Reconstructed Hamiltonian

• Reconstructed Hamiltonian is defined as an operator inferred from reduced data, enabling recovery of effective generators that capture key spectral, state, or dynamical properties.
• It leverages methods like eigenstate covariance analysis, projection-based inversion, and trajectory fitting to best approximate the underlying many-body model.
• Practical applications span quantum state inference, entanglement diagnostics, time-dependent control, and process tomography, enhancing model reconstruction accuracy.

Searching arXiv for recent and foundational papers on Hamiltonian reconstruction and reconstructed Hamiltonians. A reconstructed Hamiltonian is a Hamiltonian operator or Hamiltonian function inferred indirectly from reduced information rather than specified a priori. In current usage, the reconstructed object may be an effective model-space Hamiltonian, a full microscopic Hamiltonian recovered from an effective projection, a parent Hamiltonian inferred from a target state, an entanglement Hamiltonian $H_E=-\log\rho_A$, a time-dependent control Hamiltonian reconstructed from measurement records, or a classical quasi-Hamiltonian inferred from dissipative trajectories (Zheng, 2022, Zhu et al., 2018, Rattacaso et al., 2023, Siva et al., 2022, Dumont et al., 2024, Znojil, 28 Jun 2025). This suggests that the term designates a family of inverse constructions unified by a common objective: recover the generator that best explains selected spectral, state, dynamical, or thermodynamic data.

1. Scope and formal variants

Across the literature, the reconstructed Hamiltonian is almost always sought within a restricted ansatz space. In quantum many-body settings this may take the form

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$

or

$H_P=\sum_a w_a L_a,$

while in classical structure-preserving regression it is represented as

$$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$

and in translationally invariant effective-model reconstruction as

$$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$

The unknown coefficients are then fixed by eigenstate conditions, covariance minimization, trajectory fitting, Gibbs-state matching, or projection identities (Zhang et al., 2021, Zhu et al., 2018, Wu et al., 2019, Nandy et al., 2023).

Setting	Reconstructed object	Representative formulation
Model-space and spectral theory	Effective or full Hamiltonian	$$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$; $H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$
Quantum many-body state inference	Parent or entanglement Hamiltonian	$\rho_A=e^{-H_E}$; $H_P=\sum_a w_a L_a$
Dynamical or trajectory inversion	Time-dependent Hamiltonian or classical Hamiltonian function	$$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$; $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$0

A recurrent distinction is between exact reconstruction and best approximation. Some methods reconstruct the Hamiltonian only up to an overall scale, only up to a scalar function $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$1, or only within a local operator manifold. Others return a nearby Hamiltonian for which the supplied data are consistent, rather than the unique microscopic generator. This is not a defect of a particular algorithm; it is often a structural feature of the inverse problem itself (Zhu et al., 2018, Nandy et al., 2023, Znojil, 28 Jun 2025).

2. Spectral, secular, and projection-based reconstruction

In effective-Hamiltonian perturbation theory, the full Hamiltonian is written

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$2

with Hilbert space split into a model space $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$3 of dimension $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$4 and an exterior space $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$5 of dimension $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$6. The effective Hamiltonian $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$7 acts in $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$8-space and reproduces the selected exact eigenvalues associated with the $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$9-space states. The standard Fried–Ezra reconstruction inserts perturbative series

$H_P=\sum_a w_a L_a,$0

into

$H_P=\sum_a w_a L_a,$1

and truncates at order $H_P=\sum_a w_a L_a,$2. The improved construction introduces the relative characteristic polynomial

$H_P=\sum_a w_a L_a,$3

so that the reconstructed secular equation becomes

$H_P=\sum_a w_a L_a,$4

Because

$H_P=\sum_a w_a L_a,$5

this is equivalent to the usual reconstructed effective secular equation but avoids the large $H_P=\sum_a w_a L_a,$6-fold product and the attendant cancellations, with the practical advantage becoming especially pronounced when the $H_P=\sum_a w_a L_a,$7-space dimension is large (Zheng, 2022).

A distinct spectral notion appears in formal eigen-expansion reconstruction. For a complete orthonormal set $H_P=\sum_a w_a L_a,$8 with eigenvalues $H_P=\sum_a w_a L_a,$9, one writes

$$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$0

In infinite-dimensional Hilbert spaces, the associated coordinate-space series is typically divergent, and even the completeness relation

$$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$1

is only formal in the pointwise sense. The regularization proposed for the square well and harmonic oscillator inserts an Euler factor $$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$2, evaluates the absolutely convergent series for $$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$3, and then takes $$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$4. For the square well this yields

$$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$5

and for the harmonic oscillator the standard kernel of

$$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$6

Here reconstruction is mathematically valid only after distributional regularization (Bender et al., 2019).

The inverse Feshbach problem reverses the more familiar direction. Starting from the effective model-space operator

$$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$7

the problem is to reconstruct the full-space Hamiltonian. The reconstruction is shown feasible only under specific structural assumptions: a tridiagonal $$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$8 block, a doorway-state decoupling, and a compactification step that expresses the $$H_N(\mathbf{z})=\sum_{j=1}^{N} c_j \phi_j(\mathbf{z}),$$9-space resolvent through continued fractions. Sampling the effective scalar function at $$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$0 energies reduces the inverse problem to a finite coupled system of polynomial algebraic equations for the unknown tridiagonal parameters. The method is therefore exact only within a highly constrained class, and residual basis ambiguity remains in the off-diagonal factors (Znojil, 28 Jun 2025).

3. Reconstruction from states, entanglement, and parentage

In entanglement-Hamiltonian reconstruction, the object of interest is defined by

$$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$1

Rather than computing $$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$2 directly, one reconstructs a compact operator

$$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$3

from a single entanglement eigenstate $$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$4, chosen as the highest-weight eigenstate of $$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$5. The coefficients are obtained from the covariance matrix

$$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$6

whose smallest-eigenvalue eigenvector minimizes

$$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$7

Afterward one sets $$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$8 and determines $$H=\sum_{j,\vec\alpha} a_{\vec\alpha} O_j(\vec\alpha).$$9 by optimizing the density-matrix fidelity between $$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$0 and $$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$1. In the high-fidelity limit, the reconstructed density matrix becomes essentially the diagonal ensemble of the exact $$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$2 in the reconstructed eigenbasis. For the critical spin-$$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$3 chain at $$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$4, the smallest eigenvalue satisfies $$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$5 to $$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$6 and the density-matrix fidelity is $$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$7 or better for $$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$8 with $$\det R(\lambda)=\det[E-H_{\mathrm{eff}}(\lambda)]/\det[E-H_P]$$9. For the ladder at $H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$0, $H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$1, the reconstructed couplings are dominated by

$H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$2

with $H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$3 and fidelity above $H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$4 (Zhu et al., 2018).

Parent-Hamiltonian reconstruction from a target many-body wavefunction is formulated differently in inverse quantum annealing. One starts from a simple state $H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$5 with known $H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$6, chooses a path $H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$7 to $H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$8, and defines the pseudo-Hamiltonian

$H_{\mathrm{eff}}(E)=PHP+PHQ\frac{Q}{E-QHQ}QHP$9

A unitary generated by this projector induces an auxiliary Hamiltonian satisfying

$\rho_A=e^{-H_E}$0

Projecting onto an $\rho_A=e^{-H_E}$1-local operator basis gives a practical evolution equation in which the reconstructed local Hamiltonian is determined entirely from local expectation values of commutators in the target state. For sufficiently large annealing time $\rho_A=e^{-H_E}$2, the convergence monitor obeys

$\rho_A=e^{-H_E}$3

The method works well when the state path has finite correlation length; near criticality the required locality range can grow substantially, even linearly with system size (Rattacaso et al., 2023).

Entanglement-guided parent reconstruction also appears in the Jastrow–Gutzwiller and $\rho_A=e^{-H_E}$4 chain settings. For the Jastrow–Gutzwiller family, the reconstruction is restricted to the critical window $\rho_A=e^{-H_E}$5, where participation spectra, entanglement entropy, and correlation functions indicate a relativistic low-energy description. The Bisognano–Wichmann ansatz is then optimized by minimizing the relative entropy between the exact reduced density matrix and the BW form. Exact points are recovered at $\rho_A=e^{-H_E}$6 for the XX chain and $\rho_A=e^{-H_E}$7 for the Haldane–Shastry model, whereas $\rho_A=e^{-H_E}$8 is explicitly problematic because the exact parent is the XXZ ferromagnetic point with quadratic dispersion $\rho_A=e^{-H_E}$9, hence nonrelativistic. In the $H_P=\sum_a w_a L_a$0 chain, the correlation-matrix reconstruction and the entanglement-spectrum reconstruction both recover the Hamiltonian parameter, but the correlation-based method is reported as more robust, while the BW method is more restrictive and experimentally demanding (Turkeshi et al., 2019, Jacoby et al., 2021).

4. Reconstruction from dynamics, tomography, and trajectories

Time-dependent quantum Hamiltonians can be reconstructed directly during coherent evolution using continuous weak measurements. In a two-transmon device, the averaged resonator field tracks the qubit $H_P=\sum_a w_a L_a$1 expectation value in the adiabatic regime, so the measurement record provides $H_P=\sum_a w_a L_a$2 without interrupting the dynamics. For one qubit, the target Hamiltonian is parameterized as

$H_P=\sum_a w_a L_a$3

and the reconstruction iterates between a pseudoinverse update for the drive amplitudes and forward integration of a Lindblad equation including calibrated dephasing. Single-qubit tests reconstruct calibrated $H_P=\sum_a w_a L_a$4 pulses, out-of-phase two-quadrature pulses, and added sinusoidal error components; the reconstructed final states match tomography with about $H_P=\sum_a w_a L_a$5–$H_P=\sum_a w_a L_a$6 fidelity. In the two-qubit setting, the method recovers $H_P=\sum_a w_a L_a$7 and $H_P=\sum_a w_a L_a$8 entangling terms, while longitudinal $H_P=\sum_a w_a L_a$9, $$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$0, and $$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$1 terms require preconditioning or second-order updates. A central point is that the reconstruction reveals the Hamiltonian path, not merely the final gate, so sinusoidal amplitude errors, phase miscalibration, and off-resonant dynamical frequency shifts become visible even when final-state characterization appears benign (Siva et al., 2022).

For time-independent Hamiltonians, a quantum-process-tomography route is available. The two-step optimization algorithm prepares probe states, estimates output states via linear regression estimation, assembles the process matrix, recovers a rank-one unitary factor, and reconstructs the Hamiltonian from the propagator through Schur decomposition and phase unwrapping. In the natural basis, the method has computational complexity $$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$2, where $$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$3 is the Hilbert-space dimension, and the paper derives the error upper bound

$$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$4

with $$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$5 the resource number for tomography of each output state (Wang et al., 2016).

Variational time-series learning reconstructs a Hamiltonian from observable dynamics rather than tomography of a process matrix. The ansatz is written in a Pauli basis, the time propagation is implemented through Trotterization, and the parameters are optimized against the cost

$$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$6

Gradients are obtained from parameterized circuits, and validation is performed on unseen observables and unseen states. Benchmarks include XX and ZZ couplings and transverse-field Ising Hamiltonians, with an SU(3) extension formulated in the Gell-Mann basis (Gupta et al., 2022).

Classical and driven-dissipative reconstructions use trajectory data rather than quantum observables. In canonical Hamiltonian learning from trajectories, one reconstructs $$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$7 directly from

$$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$8

using a basis expansion and least-squares regression on numerically differentiated data. Because the learned model is enforced in Hamiltonian form, the reconstructed Hamiltonian is conserved by algebraic identity. In ringdown reconstruction, dissipation is turned into a resource: the slow-flow dynamics

$$H_t(t)/\hbar=\frac{1}{2}\vec\Omega(t)\cdot\vec\sigma$$9

permit reconstruction of a rotating-frame quasi-Hamiltonian over a large region of phase space from integrated ringdown trajectories. The same measurements also provide access to the symplectic norm $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$00, distinguishing particle-like excitations around minima from hole-like excitations around dissipation-stabilized maxima (Wu et al., 2019, Dumont et al., 2024).

5. Reconstruction as diagnostic, metric, and inverse-map geometry

A reconstructed Hamiltonian can serve as a diagnostic rather than as the final physical model. In variational many-body studies of the square-lattice spin-$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$01 $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$02-$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$03 Heisenberg model, Hamiltonian reconstruction asks for what nearby Hamiltonian a candidate wavefunction would actually be an approximate eigenstate. The covariance matrix

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$04

is diagonalized, and the lowest-eigenvalue direction gives the reconstructed Hamiltonian. Applied to CNN and RBM states on a $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$05 periodic lattice, the reconstructed Hamiltonians are typically less frustrated, show easy-axis anisotropy near the maximal frustration point $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$06, and suppress quantum fluctuations in the large-$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$07 regime. Negative $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$08 near $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$09 indicates that the learned states avoid the frustration point, while large positive $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$10 and large negative $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$11 at large $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$12 push the model toward stripe-favoring, more classical physics. The analysis isolates what energy alone cannot: whether the variational failure is dominated by anisotropy, wrong frustration, or suppressed fluctuations (Zhang et al., 2021).

The same logic underlies the Hamiltonian-reconstruction distance proposed as a success metric for VQE. A covariance matrix over a chosen operator basis is measured on the current VQE state, its lowest-eigenvalue eigenvector defines reconstructed coefficients $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$13, and the metric is

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$14

For the 11-qubit 1D transverse-field Ising model and the 6-qubit $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$15-$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$16 transverse-field Ising model, $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$17 tracks convergence more reliably than energy plateaus alone. In one simulation, the energy appears to flatten around iterations $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$18–$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$19, yet the HR distance remains around $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$20; even around iteration $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$21, fidelity is only about $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$22. The paper also reports that the usefulness of the metric depends on noise, and that on the order of $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$23 shots is needed for a reasonable signal-to-noise ratio in the examples studied. A common misconception is therefore explicitly addressed: a low HR distance indicates eigenstate-likeness within the chosen span, but it does not by itself certify proximity to the ground state, since excited states can also reconstruct the target Hamiltonian well (Moon et al., 2024).

Inverse-map geometry becomes especially visible in supervised learning from eigenstate measurements. With the ansatz

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$24

the forward map $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$25 is smooth and nearly unique for low-lying eigenstates but becomes ill-posed for middle-lying eigenstates, where trajectories in expectation-value space develop crossings and bridges. For low-lying states, a shallow feedforward network with two hidden layers of $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$26 and $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$27 neurons, Leaky ReLU activation, Adam with learning rate $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$28, and about $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$29 epochs is sufficient; using $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$30 samples of coefficient vectors, the reported fidelities are above $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$31 for most eigenstates. For middle-lying states, the paper introduces sign-sector classification and transfer learning, with a three-hidden-layer classifier of $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$32 neurons per layer and learning rate $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$33, obtaining fidelities above $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$34 in the more difficult fully connected setting. The same work also uses neural-network predictions as initialization for BFGS, greatly improving optimization success rates (Cao et al., 2020).

6. Thermal, nonequilibrium, and geometric effective reconstructions

Thermal and effectively thermal data provide another route to reconstructed Hamiltonians. A deep-learning-assisted variational scheme preprocesses local measurements with an autoencoder trained on vectors of local observables. When the data are thermal, the test error drops sharply at bottleneck size $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$35, indicating a one-dimensional latent manifold. Operators with the largest average gradient along this manifold are selected as candidate Hamiltonian terms, and the coefficients in

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$36

are fitted by matching Gibbs-state predictions. For strictly local Hamiltonians, if the measured support satisfies $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$37, the relative reconstruction error is $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$38; if $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$39, the error remains $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$40. In Floquet systems, the same method reconstructs the prethermal effective Hamiltonian without using BCH or Floquet–Magnus as prior input, and in the heating regime it reconstructs an instantaneous quasilocal Hamiltonian $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$41 whose support grows with time. The same behavior is reported for random multipolar driving (Nandy et al., 2023).

Finite-temperature equilibrium reconstruction can also be cast as a compressed-sensing problem. For an unknown sparse $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$42-qubit Hamiltonian

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$43

in the high-temperature regime

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$44

the Gibbs state has an approximately sparse Pauli polarization vector. Randomized Pauli measurements after a scrambling unitary define a compressed linear system, and the reconstruction solves

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$45

subject to the measurement constraints, then converts the estimated thermal state into

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$46

The cited sufficient compressed-sensing condition is

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$47

with $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$48 the sparsity. For $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$49- and $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$50-body cases, the reported speedups over the comparison method are about $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$51–$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$52 for $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$53, $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$54–$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$55 for $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$56, and $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$57–$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$58 for $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$59 (Rudinger et al., 2014).

A more speculative extension appears in curved spacetime, where the inverse problem starts from a macroscopic energy density profile rather than microscopic measurements. Under local thermal equilibrium with

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$60

the paper writes the density schematically as a moment of $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$61 weighted by $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$62 and proposes the effective inverse map

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$63

The equilibrium hypothesis is tied to Tolman redshift,

$H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$64

or more generally $H[\{c_n\}] = \sum_{n=1}^{N} c_n O_n,$65 when a timelike Killing field exists. Examples discussed include FLRW cosmology, polymerized Loop Quantum Cosmology, AdS/CFT, and the SYK model. The reconstruction is explicitly approximate and effective rather than universally exact, but it enlarges the notion of reconstructed Hamiltonian from laboratory inverse problems to coarse-grained gravitational settings (Momeni, 28 Jul 2025).

Taken together, these constructions show that a reconstructed Hamiltonian need not coincide with a unique microscopic generator. It may instead be an effective secular object, a parent Hamiltonian within a prescribed operator space, an entanglement generator, a local approximation to a long-range model, a time-local control Hamiltonian, or a thermodynamic inverse image of observed densities. The decisive questions are therefore not only whether a reconstruction exists, but also what data define it, what operator manifold constrains it, and in what sense—exact, approximate, local, quasilocal, or observer-dependent—it should be interpreted.