Maximum-Entropy Extension of Quantum Correlation Functions from Short Real-Time Dynamics

AI summary

Overview

Research area: Quantum many-body physics and quantum signal processing — specifically, spectral reconstruction from real-time correlation functions obtained from classical or quantum dynamics simulations.

Technical level: Advanced. The paper assumes familiarity with the Lehmann representation, positive semi-definite (PSD) functions, and stationary stochastic processes; the core results rest on Burg's maximum-entropy theorem, Yule–Walker equations, and autoregressive models.

Scope (one sentence): The paper derives a maximum-entropy framework (meAR for scalar correlators, meVAR for matrix-valued correlators) that extends short-time quantum correlation functions in a way that is positive semi-definite by construction, and demonstrates it on the retarded Green's function of a one-dimensional Fermi–Hubbard chain with open boundary conditions.

What This Paper Is About

Computing the spectrum of a quantum correlation function requires long real-time evolution, but both classical numerics and noisy near-term quantum hardware are limited to short times, and the standard post-processing tools (linear prediction, Prony, ESPRIT/MUSIC, DMD) can only resolve roughly n/2 frequencies from n complex samples and fail on structured or continuous spectra. The authors ask a different question: instead of trying to identify the individual frequency modes, which process could have generated the observed time series while committing as little as possible about the unmeasured points? They answer it with Burg's maximum-entropy principle, having first shown that samples of a stationary quantum correlator are covariance lags of an underlying wide-sense stationary process.

Key Contributions

1. Identification of correlators with covariance lags. The paper shows that any n-sample time series of a stationary correlation function can be treated as a segment of the covariance sequence of an underlying wide-sense stationary (WSS) stochastic process, because the correlator is a PSD function and its Toeplitz Gram matrix R^(n) is PSD. This reframing makes Burg's maximum-entropy theory directly applicable, without the covariance-estimation step that Burg himself had to solve with his reflection-coefficient algorithm.

2. Derivation (not postulation) of an autoregressive model from maximum entropy. The maximum differential entropy rate extension is shown to be a proper complex Gaussian AR(p) process with order at most p = n − 1, whose coefficients solve a set of Yule–Walker equations built directly from the measured correlator values. The AR form is derived from the least-committal selection principle rather than assumed, which the authors stress is the fundamental (if subtle) difference from linear prediction.

3. Guaranteed analytic structure, and removal of three practical headaches. The extension is PSD by construction, the maximum-entropy spectrum is non-negative, and the total spectral weight sum rule ∫ S_ME(ω) dω = g₀ holds automatically. Consequently there is no need to discard initial samples as transients, no need to treat the model order as a hyperparameter (it is fixed at n − 1 when R^(n) is positive definite), and no need for ad hoc pole regularization — poles lie strictly outside the unit disk.

4. Matrix generalization (meVAR) that trades simulation time for more measured correlators. A sequence of M matrix correlation functions is identified with covariance lags of a vector-valued stationary process, giving a vector autoregressive model whose coefficient matrices solve multichannel (block) Yule–Walker equations. Because all observables evolve under the same Hamiltonian and share the same excitation energies, joint modeling constrains the admissible continuation of every channel through the block-Toeplitz Gram matrix and the positivity of the matrix spectral density.

Main Findings

• Failure mode of existing methods is a data-capacity mismatch. From n complex samples one can retrieve at most about n/2 frequencies, so when the true number of contributing frequencies satisfies P ≫ n, or when spectral gaps are too small to resolve, linear prediction, subspace methods, and DMD either keep only the ~n/2 dominant frequencies or project the whole spectrum onto them. Finer sampling does not help; only longer evolution adds resolving power.

• Two regimes separated by the Gram matrix. If R^(n) is strictly positive definite, there are infinitely many admissible PSD extensions and the maximum-entropy choice picks one; if R^(n) is singular with rank P < n, the PSD extension is unique and equals a sum of P complex exponentials — exactly the assumption underlying linear prediction and subspace methods.

• AR poles need not be physical excitations. Unlike LP and subspace methods, the p poles of the maximum-entropy spectrum are not interpreted as individual modes; they collectively provide a rational approximation to the full spectral density, which is what allows broad continua and other non-discrete features to be represented efficiently.

• Fermi–Hubbard benchmark. For a chain of length L = 11 with t_h = 1, U = 4, μ = 2, and N_↑ = N_↓ = 5 fermions per spin sector, both meAR (applied pairwise, via the B = O_i + O_j and C = O_i + i O_j construction) and meVAR were applied to the spin-summed retarded correlator C^R_ij(t). Training used samples up to t = 3.

• meVAR substantially outperforms meAR in space-time. For C^R_i5(t), pairwise meAR could only extend faithfully for a limited time before the predicted signal gradually decayed toward zero — expected, since at large separation |i − j| the correlator is approximately zero through much of the training window. The meVAR extension, built from the full 11 × 11 matrix of correlators, remained in excellent agreement with exact dynamics and captured nontrivial features such as the propagating light cone and its reflections from the boundaries. The abstract reports faithful extension of the time series by nearly an order of magnitude.

• Joint modeling supplies missing information. The improvement is attributed directly to multichannel coupling: rather than extrapolating weak or nearly vanishing correlators from their own short history, meVAR lets strongly resolved channels constrain and stabilize weakly resolved ones.

• Local density of states. The abstract reports that the local density of states, A_ii(ω) = −(1/π) Im G^R_ii(ω + iη), is reconstructed to high accuracy even from short and noisy time series. The detailed LDoS figure discussion (fig. 2) is cut off in the supplied text, so the specific accuracy figures are not reported here.

• Practical regularization options. When R^(p) is near singular, the authors recommend introducing a small ridge parameter λ (chosen as the smallest value keeping the whole extension PSD up to the desired time), or restricting to a reduced Gram matrix with n′ = max{k ≤ n : R^(k) ≻ 0}. Computing S_ME(ω) directly from the rational formula is noted to require careful choice of the θ grid and can become ill-conditioned.

• Noise. The abstract states that the method enables accurate extension and spectral reconstruction from short-time data even in the presence of noise, and that the vector formulation improves robustness relative to treating correlators independently; the corresponding numerical experiments fall in the truncated portion of the text.

Methodology in Plain English

The authors do not try to fit the correlator with a finite list of oscillation frequencies. Instead they reinterpret the measured values as the first entries of a statistical object: the covariance sequence of a stationary random process. That reinterpretation matters because a huge body of theory already answers the question "given only the first few covariances, what is the most honest guess for the rest?"

The answer is Burg's maximum-entropy principle: choose the process whose entropy rate is largest, i.e., the one that assumes the most uncertainty about everything not measured. This process turns out to be an autoregressive model — each new value is a fixed linear combination of the previous n−1 values plus a random innovation — with the coefficients obtained by solving the Yule–Walker equations assembled from the measured data. Extending the series is then simple recursion, and the spectrum can be read off directly from the coefficients as a rational function. Because the construction never leaves the space of valid covariance sequences, positivity and the total-weight sum rule come for free.

For matrix-valued correlators, the same recipe is lifted to vectors: stack all M correlators into one vector-valued process, solve block Yule–Walker equations for matrix coefficients, and propagate. The validation uses exact diagonalization-style real-time data for the retarded Green's function of an open-boundary Hubbard chain, comparing against the exact dynamics. For a single scalar channel the authors use the pairwise trick (since off-diagonal correlators are not themselves PSD), while the matrix method uses the whole 11 × 11 correlator matrix at once.

Why This Matters

The method gives a principled, largely tuning-free post-processing step that sits on top of any short-time dynamics simulation — classical or quantum — and turns it into a higher-resolution spectral function without violating physical constraints. Existing tools either require the user to guess a model order, truncate to ~n/2 modes, or risk producing spectra with negative regions and broken sum rules; this one does none of those things. The matrix formulation is the more distinctive idea: it recasts the trade-off as "measure more correlators instead of evolving longer," which is attractive when several observables can be accessed at similar cost.

Real-world applications:

• Angle-resolved photoemission (ARPES) and quasiparticle spectra — the spectral function of the Green's function is the paper's own demonstration target, including site-resolved local density of states in inhomogeneous systems with boundaries, disorder, or impurities.
• Transport and optical response — quantum transport coefficients and optical/dynamical conductivities are expressed through time-dependent correlation functions whose low-frequency resolution is otherwise limited by short simulation windows.
• Scattering experiments — dynamical structure factors and susceptibilities probed by inelastic neutron or X-ray scattering, where many-particle continua make sum-of-exponentials fitting unreliable.
• Near-term quantum simulation — NISQ devices are limited to comparatively short times and are corrupted by shot noise; a PSD-preserving, noise-robust extension directly addresses that bottleneck.

Industry relevance: the authors are at Phasecraft Ltd. (Bristol and London, United Kingdom), a quantum-software company, and the work is framed around extracting useful spectral information from the shallow circuits and short evolutions that near-term hardware can actually deliver.

Future Directions

• Noise characterization in practice. The abstract promises robustness in the presence of noise, but the quantitative noise study lies in the truncated portion of the text; how the ridge parameter λ should be selected systematically under shot noise or finite bond dimension remains a practical question.
• Singular and near-singular Gram matrices. The authors themselves flag ridge regularization and the reduced Gram matrix R^(n′) with n′ = max{k ≤ n : R^(k) ≻ 0} as pragmatic workarounds rather than fully developed theory, leaving room for a more principled treatment of rank-deficient noisy data.
• Scaling of the vector formulation. meVAR requires manipulating nM × nM matrices; the paper argues this stays inexpensive up to a large number of correlators, but the crossover point for realistic M and n is not established in the supplied text.
• Finite temperature and broader observables. The derivation is presented for zero-temperature stationary correlators, with the remark that generalization to finite-temperature correlators is straightforward; verifying that in practice, and applying meVAR to other operator sets and to actual quantum-hardware data, are natural next steps.

Target Audience

Researchers and practitioners in quantum many-body simulation, quantum signal processing, and computational condensed-matter physics who need spectral functions from limited-time dynamics — particularly those working with tensor-network or quantum-computing approaches where evolution time is the binding constraint. It will also interest methodologists familiar with linear prediction, Prony/subspace methods, DMD, or maximum-entropy analytic continuation, who will appreciate the argument that the correlator is the covariance sequence rather than a signal from which covariances must be estimated.

Authors’ abstract

Correlation functions are central to understanding the behavior of quantum systems, with their spectra giving direct access to quasiparticle and collective excitations. However, the long-time evolution required for sufficient spectral resolution is challenging to reach for both classical and quantum simulations. In this work, we introduce a principled approach for extending time series of quantum correlation functions and reconstructing their spectral densities, based on their connection with covariance sequences of stationary stochastic processes. This connection enables us to apply Burg's maximum-entropy principle to find the least-committal extension compatible with the measured values of the correlation function, which corresponds to an all-pole autoregressive model whose coefficients solve a set of Yule--Walker equations. The resulting extension is positive semi-definite by construction, guaranteeing the positivity of the spectrum and the automatic satisfaction of its total spectral weight sum rule. The approach applies to both scalar and matrix-valued correlation functions, including those with highly structured or continuous spectra. The generalization to the matrix case makes it possible to trade simulation time for additional measured correlation functions, enabling accurate time-series extension and spectral reconstruction from short-time data, even in the presence of noise.