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Polynomially efficient quantum enabled variational Monte Carlo for training neural-network quantum states for physico-chemical applications

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Restricted Boltzmann machine representations of quantum states can be trained by quantum-assisted variational Monte Carlo with linear circuit width and depth, constant measurement count, and polynomial storage, covering both amplitude and…

desk verdict A genuinely new surrogate-network plus Trotterized-proposal scheme for training complex RBMs, with credible small-system results, but the optimal-efficiency claims outrun the error analysis. read the letter →

arxiv 2412.12398 v1 pith:M2RVPFOL submitted 2024-12-16 quant-ph cond-mat.str-elphysics.chem-ph

Manas Sajjan , Vinit Singh , Sabre Kais This is my paper · ORCID
classification quant-phcond-mat.str-elphysics.chem-ph
keywords neuralquantumstatesrestrictedBoltzmannmachinequantum-assistedvariationalMonteCarlosurrogatedistributionHamiltoniansimulationzero-varianceextrapolationground-stateenergyMarkovchain
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that neural-network quantum states—restricted Boltzmann machines with complex-valued parameters in the concrete implementation—can be trained by a variational Monte Carlo loop whose quantum resources are polynomially efficient. The central move is to fit a simple quadratic surrogate distribution to the RBM's diagonal probability weights from O($n^{2}$) configurations, then use a short Trotterized quantum circuit as the Markov-chain proposal for sampling that surrogate. If the claim holds, ground states of spin models and molecular electronic Hamiltonians can be learned with linear circuit width and depth, constant measurement count (independent of n), no mid-circuit measurements, and polynomial storage, with amplitude and phase both learned from the same complex parameters. The paper validates the workflow on the XXZ spin chain and on LiH and H2O potential energy surfaces, reporting near-exact energies and chemical accuracy after zero-variance extrapolation.

What carries the argument

The load-bearing object is the surrogate network $G_2$: a fully-connected Ising model over $n$ classical spins with distribution $\phi(\vec{v})\propto \exp\bigl(-\beta\sum_i l_i(\vec{X})v_i + \sum_{i,j} J_{ij}(\vec{X})v_i v_j\bigr)$, fitted to the diagonal RBM distribution $\rho_{vv}(\vec{X})$ from $O(n^2)$ configurations. Theorem II.1 guarantees that any discrete distribution $P(\vec{v})$ can be written as $Z\,\kappa(\vec{v})\,\phi(\vec{v})$ with $\phi$ of user-chosen polynomial degree $k$, which is what lets the surrogate stand in for the full RBM. The sampling machinery is the quantum proposal: a Trotterized time evolution $U(\tau,\gamma)=e^{-i(\gamma h_1+(1-\gamma)h_2)\tau}$ with $h_1$ generated by the surrogate's fields and couplings and $h_2$ a transverse-field mixer, used as the Metropolis-Hastings proposal distribution; its spectral gap decays roughly three times more slowly than classical local, uniform, or Haar-random proposals, giving faster mixing. A final piece is the kernel $\kappa(\vec{v}, \vec{X})$, which carries the residual difference between surrogate and true distribution and is included in the local-energy estimate of Eq. (8).

What would settle it

Fix a driver Hamiltonian whose dominant local-energy contributions come from configurations that the $q$-largest heuristic misses or that lie outside the fitted $O(n^2)$ set; compute $\mu_{\langle H\rangle}$ and its variance from Eq. (8) with the paper's constant-$N_s$ protocol as $n$ grows. If the estimator develops a bias that grows with $n$, or if the variance stops being compensated by the fixed measurement count, the polynomial-efficiency claim collapses.

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Extended reading notes

Core claim

The paper's central discovery is that the intractable diagonal distribution $\rho_{vv}(\vec{X})$ of a complex-valued RBM ansatz can be factored as $\kappa(\vec{v},\vec{X})\,\phi_{\vec{v}}(\vec{l}(\vec{X}),\vec{J}(\vec{X}))$, where $\phi$ is the distribution of a fully-connected quadratic Ising surrogate network and $\kappa$ is a configuration-dependent prefactor; Theorem II.1 proves such a factorization exists for any discrete distribution over $n$-bit configurations. The authors give a constructive, data-driven recipe that fixes the surrogate parameters $(\vec{l},\vec{J})$ by weighted least-squares plus BFGS refinement on only $O(n^2)$ sampled configurations, choosing the $q$ largest-probability configurations plus random ones. Sampling from $\phi$ is then done by Metropolis-Hastings with a quantum proposal $U(\tau,\gamma)=e^{-i(\gamma h_1+(1-\gamma)h_2)\tau}$ built from the surrogate couplings $h_1=\sum_i l_i \sigma_z^{(i)}+\sum_{i,j} J_{ij}\sigma_z^{(i)}\sigma_z^{(j)}$ and a transverse mixer $h_2=\sum_i \sigma_x^{(i)}$, implemented with Trotterization. The paper claims this yields linear circuit width $O(n)$, linear circuit depth $O(\tau n)$, $O(N_s)$ queries independent of $n$, no mid-circuit measurements, $O(mn)$ storage, and analytical gradients that involve both amplitude and phase; benchmark results on XXZ, LiH, and H2O are reported to match exact diagonalization and CASSCI, with zero-variance extrapolation bringing errors to or below chemical accuracy.

Load-bearing premise

The quadratic surrogate distribution $\phi$ fitted from $O(n^2)$ configurations faithfully reproduces the RBM diagonal distribution over the configurations that dominate the energy and gradient estimates, with the residual captured by $\kappa(\vec{v})$; the paper supplies no error bound for the fit, and Appendix D concedes that the $q$-largest-configuration selection is heuristic and can fail on specially curated instances.

Editorial extensions

If this is right

  • Training an RBM neural quantum state with quantum-assisted sampling needs $O(n)$ qubits, $O(\tau n)$ depth per Trotter layer, and $O(N_s)$ queries with $N_s$ independent of $n$, so the end-to-end resource count is polynomial in system size.
  • Because the RBM parameters are complex-valued and the surrogate maps them holistically, both amplitude and phase of the target state are learned without a separate classical phase-preprocessing step, enlarging the trial space over earlier amplitude-only quantum training.
  • Quantum proposals built from the surrogate Hamiltonian show a spectral gap that decays about three times slower than classical local, uniform, or Haar-random proposals, which translates into shorter mixing time and roughly five-fold smaller $\ell^2$ sampling error in the paper's $n=8$ tests.
  • On the XXZ model, relative ground-state energy errors remain below $5\times10^{-3}$ across antiferromagnetic, XY, and ferromagnetic phases when zero-variance extrapolation is applied.
  • The same workflow reproduces CASSCI ground-state energies for LiH and H2O over stretched bonds and distorted angles, with errors at or below the chemical-accuracy threshold ($10^{-3}$ a.u.) for most points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The constant-measurement-count claim is only as strong as the surrogate fit: if the residual kernel $\kappa(\vec{v})$ varies strongly over configurations that carry large local energy, the variance of Eq. (8) could grow with $n$ and force $N_s$ to grow; a stress test on non-local (Jordan-Wigner) molecular Hamiltonians with fixed $N_s$ would settle this.
  • The surrogate construction is distribution-agnostic, so the same fitting-plus-quantum-proposal recipe could be applied to autoregressive or feed-forward neural quantum states, not just RBMs; the paper proves generality but only demonstrates it on RBMs.
  • The $q$-largest configuration selection is a heuristic, and the paper's own Appendix D acknowledges adversarial instances where it fails; one can probe this by constructing RBM parameter sets with two well-separated probability peaks and checking whether the fitted surrogate keeps the second peak accurate.
  • Since ZVE corrects only the energy estimate, not the trained parameters, claims about the quality of the final state rest on the variance diagnostic; an independent fidelity check against the exact state would be a stronger validation than energy alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a quantum-assisted variational Monte Carlo protocol for training Restricted Boltzmann Machine neural quantum states. A quadratic Ising surrogate distribution phi(v) is fitted from O(n^2) configurations of the RBM diagonal rho_vv; quantum circuits based on Trotterized time evolution are used as MCMC proposal distributions to sample phi(v); energy and gradients are estimated from these samples; ground states of the XXZ spin model and LiH/H2O electronic Hamiltonians are computed and benchmarked against exact diagonalization and CASSCI. The abstract claims that the algorithm scales linearly in circuit width and depth, needs a constant number of measurements, avoids mid-circuit measurements, requires polynomial storage, and is optimally efficient.

Significance. If the efficiency claims were rigorously supported, the paper would be a valuable contribution to quantum-assisted NQS training: the idea of using a surrogate distribution to make the RBM diagonal samplable, and the quantum-circuit proposal for faster mixing, are appealing. The numerical benchmarks for small systems show good agreement with exact references, and Appendix B gives a standard but correct monomial-expansion proof of representability. However, the central resource claims are not backed by a control on the surrogate fitting error, so the paper's main advertised contribution is currently not established.

major comments (4)
  1. [Section III, Eq. (8); Appendix D] The central estimator is not connected to Eq. (6) by any controllable approximation. Eq. (5) defines rho_vv(X) ∝ kappa(v,X) phi_v(l(X),J(X)), and Eq. (8) samples from phi with kernel kappa. However, the fitting protocol in Appendix D fits log rho_vv to a quadratic polynomial and terminates with rho_vv = phi*kappa + epsilon(v), where epsilon(v) is called the fitting error; the algorithm never constructs or bounds kappa(v). If the applications implicitly omit kappa and treat phi as 'a close approximant' to rho_vv, then Eq. (8) is biased by an uncontrolled fitting error. If kappa is instead included through the known RBM functional form, the estimator is unbiased only with perfect kappa, and its variance is governed by the importance ratio rho_vv/phi, which is not bounded. Theorem II.1 guarantees only the existence of some high-degree polynomial with a prefactor kappa bounded by the exponentially large expression M(2^n − sum_j C(n,j)); it says nothing about the k=2 truncation actually used. Appendix D itself labels the q-largest configuration selection 'heuristic' and admits 'it might be possible to curate special instances where it might fail.' Thus the polynomial-efficiency and constant-measurement claims do not follow from the presented analysis.
  2. [Section III, step 2; Section IV] The 'constant in measurement count' claim is not established by the paper's own sample-complexity formula. Step 2 of Section III gives N_s = O(Var(mu_H)/epsilon^2), and Section IV states that the number of circuit queries is O(N_s) and is 'independent of system size n.' But Var(mu_H) is never shown to be independent of n; because mu_H depends on the ratio rho_vv/phi through kappa, the variance may grow with n and with the misfit of the quadratic surrogate. Without a bound on Var(mu_H), the statement that the measurement count is constant in system size is unsupported.
  3. [Section IV; Abstract] The abstract's claim that the algorithm 'scales linearly with circuit width and circuit-depth' is inconsistent with the Trotter analysis in Section IV. The Trotter number is N_trot approximately O((sum_{i=1}^{n^2+2n} ||chi_i||_1 tau)^{1+1/p}/eps^{1/p}), where the sum runs over O(n^2) terms. Even if a single Trotter layer has depth O(n), the total circuit depth is O(n) N_trot and therefore grows superlinearly with n unless the couplings l_i and J_ij are assumed to vanish or are uniformly bounded in a way not stated in the paper. If 'linear in circuit-depth' is intended per Trotter layer, this should be stated explicitly rather than as an overall scaling claim.
  4. [Section V] The numerical demonstrations are limited to n=8 spins and to active spaces of 8 and 12 spin orbitals, and they do not test the resource scaling advertised in the abstract. In particular, they provide no evidence that the O(n^2)-configuration fitting step of Appendix D yields a reliable surrogate as n grows, especially in view of the admitted heuristic nature of the q-largest configuration search. The agreement with exact diagonalization and CASSCI is encouraging for small n, but it is not a substitute for an error bound or for scaling tests.
minor comments (5)
  1. [Section II; Appendix A] The notation shifts between v_i in {1,-1} and integer-indexed binary configurations; Appendix A should state the explicit mapping between the two conventions.
  2. [Section III; Appendix D] There are several typos and incomplete formulas: 'doesnt', 'precludes the need to include the factor ... in Eq.9' has mismatched parentheses, and 'See SectionFinAppendix' is missing a space.
  3. [Appendix D.2] The algorithm is described as returning the q-largest configurations, but the text later admits the procedure is heuristic and may fail; the output should be labeled as an approximate list to avoid ambiguity.
  4. [Appendix F] The text states that a k-qubit R_zz...z gate requires 'only 2k two-qubit entangling gates,' but the CNOT decomposition in Fig. 7 uses 2(k-1) CNOT gates; this should be corrected for consistency with the main-text statement about R_zz.
  5. [Section IV] The notation O(2(nm+n+m)+1) for the total number of observables is confusing; since the leading term is O(nm), the display should be simplified or clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation; the central energy estimates are validated against external exact diagonalization and CASSCI benchmarks.

full rationale

The algorithm's load-bearing quantities are the RBM density rho_vv and the surrogate phi fitted to it. The fitting target is the NQS's own diagonal distribution, not the energy; the energy estimator (Eq. 8) additionally involves the local energy E_loc, which is evaluated from the driver Hamiltonian and the RBM off-diagonal elements, so the fitted phi does not by construction determine the reported ground-state energies. Theorem II.1 is a generic existence statement for P = kappa*phi over the Boolean cube, proved in Appendix B from an orthonormal monomial basis, and is not imported from the authors' prior work. Self-citations (e.g., [28,43,45]) describe the earlier algorithm being replaced and are contextual, not load-bearing for the polynomial-resource claim. The numerical benchmarks are against exact diagonalization and CASSCI, external references. A correctness gap exists: the paper never explicitly constructs kappa(v) or bounds the fitting error epsilon(v) from Appendix D, so the unbiased form of Eq. 8 is not shown to be realized by the implemented sampler; this is an unsupported approximation, not a circular reduction of the claim to its inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central efficiency claim rests on the surrogate approximation and the fixed Trotterized proposal, neither of which carries a proof of approximation error or scaling with system size. The main formal result (Theorem II.1) is a standard expansion with a vacuous bound on kappa.

free parameters (4)
  • surrogate polynomial degree k = 2
    Choice of quadratic Ising surrogate; higher k would require k-qubit gates, but no approximation guarantee is given for k=2.
  • fitting sample mix ratio = 25% top / 75% random
    Hyperparameter controlling surrogate fit; no systematic study is provided.
  • quantum proposal parameters (tau, gamma, dt) = tau=11, gamma=0.425, dt=0.2
    Hand-picked after comparing variants D-H on the same class of distributions; this selection loads the spectral-gap claim.
  • number of fitting configurations = O(n^2), exact count unspecified
    Used to fit l,J; no error analysis for how this count controls approximation error.
assumptions (5)
  • standard math Monomial functions F_a(v) form a basis for functions on {-1,1}^n.
    Used in Lemma B.1 and B.3; this is the standard parity/Fourier basis on the Boolean cube.
  • standard math Metropolis-Hastings with a symmetric proposal converges to the target distribution under ergodicity.
    Used in Eq. (9) to justify sampling from phi(v); standard MCMC theory.
  • domain assumption rho_vv is sufficiently close to a quadratic exponential family so that kappa(v) has small variance.
    The entire surrogate construction in Section II and Appendix D assumes this; no bound is proven for the fitted l,J.
  • domain assumption Trotterized evolution with N_trot=55 and dt=0.2 approximates the exact proposal unitary well enough for sampling.
    Section IV gives a general Trotter error bound but does not evaluate it for the fitted l,J values at n=8 or n=12.
  • domain assumption The spectral-gap advantage measured for n=8 random l,J instances persists for larger n and for actual surrogate distributions during training.
    Figure 2 uses n=8 and random parameter instances; no scaling study is provided.
invented entities (1)
  • Surrogate Ising network G2
    purpose: Provide a simple, explicitly known distribution phi that approximates the NQS diagonal and can be sampled by a quantum circuit; also defines h1 for the quantum proposal.
    It is an algorithmic construct, not independently observable; its validity depends entirely on the fitting quality.

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Cite this review

Pith. "Pith review of Polynomially efficient quantum enabled variational Monte Carlo for training neural-network quantum states for physico-chemical applications." pith.science (2026). https://pith.science/paper/M2RVPFOL

@misc{pith2026241212398,
  author       = {Pith},
  title        = {Pith review of: Polynomially efficient quantum enabled variational Monte Carlo for training neural-network quantum states for physico-chemical applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2RVPFOL}},
  note         = {Machine review of arXiv:2412.12398}
}
read the original abstract

Neural-network quantum states (NQS) offer a versatile and expressive alternative to traditional variational ans\"atze for simulating physical systems. Energy-based frameworks, like Hopfield networks and Restricted Boltzmann Machines, leverage statistical physics to map quantum states onto an energy landscape, functioning as memory descriptors. Here, we show that such models can be efficiently trained using Monte Carlo techniques enhanced by quantum devices. Our algorithm scales linearly with circuit width and depth, requires constant measurements, avoids mid-circuit measurements, and is polynomial in storage, ensuring optimal efficiency. It applies to both phase and amplitude fields, significantly expanding the trial space compared to prior methods. Quantum-assisted sampling accelerates Markov Chain convergence and improves sample fidelity, offering advantages over classical approaches. We validate our method by accurately learning ground states of local spin models and non-local electronic structure Hamiltonians, even in distorted molecular geometries with strong multi-reference correlations. Benchmark comparisons show robust agreement with traditional methods. This work highlights the potential of combining machine learning protocols with near-term quantum devices for quantum state learning, with promising applications in theoretical chemistry and condensed matter physics.

Figures

Figures reproduced from arXiv: 2412.12398 by the authors.

Figure 1
Figure 1. FIG. 1. (a) The initial neural-network quantum state, shown on the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Various proposal matrices for an [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The relative errors of the estimated ground-state energy [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The ground-state energy (a.u.) vs.bond length ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) The ground-state energy (a.u.) vs. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The figure demonstrates how the choice of configuration significantly impacts the quality of the model’s predictions. In both plots, [PITH_FULL_IMAGE:figures/full_fig_p029_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Decomposition of [PITH_FULL_IMAGE:figures/full_fig_p034_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Decomposition of [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]

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Forward citations

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    It is always possible to find a representation of𝑃(⃗ 𝑣)as 𝑃(⃗ 𝑣) = 𝜅(⃗ 𝑣) 𝑒 ∑ ⃗ 𝑎∈{0,1}𝑛,𝐻(⃗ 𝑎)≤𝑘𝐶⃗ 𝑎𝐹⃗ 𝑎(⃗ 𝑣) ∑ ⃗ 𝑣𝑒 ∑ ⃗ 𝑎∈{0,1}𝑛,𝐻(⃗ 𝑎)≤𝑘𝐶⃗ 𝑎𝐹⃗ 𝑎(⃗ 𝑣) = 𝜅(⃗ 𝑣)𝜙(⃗ 𝑣) (B6) where𝐹⃗ 𝑎(⃗ 𝑣)= ∏ 𝑖𝑣𝑎𝑖 𝑖 ,⃗ 𝑎=(𝑎1,𝑎2....𝑎𝑛)∈{0 ,1}𝑛, ⃗ 𝑣=(𝑣1,𝑣2,...𝑣𝑛) ∈ 𝕍, 𝐶⃗ 𝑎∈ ℝ,and 𝐻(⃗ 𝑎)= ∑𝑛 𝑖 𝑎...

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    In the mostgeneric case,∃𝑀 ∈ ℝ ≥0such that|log(𝜅(⃗ 𝑣))|is upper bounded as |log(𝜅(⃗ 𝑣))| ≤𝑀 ( 2𝑛− 𝑘∑ 𝑗=0 𝑛𝐶𝑗 ) (B7) Proof

    𝜅(⃗ 𝑣)is a configurationdependant non-negative prefactor. In the mostgeneric case,∃𝑀 ∈ ℝ ≥0such that|log(𝜅(⃗ 𝑣))|is upper bounded as |log(𝜅(⃗ 𝑣))| ≤𝑀 ( 2𝑛− 𝑘∑ 𝑗=0 𝑛𝐶𝑗 ) (B7) Proof. We establish each of the assertions as follows:

  159. [168]

    Statement 1:𝑃(⃗ 𝑣)= 𝜅(⃗ 𝑣) 𝑒 ∑ ⃗ 𝑎∈{0,1}𝑛,𝐻(⃗ 𝑎)≤𝑘𝐶⃗ 𝑎𝐹⃗ 𝑎(⃗ 𝑣) ∑ ⃗ 𝑣𝑒 ∑ ⃗ 𝑎∈{0,1}𝑛,𝐻(⃗ 𝑎)≤𝑘𝐶⃗ 𝑎𝐹⃗ 𝑎(⃗ 𝑣) = 𝜅(⃗ 𝑣)𝜙(⃗ 𝑣) Subproof. We can prove this assertion the following way 𝑃(⃗ 𝑣) = 𝑒 ∑ ⃗ 𝑎∈{0,1}𝑛,⃗ 𝑎≠⃗0𝐶⃗ 𝑎𝐹⃗ 𝑎(⃗ 𝑣) ∑ ⃗ 𝑣𝑒 ∑ ⃗ 𝑎∈{0,1}𝑛,⃗ 𝑎≠⃗0𝐶⃗ 𝑎𝐹⃗ 𝑎(⃗ 𝑣) (𝑆𝑒𝑒 𝐿𝑒𝑚𝑚𝑎𝐵.4)...

  160. [169]

    It is clear from the functional form of𝜅(⃗ 𝑣)inStatement 1above that𝜅(⃗ 𝑣)= 𝑒 ∑ ⃗ 𝑎∈{0,1}𝑛,⃗ 𝑎≠⃗0,𝐻(⃗ 𝑎)>𝑘𝐶⃗ 𝑎𝐹⃗ 𝑎(⃗ 𝑣) ≥0and is also 22 explicitly configuration⃗ 𝑣dependant

    Statement2: 𝜅(⃗ 𝑣)isaconfigurationdependantnon-negativeprefactorupperboundedas |log(𝜅(⃗ 𝑣))| ≤𝑀 ( 2𝑛−∑𝑘 𝑗=0 𝑛𝐶𝑗 ) Subproof. It is clear from the functional form of𝜅(⃗ 𝑣)inStatement 1above that𝜅(⃗ 𝑣)= 𝑒 ∑ ⃗ 𝑎∈{0,1}𝑛,⃗ 𝑎≠⃗0,𝐻(⃗ 𝑎)>𝑘𝐶⃗ 𝑎𝐹⃗ 𝑎(⃗ 𝑣) ≥0and is also 22 explicitly confi...

  161. [170]

    Ideally,wewoulduseallpossibleconfigurationstofittheseparameters

    Sampling configurations for fitting To ensure that the surrogate distribution closely approximates the actual distribution, we aim to fit the parameters of the polynomialfunctionthroughanon-linearfittingprocess. Ideally,wewoulduseallpossibleconfigurationstofittheseparameters. ...

  162. [171]

    Algorithm for finding q-largest configurations Givenaprobabilitydistribution 𝜌𝑣 𝑣whichhasitssupportover 𝑂(2𝑛)spinconfigurationsas ⃗ 𝑣∈ 𝕍. Weneedtofind 𝑞-spinconfig- urationsthatmaximize 𝜌𝑣 𝑣 whichfromthemaintextisdefinedas: 𝜌𝑣 𝑣= exp ( −2𝛽∑𝑛 𝑖=1𝑅𝑒(𝑎𝑖)𝑣𝑖 )∏𝑚 𝑗=1|cosh(𝑏𝑗+∑𝑛 𝑖=1𝑤...

  163. [172]

    Initialize a listconfig to store the candidate spin configurations and another listresultto store the spin configuration with the largest𝜌𝑣 𝑣

  164. [173]

    These seeds are chosen because they individually maximize different terms in𝜌𝑣 𝑣

    Choose{-sgn(Re(a)),sgn(Re(𝑤𝑗)),and-sgn(Re( 𝑤𝑗))}asthefirst(2m+1)contenderstothelargestspinconfigurationsand store them inconfig. These seeds are chosen because they individually maximize different terms in𝜌𝑣 𝑣

  165. [174]

    Compute𝜌𝑣 𝑣 (upto normalization) for each (2m+1) configuration, sort them in descending order of their𝜌𝑣 𝑣, and remove any duplicates

  166. [175]

    Remove it fromconfig

    Run q iterations: (a) Select a configuration𝑣𝑖∈configwith the largest𝜌𝑣 𝑣 and store it inresult. Remove it fromconfig. (b) Perform single-site perturbations on𝑣𝑖, flipping each spin to generate𝑛new candidate configurations{𝑣𝑗 𝑖}𝑛 𝑗=1. (c) For each new configuration𝑣𝑗 𝑖 not alr...

  167. [176]

    Time Complexity: The algorithm runs in𝑂(𝑞𝑛)

    Returnresult- the approximate q-best configurations. Time Complexity: The algorithm runs in𝑂(𝑞𝑛). At each iteration, we generate𝑂(𝑛) new configurations and insert each into a sorted list. Depending on the choice of data structure, the insertion operation can take𝑂(log(𝑞)) time...

  168. [177]

    First,wefitthelogarithmofthe 𝜌𝑣 𝑣 distributiontoapolynomial model through a weighted least-squares (LSQ) fitting procedure

    Final Fitting Algorithm Weachievethebestfitusingatwo-foldoptimizationprotocol. First,wefitthelogarithmofthe 𝜌𝑣 𝑣 distributiontoapolynomial model through a weighted least-squares (LSQ) fitting procedure. Then, we refine the results using non-linear optimization techniques, such...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.