order responses recover edge traces.The final round also uses single-vertex second-order responses at vertices of degree at most two.Source: D Learning NN Hamiltonians on graphs… · p. 31

D.1 First-order reconstruction and its complete kernel

First-order Taylor data recover the fields, symmetric traceless edge couplings, and vertex divergences, but leave a precisely characterized kernel. Its remaining freedom comprises cycle-like skew-coupling changes and arbitrary edge traces.

D.1 First-order reconstruction and its complete kernel

  • First-order Taylor coefficients determine every single-site field, symmetric traceless edge coupling, and vertex divergence.With fixed settings, linear systems recover these quantities with errors bounded by Cη for the couplings and C(Δ + 1)η for fields and divergences.Source: §D.1 · p. 31
  • The complete first-order kernel has Δh = 0, ΔT = 0, (B_G ⊗ I_3)Δd = 0, and arbitrary Δλ.Skew-vector changes sum to zero at every vertex, while each edge trace remains unconstrained by first-order data.Source: §D.1 · p. 32
  • The skew-coupling ambiguity has three parameters per independent cycle, and edge traces add one free parameter per edge.For a graph with c connected components, the total first-order kernel dimension is |E| + 3(|E| − |V| + c); later rounds remove these ambiguities for nearest-neighbor Hamiltonians.Source: §D.1 · p. 32

D.2 Removing curl gauges with second-order data

This section develops recovery systems that remove gauge ambiguities in edge couplings and traces using second-order data and third-order four-site moments. The systems cover all edges with O(n + |E|) equation systems, with a separate affine system for isolated triangles.

D.2 Removing curl gauges with second-order data

  • Connected three-site second-order moments yield linear equations in skew-symmetric edge vectors, enabling reconstruction on induced paths, triangles, and four-vertex cliques for generic couplings.The equations remove known symmetric-traceless contributions without requiring knowledge of the edge traces.Source: §D.2 · pp. 32, 34, 35
  • The path, triangle-completion, and clique systems have full column rank generically, and the resulting local systems recover all edges outside isolated triangles using O(n + |E|) systems.Edges in induced paths are solved first; remaining edges in non-complete components are recovered by triangle systems, while larger complete components use K4 systems.Source: §D.2 · pp. 35–36
  • Six single-vertex second-order measurements form a generically invertible affine system that recovers the curl gauge and all three traces of an isolated triangle.The six unknowns are the triangle's curl parameter and three edge traces; the reported specialization has det B_triangle = -64.Source: §D.2 · pp. 36–38

D.3 Trace recovery from four-site third-order moments and single-site second moments

  • After the second round, edge traces are recovered from connected four-site third-order moments when an endpoint has degree at least three, and from single-vertex second-order moments for remaining edges.Each recovery system has at most six unknowns, independent of the graph's maximum degree.Source: §D.3 · p. 38
  • Adding trace terms edge by edge to the known contribution proves the recovery equation without products of trace variables.The relevant coefficient depends only on T.Source: §D.3 · p. 40
  • Nonzero determinants establish generic trace recovery for four-site stars, triangle-with-pendant, diamond, and K4 configurations.The selected determinant degrees in the original independent Pauli coefficients are 6, 6, 10, and 12, respectively.Source: §D.3 · p. 41
  • First-order data directly determine the trace matrix, while single-vertex second-order moments suffice for remaining edges of degree at most two.Only traces on edges incident to the vertex enter; at degree one, one orthogonal setting solves the incident trace, and at degree two, two settings solve both.Source: §D.3 · p. 42
  • Four-site and single-vertex systems recover every edge trace using at most O(n + |E|) equation systems.For edges incident to a vertex of degree at least three, use a four-site system; when both endpoints have degree at most two, use a single-vertex system. For isolated triangles, retain estimates from the second round.Source: §D.3 · p. 42

D.4 Conditioning, precision, and complexity

The analysis bounds conditioning of the reconstruction systems under Gaussian smoothing, propagates measurement error, and accounts for sampling and classical costs.

D.4 Conditioning, precision, and complexity

  • The protocol reconstructs every field and edge matrix to error ε with probability at least 1−δ for independently Gaussian-smoothed nearest-neighbor Hamiltonians.The theorem applies on a known simple graph with maximum degree Δ.Source: §D.4 · p. 42
  • Gaussian smoothing gives simultaneous conditioning bounds because the reconstruction determinants are nonzero polynomials, with uniform bounds also established for degree-one and degree-two vertex systems.The systems have O(n + |E|) total count, and their determinant degrees are specified for the path, clique, triangle, and trace systems.Source: §D.4 · p. 43
  • Measurement errors propagate through the skew-vector and trace solves, yielding final coefficient error at most C(Δ + 1)^4R^4η/κ^3.The trace reconstruction uses four-site systems for higher-degree vertices and single-vertex systems for the remaining edges.Source: §D.4 · p. 44
  • The polynomial exponents in the resource bounds are independent of graph degree, and each experiment lasts at most C/((Δ + 1)R).The total evolution time is bounded by C N_shot/((Δ + 1)R).Source: §D.4 · p. 45
  • At each setting, one bitstring supplies samples for every required moment, so simultaneous accuracy adds only a logarithmic factor and needs no separate batches for different site sets.The graph operations, constant-size solves, moment accumulation, and interpolation give polynomial classical post-processing, with total failure probability at most δ.Source: §D.4 · p. 45

E.1 Isotropic TI NN Hamiltonians from all-zero input

For isotropic translation-invariant nearest-neighbor IXZ Hamiltonians, all five parameters can be recovered from short-time data on open lattices in dimensions one through three. Generic solutions are identifiable up to two independent common-sign ambiguities.

E.1 Isotropic TI NN Hamiltonians from all-zero input

  • Second- and fourth-order Z-moment derivatives recover the five parameters on open chains, square lattices, or cubic lattices, up to two independent common signs.The protocol estimates each parameter to error ε with probability at least 1 − δ under independent Gaussian smoothing.Source: §E.1 · pp. 47–48
  • All-zero preparation and computational-basis measurements on constant-size regions supply the local derivatives used for reconstruction.The protocol combines second-order data with four fourth-order derivatives from bulk and boundary measurement regions.Source: §E.1 · p. 47
  • The four generic solutions differ by independent common sign changes of (h_X, J_XZ) and (h_Z, J_ZZ, J_XX).Table 2 lists these gauge-transformed configurations.Source: §E.1 · p. 48, Table 2 · p. 48
  • The sample complexity, total evolution time, and classical post-processing cost scale as e O(ε^-2 poly(R, σ^-1, δ^-1)).The stated bound follows from the derivative estimates and refinement procedure.Source: §E.1 · pp. 47–48

E.2 TI NN Hamiltonians from arbitrary 𝑍input

With arbitrary computational-basis inputs, the protocols recover translation-invariant nearest-neighbor IXZ parameters on chains and lattices, and extend to three-local translation-invariant chains. These reconstructions use local short-time data and retain only the stated common-sign ambiguities.

E.2 TI NN Hamiltonians from arbitrary 𝑍input

  • For generic translation-invariant nearest-neighbor IXZ Hamiltonians on open chains, square lattices, or cubic lattices, constant-size Z-basis experiments recover every parameter up to two common signs.The protocol estimates parameters to error ε with probability at least 1 − δ, and its resource bound is independent of the number of sites.Source: §E.2 · p. 48
  • Second-order data recover single-X coefficients and XX magnitudes; fourth-order data resolve relative XX signs and, after K-only subtraction, determine diagonal coefficients.The two remaining common signs arise from conjugation by Z⊗n and time reversal.Source: §E.2 · p. 48

E.3 TI three-local chains

  • For translation-invariant three-local IXZ chains, constant-size local Z-basis experiments estimate all parameters up to an independent common sign for each of two coefficient families.The sample, evolution-time, and post-processing costs are each e O(ε^-2 poly(R, 1/σ, 1/δ)).Source: §E.3 · p. 51
  • Two fourth-order moments distinguish the remaining off-diagonal candidates, and a 14 × 14 linear system determines the diagonal parameters.The proof establishes explicit coefficient choices that separate the signs and make the system determinant nonzero.Source: §E.3 · p. 51
  • For higher fixed locality, the method requires a fixed derivative set that determines parameters up to finitely many gauges and a nonzero certificate; this is not established for every locality.These are the stated conditions for applying the same method at higher fixed locality.Source: §E.3 · p. 51

E.4 Spatially varying NN Hamiltonians

Local Z-basis experiments reconstruct generic spatially varying nearest-neighbor IXZ Hamiltonians on chains and square lattices, subject to the sign ambiguities in Eq. (28). The procedure combines local off-diagonal recovery with window- or vertex-based diagonal solves.

E.4 Spatially varying NN Hamiltonians

  • The sign ambiguities arise from sitewise Pauli-Z conjugations and the global time-reversal transformation H → −H.These transformations leave Z-basis transition probabilities unchanged.Source: §E.4 · p. 51
  • For generic spatially varying nearest-neighbor Hamiltonians, local Z-basis experiments determine all coefficients up to exactly the sign choices in Eq. (28).The result covers chains meeting the stated open- or periodic-size conditions and square lattices meeting the stated side-length or period conditions.Source: §E.4 · p. 52
  • The method recovers and sign-aligns off-diagonal coefficients first, then uses a shared representative of K for local diagonal solves.It uses four-site chain windows or a square-lattice vertex with its neighbors.Source: §E.4 · pp. 52–53
  • Boundary reconstruction uses nonadjacent-pair fourth-order transitions on chains and separate three-neighbor and linear equations on rectangle sides and corners.The chain diagonal certificate remains nonzero when either outside edge of a window is absent.Source: §E.4 · p. 53
  • The stated minimum system sizes preserve the local equations used by the reconstruction, including on periodic lattices.The proof bounds the equations to local neighborhoods and shows that periods of at least nine avoid identifying sites or creating extra nearest-neighbor edges.Source: §E.4 · p. 53

F Deferred proofs

The deferred proofs establish derivative-estimation bounds and verify the algebraic conditions behind the Z-basis reconstructions. Stable polynomial interpolation extracts short-time derivatives with locality-controlled errors independent of total system size.

F Deferred proofs

  • The proof section first bounds accurate short-time derivative estimation, then verifies the identities, uniqueness, and stability conditions used in the Z-basis reconstructions.The reconstruction proofs use explicit certificates to apply the general error bounds to nonlinear systems.Source: F Deferred proofs · p. 53

F.1 Derivative estimation by stable polynomial interpolation

  • Finite-time polynomial interpolation estimates derivatives while locality bounds the Taylor remainder independently of total system size.The relevant scale is the local interaction strength rather than the norm of the full Hamiltonian.Source: §F.1 · pp. 53–54
  • The estimation guarantee applies for fixed interaction size, observable support, and derivative order, with bounded local interaction strength and measurements supplying bounded outcomes.The same shots can provide many local observables; third-order derivatives serve uniform-control reconstruction and fourth-order coefficients serve Z-basis protocols.Source: §F.1 · pp. 54, 56
  • Chebyshev sampling controls interpolation-error amplification logarithmically in the polynomial degree, while extrapolation and derivative extraction bound errors at zero.The sampling times are chosen near the interval endpoints, with a degree choice that makes the Taylor remainder small.Source: §F.1 · pp. 55–56

F.2 Transition identities and gauge fixing

This section develops transition identities, gauge-fixing procedures, and reconstruction equations for recovering model parameters from short-time data, including structured real amplitudes from their squares up to a common sign. It uses certificate determinants to establish generic identifiability while accounting for four sign-related solutions from symmetry, and uses Gaussian smoothing to control estimation errors.

F.2 Transition identities and gauge fixing

  • For TI models, fourth-order interference fixes relative XX signs; nonadjacent-pair tests also resolve endpoint ambiguities and work at open boundaries.Bulk Fourier tests isolate products of incident XX couplings when extended neighbors are present; endpoint tests use a known interior block and recover the endpoint block after fixing its gauge.Source: §F.2 · pp. 57–59
  • Fourth-order data for a nonadjacent two-site flip isolate the squared two-step transition amplitude, yielding products of XX couplings without any diagonal coefficient.The coefficient can be obtained from local output probabilities, and varying the input bits separates the known single-flip contribution from the coupling-product sum.Source: §F.2 · pp. 58–59
  • On rectangular square lattices, incident-edge products determine all XX signs up to one common sign, which remains as a time-reversal and Z-conjugation gauge.The products determine relative signs at each vertex, and connectivity propagates them across the lattice; plaquette tests recover products even at sides and corners.Source: §F.2 · p. 60
  • With known allowed character support satisfying the stated symmetric-difference condition, squared amplitude values generically determine all coefficients up to a common sign.The only competing admissible sign functions are constant; the reconstruction procedure tests sign assignments by retaining those whose Boolean Fourier coefficients vanish outside the allowed support.Source: §F.2 · p. 61

F.3 Learning TI Hamiltonians

  • The TI reconstruction section treats isotropic models, general nearest-neighbor models, and three-local chains, deriving equations and checking generic recovery and stability conditions.For isotropic models in dimensions 1, 2, and 3, second- and fourth-order data form reconstruction equations whose solutions are checked against the four sign-gauge choices.Source: §F.3 · p. 62, §F.3.1 · p. 62

F.3.1 Isotropic TI systems and their certificates

  • For isotropic TI IXZ models in dimensions 1, 2, and 3, single-site and adjacent-pair second derivatives supply equations, while fourth derivatives constrain remaining parameters.The equations are verified to have only the four solutions allowed by the sign gauges; Table 3 lists monomial coefficients of the four fourth derivatives.Source: §F.3.1 · p. 62, Table 3 · p. 63
  • The certificate uses measured-equation differences, monomial multiples through degree L, and a constant row to test for a unique solution.Its columns are all monomials of degree at most L; each row records coefficients of x^α E_j(x) for admissible α.Source: §F.3.1 · p. 63
  • Nonzero certificate determinants imply generic uniqueness of the three remaining parameters even over complex candidates; symmetries yield exactly four sign-related solutions.Time reversal and Z⊗n conjugation relate the four sign choices for S_D and J_XX.Source: §F.3.1 · p. 64
  • Gaussian smoothing controls errors in the remaining parameters by conditioning on S_D and J_XX and applying Corollary 1 in the original five coordinates.This accounts for correlated coefficients and errors from estimating S_D, J_XX, and fourth derivatives.Source: §F.3.1 · p. 64

F.3.2 Learning TI NN IXZ Hamiltonians on lattices with 𝑍-basis access

These results develop computational-basis methods for reconstructing nearest-neighbor IXZ lattice Hamiltonians and chain Hamiltonians, including spatially varying chains, up to unavoidable gauges. They establish generic uniqueness and stability bounds for these reconstructions.

F.3.2 Learning TI NN IXZ Hamiltonians on lattices with 𝑍-basis access

  • In 1D, second- and fourth-order data recover the off-diagonal and diagonal coefficients, respectively, uniquely up to unavoidable gauges under generic nonzero conditions.The off-diagonal terms are determined up to a simultaneous sign flip; fourth-order equations then recover the diagonal coefficients.Source: §F.3.2 · pp. 64–65
  • Gaussian smoothing stability bounds yield parameter error at most ε using derivative precision ε/poly(R, σ^-1, δ^-1), preserving eO(ε^-2) precision dependence.The bounds cover the nonzero reconstruction polynomials, determinants, and errors in subtracted terms and matrix entries.Source: §F.3.2 · pp. 66, 68
  • In 2D, five weighted averages over neighbor bits isolate products of diagonal couplings, which are uniquely determined under generic nonzero conditions.Sign products involving three or four neighbors eliminate terms of degree at most two in the neighbor signs.Source: §F.3.2 · p. 66
  • In 3D, nine weighted averages yield an invertible system generically, recovering diagonal couplings up to a common sign before a measurement fixes their relative sign.The nine-equation matrix has determinant 777600 at the stated parameter point.Source: §F.3.2 · pp. 67–68

F.3.3 Learning TI 3-local IXZ Hamiltonians on chains with 𝑍-basis access

  • On TI 3-local chains, second-order data recover off-diagonal coefficients, and fourth-order measurements generically distinguish eight candidates, leaving two independent sign gauges.Single-flip and adjacent-pair amplitudes determine coefficient families up to common signs; distance-two and triple flips provide additional constraints.Source: §F.3.3 · pp. 68–70
  • Fourth-order equations determine the four diagonal parameters through a generically full-rank system, with the remaining signs corresponding to the odd- and even-X gauges.The system recovers fourteen quantities, including the four energy differences used to obtain the diagonal coefficients.Source: §F.3.3 · p. 70

F.4 Spatially varying chains

  • For spatially varying chains, on the conditioning event, off-diagonal reconstruction has coefficient error at most poly(n, R, σ^-1, δ^-1) η_data.The bound applies to a uniform additive error on estimated second- and fourth-order local derivatives.Source: §F.4 · p. 71
  • Ten fourth-order polynomial constraints uniquely determine the five diagonal variables in each window for generic Hamiltonian coefficients.The certificate’s 56 × 56 matrix has determinant 765893 modulo 1000003 at an explicit specialization, proving its determinant is not identically zero.Source: §F.4 · pp. 73–74
  • The same uniqueness certificate applies to interior and boundary windows, including length-four open chains with both outside edges absent.The boundary specialization gives determinant 181025 modulo 1000003; absent outside diagonal couplings are set to zero, so endpoint combinations yield boundary fields.Source: §F.4 · pp. 72, 74
  • Applying the local reconstruction across every window recovers all fields and ZZ couplings, and retaining any local estimate prevents error accumulation along the chain.For open chains, endpoint fields come from the first and last windows; periodic-chain fields appear as middle-site fields in some window.Source: §F.4 · p. 75
  • The reconstruction achieves soft ε^-2 bounds on measurements, total evolution time, and classical post-processing while controlling errors across all windows.Simultaneous conditioning over O(n) windows and stability estimates bound every reconstructed coefficient by a polynomial in n, R, σ^-1, and δ^-1 times the data error.Source: §F.4 · p. 75

F.5 Spatially varying square lattices

For spatially varying square lattices, local transition data and polynomial systems reconstruct diagonal parameters after off-diagonal coefficients are recovered and aligned. Polynomial certificates establish generic uniqueness and support stable, efficient reconstruction.

F.5 Spatially varying square lattices

  • After recovering and globally aligning off-diagonal coefficients, the method subtracts their known contribution and uses Boolean Fourier coefficients to eliminate neighbor-side sums.The resulting polynomial equations involve only the center field and its incident ZZ couplings.Source: §F.5 · pp. 75–77
  • At an interior vertex, seven quadratic equations determine the center field and four incident ZZ couplings uniquely for generic coefficients.The equations contain five unknowns after auxiliary neighbor-side sums are eliminated.Source: §F.5 · pp. 77–78
  • At straight boundaries, an additional three-neighbor Fourier coefficient supports generic uniqueness; at corners, known incident couplings leave one field recoverable by a linear equation.The corner recovery is generic when L1 is nonzero.Source: §F.5 · pp. 78–79
  • All Z fields and ZZ couplings are recovered on periodic lattices and open rectangles, with no additional diagonal gauge to align between local reconstructions.The procedure reconstructs interior and straight-boundary centers before recovering corner fields.Source: §F.5 · p. 79
  • With the stated conditioning, reconstruction error can be reduced to ε using O(n) constant-size local systems, preserving the soft ε^-2 cost of derivative estimation.The error bound depends polynomially on n, R, σ^-1, δ^-1, and the derivative-data error.Source: §F.5 · p. 79

G Certification of identifiability of polynomial systems

  • Polynomial-system certificates address global uniqueness, closeness of noisy approximate solutions, and computational cost; a full-rank Jacobian alone guarantees neither uniqueness nor closeness.The appendix assigns these guarantees to separate certificate, stability, and algorithmic results.Source: G Certification of identifiability of… · p. 79

G.1 Exact certification

This part develops exact certification of unique parameter recovery from polynomial equations, extends the certificate to noisy data, and uses it to support coarse-grid and Gauss–Newton reconstruction. It also describes a coarse-search and refinement solver with sample, evolution-time, and classical-processing dependence of \(\tilde O(\varepsilon^{-2})\), together with parameter bounds and failure-probability allocation for local systems.

G.1 Exact certification

  • When the determinant certificate Δ is nonzero, the true tuple t is the unique complex solution to P_θ(x) = P_θ(t).The construction linearizes residual equations using monomials and their products; a rational witness with nonzero determinant certifies that Δ is not identically zero.Source: §G.1 · pp. 80–82

G.2 Quantitative bounds

  • With bounded parameters and |Δ| bounded below, a candidate’s residual bounds its distance from the true parameters; Gaussian smoothing ensures these conditions with high probability.The probability argument writes Δ as a polynomial in the independent smoothed Hamiltonian coefficients, including when the solved quantities and remaining unknowns depend on shared coefficients.Source: §G.2 · pp. 82–84

G.3 Coarse search and local refinement

  • A coarse grid of size independent of ε followed by Gauss–Newton refinement reaches parameter error ε in at most log_16(1/ε) iterations.The refinement uses the Gauss–Newton method because the system has more equations than unknowns; precision affects the iteration count logarithmically.Source: §G.3 · pp. 84–86
  • Coarse search and refinement use polynomially many operations in κ, L, and R̄, times 1 + log(1/ε), preserving tilde O(ε^-2) dependence for samples, evolution time, and classical processing.Other classical costs are linear in the number of experimental samples and O(d^2) for interpolation.Source: §G.3 · p. 86
  • Bounds on κ, L, r, and the coarse grid are polynomial in R, β^-1, and ς^-1, allowing data errors ηθ, ηy ≤ ε/poly(R, (βς)^-1).The bounds can be determined without knowing t*; the determinant lower bound holds with failure probability at most β.Source: §G.3 · pp. 86–87
  • Normalizing H by R keeps local reconstruction systems fixed in size and degree, adding only fixed powers of R to bounds and polylogarithmic factors in N/δ for base coefficients of magnitude O(1).The normalization rescales smoothing to ς/R and target coefficient error to ε/R.Source: §G.3 · p. 87
  • A union-bound allocation across radius, conditioning, and derivative-estimation events gives total failure probability at most δ, without requiring independence between local systems.Fresh samples are used when later-stage precision depends on earlier outcomes; the same allocation applies to TI protocols, and isotropic sign choices add only constantly many tests.Source: §G.3 · p. 87
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