PaperlayerView exact revision on arXiv

Source-linked AI summary

Robust Online Hamiltonian Learning

Christopher E. Granade, Christopher Ferrie, Nathan Wiebe, D. G. Cory

arXiv:1207.1655v2quant-phcs.LG

TL;DR

The paper addresses Hamiltonian learning when quantum-system parameters may fluctuate between experiments and noise processes may be unknown. It combines Bayesian experimental design with sequential Monte Carlo in an online algorithm that also estimates uncertainty and the Cramer-Rao lower bound. The reported results show accurate learning with few experiments in several adverse settings, including unknown decoherence time and fluctuating parameters.

  • Problem

    Hamiltonian learning must handle dynamical parameters that can vary between experiments and additional unknown noise processes, while avoiding worst-case tomography assumptions.

  • Method

    The algorithm combines Bayesian experimental design and sequential Monte Carlo for online, model-informed estimation of Hamiltonian parameters, hyperparameters, and uncertainty regions.

  • Results

    The algorithm learns parameters with few experiments, including cases with fluctuating hyperparameters and unknown decoherence time, while estimating region coverage near 99.7%.

  • Takeaways & Limitations

    The method can learn an unknown frequency and unknown T2 simultaneously and provide uncertainty regions during online data collection.

Abstract

from arXiv · show

In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer-Rao lower bound, certifying its own performance.

I. INTRODUCTION

The paper frames Hamiltonian estimation as a model-informed, adaptive alternative to black-box process tomography, designed to learn dynamical parameters online with fewer experiments. Its framework combines Bayesian experimental design with sequential Monte Carlo and extends estimation to fluctuating parameters and hyperparameters.

  • Motivation: Hamiltonian estimation targets the dynamics of closed quantum systems rather than a single process snapshot.The Hamiltonian is parameterized as H(x_1, ..., x_d), reducing the task to estimating a parameter vector.
  • Motivation: Model knowledge can reduce the number of unknowns compared with full quantum process estimation of a black box.The paper calls this model-informed setting partial process estimation or partial tomography.
  • Approach: Adaptive online design chooses later experimental controls from earlier measurement outcomes instead of relying on fixed controls and post-processing.The paper names its approach sequential Monte Carlo Bayesian experimental design.
  • Scope: The framework estimates parameter regions and learns hyperparameters describing distributions of Hamiltonian parameters that fluctuate between experiments.This generalizes the task beyond constant Hamiltonian parameters.
  • Approach: The algorithm uses expected utility to identify more informative experiments, considering information gain and negative variance as canonical choices.The posterior distribution feeds back into the next iteration of the algorithm.

III. UTILITY FUNCTIONS AND THE CRAMER-RAO LOWER BOUND

The paper evaluates experiment choices through utility functions tied to posterior information or estimation risk, and uses Fisher information to establish Cramer-Rao lower bounds. These bounds provide a statistical benchmark for the achievable estimation performance.

  • Utility functions: Information gain selects experiments that reduce the expected entropy of the posterior distribution over model parameters.The associated utility is written as an expected log posterior for possible future data.
  • Utility functions: Negative variance is another utility choice, and the paper states that it is optimal for minimizing the protocol’s mean squared error.The estimator’s quality is evaluated using quadratic loss and its associated Bayes risk.
  • Risk and estimation: Under quadratic loss, the Bayes estimator is the mean of the posterior distribution.The paper identifies the posterior mean as the unique Bayes estimator for quadratic loss.
  • Cramer-Rao bound: The Bayesian Cramer-Rao bound lower-bounds achievable estimation risk through the Fisher information, including for sequentially accumulated measurements.The multivariate standard bound is Cov(ˆx) ≥ I(x; C)^−1 when the Fisher information matrix is nonsingular.
  • Cramer-Rao bound: Numerical integration and an iterative procedure are used to monitor the lower bound for specific experimental-control choices.The bound is evaluated sequentially as measurements are added.

IV. SEQUENTIAL MONTE CARLO ALGORITHM

The section develops a sequential Monte Carlo approximation for Bayesian online experiment design, representing parameter distributions with weighted particles and updating them as data arrive. Resampling and likelihood-call approximations make the procedure more practical, while numerical precision and quantum simulation remain important constraints.

  • Particle representation: Sequential Monte Carlo approximates posterior distributions over Hamiltonian parameters using finitely many weighted particles.The particle representation supports sequential Bayesian updates while avoiding costly multidimensional integrations.
  • Resampling: Resampling is triggered using effective sample size, with resample_threshold=0.5 of the original particle count.This postpones the zero-weight problem by moving particles toward higher-weight regions.
  • Particle representation: Each new datum updates particle weights through its likelihood under the applied experimental control, followed by normalization.The update uses the previous weights and Pr(D|x_i, C), producing a new particle-weight distribution.
  • Resampling: Resampling replaces particles with weighted draws, perturbs their locations, and resets all new-particle weights to 1/n.The perturbation distribution combines sampled particle locations with the expected model and uses covariance-based random spreading.
  • Computational efficiency: Approximating utility-function expectations with only the highest-weight particles reduces likelihood evaluations without reducing parameter-estimation accuracy.The approximation affects optimal-experiment selection rather than the accuracy of parameter estimates from collected experiments.
  • Computational efficiency: The complete adaptive algorithm combines the SMC approximation with Bayesian experimental design and leaves the local optimizer selectable for the experimental model.The paper compares Newton conjugate-gradient, nonlinear conjugate-gradient, and null optimization strategies.

V. REGION ESTIMATION

This section constructs credible region estimates for Hamiltonian parameters from the current SMC state and relates covariance-based regions to approximately Gaussian posteriors. Region size and probability coverage are evaluated against the Bayesian Cramer-Rao bound.

  • Region objectives: Small-volume, high-probability regions are useful because robust optimal-control designs must work across every model in the estimated range.The cost of that optimization increases with the volume of the region estimate.
  • Region objectives: Credible regions quantify the probability that the true model lies within an estimated parameter region for a single data record.This differs from confidence regions, which describe long-run coverage over data records.
  • SMC region construction: The probability mass of a region is estimated as an SMC expectation of its indicator function.The indicator equals 1 inside the region and 0 outside, allowing Algorithm 2 to compute the region probability from particle weights.
  • SMC region construction: Any region containing particles with total weight at least r has an approximate probability mass of at least r.Convex hulls and minimum-volume enclosing ellipses of sufficiently weighted particles provide efficiently computable region constructions.
  • Gaussian covariance regions: When the posterior is approximately N(µ, Σ), the covariance matrix defines an error ellipse containing approximately 0.682^d of the probability mass.This Gaussian approximation is expected after sufficient data when the posterior is sharply peaked and Fisher information is non-singular.
  • Gaussian covariance regions: Performance is assessed by testing Gaussian-posterior coverage and comparing covariance-ellipse size with the Bayesian Cramer-Rao bound.The comparison uses Eπ[Cov(ˆx)] ≥ J(π; C)^−1 as a performance metric.

VI. HYPERPARAMETER ESTIMATION

The section extends Hamiltonian learning to hyperparameters that describe experiment-to-experiment variation in model parameters. The same SMC framework can learn these hyperparameters and convert their region estimates into regions for the underlying parameters, at increased likelihood-evaluation cost.

  • Motivation: Hyperparameters generalize Hamiltonian learning to quantum systems whose model parameters vary between experiments.This addresses variability arising from experimental errors.
  • Hyperparameter model: The distribution of model parameters is represented conditionally as Pr(x|y), where y denotes the hyperparameters.Despite their different interpretation, hyperparameters can be learned with Algorithm 7 in the same way as x.
  • Computational cost: Hyperparameter likelihood evaluations can be substantially more expensive because they average data probabilities over sampled experimental parameters.The sampling error scales as 1/√N_s, so some cases may require many samples.
  • Computational cost: Analytical marginalization over x can avoid this computational drawback in special cases without changing the underlying SMC algorithm.The paper identifies a particular analytically tractable case for discussion.
  • Hyperparameter model: Region estimates for hyperparameters can be converted into region estimates for model parameters x.The conversion uses expectation and covariance relationships between the posterior over y and the intermediate distribution Pr(x|y).
  • Region conversion: As Cov_y(ˆx) decreases, the estimated model region approaches the true conditional model region Cov_x|y(x|y).This describes the limiting behavior of the model-region estimate as the estimated mean model becomes more precise.

VII. TEST CASES

The numerical test cases increase in complexity to evaluate Hamiltonian learning under known and unknown decoherence conditions. They begin with a single-parameter experiment and then allow T2 to be unknown.

  • Test cases: The first test case learns a single parameter in an experiment with a known decoherence time.
  • Test cases: The second test case extends the first by treating T2 as unknown.

A. Single Parameter Hamiltonian

The section introduces single-parameter Hamiltonian estimation for a qubit, using controlled evolution times and measurements, then extends the model to include known dephasing noise.

  • A qubit evolves under an internal Hamiltonian with unknown frequency ω, which is the parameter to estimate.
  • Each experiment prepares |+⟩, evolves for a controllable time t, and measures in the σx basis, recording outcomes d_k ∈ {0, 1}.
  • A T2 dephasing process introduces decay in the information extractable from each measurement.
  • With finite T2, risk scaling is exponentially suppressed when measurement times reach t = T2.

B. Two Parameter Model with Single Control

The two-parameter model estimates both the precession frequency and inverse decoherence time using a single controllable time, but standard Fisher-information and asymptotic methods become inadequate.

  • The model treats both ω and T2 as unknown while retaining t as the only experimentally controllable parameter.
  • For numerical convenience, the unknown vector is parameterized as x = (ω, T2^-1), giving both parameters the same dimensions.
  • The Fisher matrix of any single measurement is singular, so the standard Cramer-Rao bound and normal asymptotic approximations do not apply.
  • Because the parameters need not share a natural scale, the loss function requires a semidefinite matrix Q.

C. One Parameter Model with Hyperparameters

The section extends the SMC approach to hyperparameter models by marginalizing an intermediate frequency drawn from Gaussian or Lorentz distributions, thereby estimating distributional parameters instead.

  • A Gaussian-distributed frequency ω is marginalized so the data are conditioned on hyperparameters y = (μ, σ).
  • After marginalization, ω is removed and the task becomes estimating the mean and variance of an unknown normal distribution.
  • A Lorentz-distributed ω is described by location and scale hyperparameters ω0 and γ.
  • The Lorentz hyperparameter model is identical to the decoherence model in Equation (10), linking decoherence processes with formalized parameter uncertainty.
  • The same SMC algorithm estimates unknown distributions under assumed distributional forms and supports region estimation for the intermediate parameter ω.

VIII. RESULTS AND DISCUSSION

Numerical results show that the adaptive Bayesian algorithm learns parameters with few experiments, including cases involving fluctuating parameters or unknown decoherence time. For the known-T2 model, 11 experiments produce a tight approximately Gaussian posterior over ω.

  • The adaptive Bayesian algorithm learns model parameters using a very small number of experiments, including adverse cases requiring hyperparameters.
  • Figure 2 compares likelihood functions after N = 1, 6, and 11 simulated measurements, showing true ω and n = 100 SMC particles.
  • Figure 3 compares performance across n = 100, 1 000, and 10 000 particles, with and without NCG optimization, against the Bayesian Cramer-Rao bound.
  • Its performance is especially noteworthy when the decoherence time T2 is unknown.
  • 11 experiments yield a tight approximately Gaussian posterior over ω for the known-T2 model.

A. Results for Known T2 Model

For the known-T2 model, the algorithm approaches the BCRB with sufficient particles, while more guesses and NCG optimization improve experiment selection and reduce error. Region estimates become reliable as the posterior approaches Gaussianity, and the unknown-T2 benchmark shows that 30 guesses can learn ω accurately without a well-characterized T2.

  • Known T2: The MSE remains close to the BCRB with 1,000 or 10,000 particles without optimization, but 10,000 particles are needed when NCG is used.This indicates near-optimal MSE scaling when the SMC approximation uses sufficiently many particles.
  • Known T2: Choosing the best of 30 guessed experiments reduces MSE relative to one random guess, while NCG improves approximation quality when approx ratio = 1.With approx ratio = 0.1, 30 guesses plus NCG can perform worse than one random guess without optimization.
  • Known T2: Without NCG optimization, individual-trial MSE can fall outside the 68% confidence interval because uninformative guesses produce a skewed utility distribution.NCG pushes guesses toward more informative experiments, although the MSE remains closer to the interval’s upper limit.
  • Known T2: Poor guess heuristics are mitigated by increasing the number of guesses and using local optimization, while posterior variance or region estimates identify pathological trials.The median square-error performance is better than the mean because a small fraction of trials select very bad guesses.
  • Region estimation: The covariance-region estimator targets approximately 0.9973 probability mass for the one-parameter Gaussian posterior when Z = 3.The corresponding two-parameter target is approximately 0.9946.
  • Region estimation: The approximate enclosed probability mass approaches 0.9973 as experiments increase, supporting covariance ellipses as region estimators for the known-T2 posterior.Early deviations reflect insufficient data for the posterior to become approximately Gaussian; average differences after each experiment are about 0.01%.
  • Unknown T2: With unknown T2 and no local optimization, 30 guesses require only 50 experiments on average to learn ω within a 0.9% error.Increasing guesses also improves estimation of 1/T2, but the improvement is smaller because T2 has weaker contrast.
  • Unknown T2: With NCG, ω’s MSE continues shrinking as experiments increase, whereas one guess provides little benefit and T2 estimation improves more subtly.The results suggest NCG can find informative experiments near uninformed guesses when the local landscape lacks substantial local optima.

D. Hyperparameter Region Estimation Performance

The hyperparameter-region estimator remains robust when Hamiltonian parameters fluctuate between experiments, providing probability regions and variance estimates that approach the intended behavior as data accumulate.

  • Benchmarking: Figure 7 compares one versus 30 random initial guesses with NCG optimization, using 1,023 and 930 trials respectively.Dashed curves represent single-guess trials, solid curves represent 30-guess trials, and red shading indicates performance-estimation errors.
  • Experimental setup: The algorithm benchmarks hyperparameter-region estimation under a Gaussian model where the unknown frequency follows ω ∼ N(µ, σ2).The model uses µ and σ2 as hyperparameters governing the frequency distribution.
  • Region coverage: 99.4% of hyperparameter probability mass lies within the estimated region at worst, approaching the ideal 99.7% for optimized data as N increases.The 99.7% target follows from the assumed Gaussian posterior and a Z-score of 3.
  • Variance estimation: The inferred variance of the unknown frequency systematically overestimates the true variance on average, but this bias vanishes as the number of experiments increases.The variance is computed from uncertainty in the estimated mean and variance hyperparameters.

E. Computational Cost

The algorithm exposes a trade-off between experimental and computational cost: heuristics that reduce experiments can require more likelihood calls, while inexpensive heuristics may stop improving accuracy. Parallelization and lower-precision simulation offer ways to reduce computational time.

  • Computational cost: 1.05 × 10^7 to 1.5 × 10^9 likelihood calls were required on average across the datasets in Figures 4 and 11.Each likelihood call evaluated the model likelihood, leading to total computational times on the order of a second to a minute.
  • Experimental–computational trade-offs: If experiments run much faster than 200 Hz, the algorithm’s benefit for accelerating data collection may be lost; comparable rates create computational–experimental trade-offs.The balance depends on whether computational or experimental time is the primary resource.
  • Experimental–computational trade-offs: Strategies with the lowest mean–square error per experiment tend to have the highest mean–square error per likelihood call.Some of these strategies saturate after a finite number of experiments, creating an unfavorable computational-time scaling.
  • Stability: Poor randomized guesses can leave the algorithm stuck at large mean–square error, while NCG and a large approximation ratio can improve stability at higher computational cost.The 84th-percentile analysis indicates that most trials continue improving even when the expected loss appears saturated.
  • Reducing computational cost: Independent guess calculations can be parallelized with little inter-node communication, and single-precision simulations may improve performance because high precision is unnecessary.The paper also identifies more advanced optimization and resampling heuristics as future routes to reducing computational cost.
  • Experimental–computational trade-offs: Heuristics that minimize experimental time often require more computational time to reach a target accuracy.The preferred heuristic changes depending on the relative rates and costs of experiments and computation.
Loading 1207.1655v2…