Source-linked AI summary
Scalable Bayesian Hamiltonian learning
Tim J. Evans, Robin Harper, Steven T. Flammia
TL;DR
Hamiltonian-learning approaches can be difficult to use because they are poorly conditioned and recover the system only up to a scalar factor. The paper develops Bayesian Hamiltonian Learning with prior information and a model that accounts for dependent approximation errors. Numerically, the Bayesian approach outperforms the naive approximate kernel estimate.
Problem
Existing Hamiltonian estimators are difficult to analyze when prior information affects the model, and the Bayesian estimator’s sample complexity is hard to bound.
Source: §3 · p. 8Method
BHL uses a Bayesian model that carries the conditional distribution of approximation error because that error depends on the unknown parameter vector.
Source: §III · p. 9, §1 · p. 10Results
The Bayesian approach numerically outperforms the naive approximate kernel estimate.
Source: §3 · p. 8Takeaways & Limitations
BHL provides point estimates of couplings and posterior covariance as a measure of uncertainty.
Source: §III · p. 9A sample-complexity bound for the Bayesian MAP estimator remains an open question; the analysis instead bounds the naive kernel estimator.
Source: Appendix D · p. 14
Abstract
from arXiv · showhide
As the size of quantum devices continues to grow, the development of scalable methods to characterise and diagnose noise is becoming an increasingly important problem. Recent methods have shown how to efficiently estimate Hamiltonians in principle, but they are poorly conditioned and can only characterize the system up to a scalar factor, making them difficult to use in practice. In this work we present a Bayesian methodology, called Bayesian Hamiltonian Learning (BHL), that addresses both of these issues by making use of any or all, of the following: well-characterised experimental control of Hamiltonian couplings, the preparation of multiple states, and the availability of any prior information for the Hamiltonian. Importantly, BHL can be used online as an adaptive measurement protocol, updating estimates and their corresponding uncertainties as experimental data become available. In addition, we show that multiple input states and control fields enable BHL to reconstruct Hamiltonians that are neither generic nor spatially local. We demonstrate the scalability and accuracy of our method with numerical simulations on up to 100 qubits. These practical results are complemented by several theoretical contributions. We prove that a $k$-body Hamiltonian $H$ whose correlation matrix has a spectral gap $\Delta$ can be estimated to precision $\varepsilon$ with only $\tilde{O}\bigl(n^{3k}/(\varepsilon \Delta)^{3/2}\bigr)$ measurements. We use two subroutines that may be of independent interest: First, an algorithm to approximate a steady state of $H$ starting from an arbitrary input that converges factorially in the number of samples; and second, an algorithm to estimate the expectation values of $m$ Pauli operators with weight $\le k$ to precision $\epsilon$ using only $O(\epsilon^{-2} 3^k \log m)$ measurements, which quadratically improves a recent result by Cotler and Wilczek.
I. INTRODUCTION
Scalable Hamiltonian learning can reveal coherent noise sources, but existing approaches face practical limits in conditioning, scale ambiguity, and state preparation. The paper frames robust reconstruction and uncertainty estimation under experimental and preparation errors as central challenges.
I. INTRODUCTION
- Hamiltonian learning can diagnose coherent noise, whereas randomized benchmarking mainly provides performance metrics and limited insight into noise sources.More detailed scalable error models can remove coherent noise terms, while state-reconstruction methods do not reveal system dynamics.Source: §I · p. 1
- Reconstructing a Hamiltonian can support noise reduction and pulse design, but GRAPE requires a good characterization that is prohibitive for larger devices.The paper identifies Hamiltonian learning as a route to characterizing coherent noise sources.Source: §I · p. 1
A. Problem statement
- Few-body Hamiltonians have polynomially many parameters, but preparing useful steady states is challenging because informative states can be difficult to prepare and easy states may fit multiple Hamiltonians.The problem statement also identifies SPAM errors as a source of bias and uncertainty in the estimate.Source: §I · p. 1, A. Problem statement · p. 2
I. INTRODUCTION
- Prior methods reconstruct generic spatially local Hamiltonians from one commuting state, but are ill-conditioned, recover the Hamiltonian only up to scale, and exclude some non-generic or nonlocal cases.These limits make reconstruction sensitive to noise and difficult to implement practically.Source: §I · p. 1
A. Problem statement
- A small spectral gap can make otherwise unique reconstruction impractical, and a null-space estimate also leaves an unresolved overall scalar factor.The paper notes a prior example requiring noise standard deviation 10^-12, corresponding to about 10^24 or more measurements.Source: A. Problem statement · p. 2
II. RESULTS
The paper presents BHL as a Bayesian framework for practical Hamiltonian reconstruction, using prior information and optional multiple preparations or control fields. Its multiple-state simulations show that adding an eigenstate can make reconstruction unique, while control-field constraints can retain degeneracies unless supplemented by further preparations.
A. Summary of main results
- BHL combines prior information, optional multiple preparations or calibrated controls, and Bayesian uncertainty quantification to address practical Hamiltonian reconstruction.The framework handles approximation errors in the likelihood to prevent overfitting and returns a point estimate with posterior covariance.Source: A. Summary of main results · p. 3
- BHL also supports online estimation, with real-time use conditioned on quickly preparing the input steady state.The paper further reports robustness to a large class of measurement errors and methods for addressing state-preparation errors.Source: A. Summary of main results · p. 3
B. Using multiple input states
- Stacking constraints from multiple input states produces a composite matrix whose spectral gap is at least that of its best constituent constraint matrix.The bound follows from Weyl’s inequality.Source: B. Using multiple input states · p. 3
- One additional eigenstate lifts the degeneracy in the simulated nonlocal two-body model, making reconstruction unique up to numerical precision.For random Hamiltonians, the spectral gap improves further as more eigenstates are added.Source: Figure 1 · p. 3, B. Using multiple input states · p. 4
C. Using multiple control fields
- Control fields can be estimated alongside the bare Hamiltonian, and their constraints can be combined with multiple input states.Well-characterized controls can serve as prior information to select a preferred solution within a degenerate space.Source: C. Using multiple control fields · p. 4
- With N control fields, the combined matrix has at least N + 1 independent consistent solutions, so controls alone do not guarantee a unique reconstruction.In the generic case, ℓ = ceil(N/(m − 1)) extra state preparations suffice to break the degeneracy.Source: C. Using multiple control fields · p. 4
D. Bayesian model
Bayesian Hamiltonian Learning (BHL) uses prior information and a Bayesian model to make Hamiltonian reconstruction more robust and quantify uncertainty. Multiple controls and sequential updates support reconstruction as data arrive.
D. Bayesian model
- BHL uses a Bayesian model to infer a vector in an operator's kernel, leveraging prior information and quantifying uncertainty in the estimate.The posterior covariance provides a direct measure of estimate uncertainty.Source: D. Bayesian model · pp. 4–5
- Unlike prior approximate-null-space methods, BHL provides a point estimate and eliminates the ambiguity in the overall scale factor.The prior methods resolve parameters only up to a linear subspace; the scale factor had to be added by hand.Source: D. Bayesian model · pp. 4–5
- BHL models uncertainty in the forward operator as state-dependent approximation error, with a Gaussian noise model that can also account for correlated errors.A Gaussian approximation is used for experimental noise, although the passage notes that binomial noise may be more realistic.Source: D. Bayesian model · p. 5
- BHL updates its posterior sequentially, using each partial dataset as the prior for the next update and allowing later measurements to target system subregions.Subregion couplings depend only on measurements from that subregion.Source: D. Bayesian model · pp. 5–6
- With four added control fields, the 100-qubit reconstruction posterior contracts around the true parameters while its 2σ ellipse contains them at each iteration.The full parameter space has 1,191 dimensions; the figure displays two parameters.Source: D. Bayesian model · p. 6, Figure 2 · p. 6
- At constant experimental cost, spreading measurements across more control settings improves reconstruction, with diminishing returns as N increases from 4 to 8.The prior uncertainty bounds the posterior error; the plotted estimator's reconstruction error is a best-case comparison because it requires the unknown true norm.Source: D. Bayesian model · pp. 6–7
E. Dealing with errors
The paper addresses measurement, statistical, and state-preparation errors through parallel Pauli estimation and time-averaged state preparation.
E. Dealing with errors
- The method measures m weight-k Pauli expectation values in parallel in O_k(log m) time to fixed precision, rather than the naive O(m) scaling.The paper states that the constant in O_k depends on k.Source: E. Dealing with errors · p. 7
- State-preparation errors are addressed by using time averaging to move an input closer to a steady state, then quadrature rules to approximate that average.The section treats state-preparation errors alongside measurement errors and finite-sampling statistical errors.Source: E. Dealing with errors · p. 7
1. Measurement errors
Measurement errors can preserve the kernel under a uniform error rate, while state-preparation errors are reduced through time averaging and quadrature. The sample-complexity analysis gives sufficient recovery guarantees under a spectral-gap condition.
1. Measurement errors
- A uniform measurement error rate rescales K without changing its kernel, whereas fluctuations between error rates contribute bias; errors still require more samples for equal statistical resolution.The Bayesian model can incorporate a statistical description of measurement errors to correct them.Source: §1 · p. 7
2. State preparation errors
- Time-averaged states approximately commute with the Hamiltonian, and a warm start ε-close to a steady state converges as O(ε/t); quadrature sampling rapidly approximates the time average.The preparation errors arise because the target state is not known in advance and the prepared input is therefore approximate.Source: §2 · p. 7
3. Sample complexity upper bound
- For coefficients bounded by O(1), the stated recovery guarantee achieves infidelity at most ε under a sample bound governed by the noisy forward operator's spectral gap Δ.The bound implies polynomial-time estimation of k-body Hamiltonians when Δ is not too small, but supplies sufficient recovery conditions rather than necessary ones.Source: §3 · p. 8
F. Example: Long-range Ising Hamiltonian
The long-range Ising example illustrates BHL with multiple prior-eigenstate preparations and online posterior updates. The discussion presents BHL’s uncertainty-aware framework, adaptive use, and theoretical sample-complexity basis, while noting that the tightness of its bounds remains unclear.
F. Example: Long-range Ising Hamiltonian
- The excited prior eigenstate’s time-averaged state vanishes periodically, with its persistent frequency component giving an approximate period of 2.Both example states’ time averages approach commuting states with 1/t scaling; high-frequency components decay fastest.Source: F. Example: Long-range Ising Hamiltonian · p. 9
- 3 of 66 couplings fell outside the posterior 2σ error bars, while remaining within the expected 95% credible region.The example used 16 prior eigenstates, time-averaged for t = 3π, and accounted for approximation error when using non-commuting states.Source: F. Example: Long-range Ising Hamiltonian · p. 9
III. DISCUSSION
- BHL combines prior information with multiple input states and calibrated control fields to improve stability and accuracy while providing a point estimate and posterior uncertainty.Its posterior covariance includes off-diagonal correlations as well as marginal variances.Source: §III · p. 9
- Online BHL updates estimates as data arrive and can adapt measurements to maximize estimation accuracy.Source: §III · p. 9
- BHL models approximation errors in the likelihood to prevent overfitting and provide robust uncertainty quantification.Source: §III · p. 9
- The proposed steady-state and low-weight Pauli-estimation subroutines support a polynomial sample-complexity bound for k-body Hamiltonians when K’s spectral gap is at least 1/poly(n).The authors say it remains unclear how tight the derived bounds are and whether they can be improved.Source: §III · p. 9
Appendix A: Bayesian Preliminaries
The likelihood construction treats uncertainty in the measured operator as an approximation error that depends on the unknown coefficients. Marginalizing that error therefore requires retaining its conditional distribution.
1. Construction of the likelihood
- The measured operator differs from the ideal operator by an additive noise matrix, producing an approximation error that depends on the unknown coefficients.Source: §1 · p. 10
- Assuming the joint distribution of the operator, coefficients, and approximation error is normal allows the error to be marginalized to obtain the likelihood.Source: §1 · p. 10
- Unlike a standard linear model with independent additive noise, this likelihood retains the conditional distribution of the approximation error given the unknown coefficients.Source: §1 · p. 10
2. Gaussian Posterior
This section derives a maximum a posteriori estimator using a conjugate Gaussian prior for the Gaussian likelihood. It then obtains the posterior mean and covariance.
2. Gaussian Posterior
- The section derives a maximum a posteriori estimator from a conjugate Gaussian prior, which makes the posterior exactly computable for a Gaussian likelihood.Source: §2 · p. 10
- The coefficient vector is assigned a Gaussian prior centered at its prior mean with covariance representing prior uncertainty.Source: §2 · p. 10
- A Cholesky decomposition of the positive definite covariance matrices is used in the posterior derivation.Source: §2 · p. 10
- The derivation gives expressions for the posterior mean and covariance.Source: §2 · p. 10
3. Distribution of approximation error
The approximation-error statistics are derived under a Gaussian measurement-noise model and used to form conditional distributions. The appendix also specializes the covariance construction to multiple input states and control fields.
3. Distribution of approximation error
- Measurement averaging is modeled as zero-mean Gaussian noise, with the central limit theorem cited as justification.Source: §3 · p. 10
- Under the stated independence assumption, the approximation error has zero mean and its covariance is computed from the rows of the noise matrix.Source: §3 · pp. 10–11
- A Schur complement gives the conditional Gaussian distribution of the approximation error given the unknown coefficients.Source: §3 · p. 11
- For multiple input states and control fields, the noise covariance is organized in block-diagonal form using the corresponding conditional covariances.Source: §3 · p. 11
Appendix B: Approximate state preparations
Appendix B analyzes how approximate state preparations affect Hamiltonian reconstruction. It bounds both the discretization error from time averaging and the resulting kernel-estimator error.
Appendix B: Approximate state preparations
- A δ-approximate steady state still leaves the true coupling vector approximately in the kernel of K, and Theorem 2 bounds the least-singular-vector estimate’s error using the spectral gap.The theorem’s conditions include the dimension m and the gap Δ of K(ρ).Source: Appendix B · pp. 11, 13
- The continuous time average becomes a better approximate steady state as time increases, and its matrix-element approximation error can be incorporated into finite-sampling bounds.The analysis uses the time-averaged state’s commutator with H and bounds quadrature error in the 1-norm.Source: Appendix B · pp. 11–13
- A discrete weighted average over intermediate times approximates the unphysical continuous time-averaged state, allowing experiments to use physically preparable states.Gauss-Legendre quadrature gives a provable approximation guarantee; nested quadrature could support online convergence checks while reusing data.Source: Appendix B · p. 12
- Theorem 2’s error scaling agrees with Ref. and makes explicit its dependence on the length of the true Hamiltonian vector, without uncontrolled sub-leading error terms.A corollary extends the estimator bound to additive entrywise errors in K.Source: Appendix B · pp. 13–14
Appendix C: Pseudocode for the main algorithms
Appendix C presents pseudocode for estimating low-weight Pauli expectation values and for estimating the relevant values on a time-averaged state.
Appendix C: Pseudocode for the main algorithms
- The appendix provides two algorithms: one estimates k-body Pauli expectation values from an input state, and the other uses those estimates for a time-averaged state.The pseudocode is presented in Table I, with performance guarantees analyzed in Appendices B and D.Source: Appendix C · p. 14
Appendix D: Sample complexity
Appendix D bounds measurement costs for estimating Pauli expectations and reconstructing a Hamiltonian with a kernel estimator. Its guarantees depend on the precision, failure probability, and spectral gap.
Appendix D: Sample complexity
- The randomized Pauli-measurement algorithm avoids perfect hash families and improves the scaling with m over the alternative based on Ref. [10].The theorem considers m Pauli operators on n qubits, each with weight at most k.Source: Appendix D · p. 14
- Choosing T = 2ε^-2 log(3m/δ) suffices to make the total failure probability less than δ, while a union bound controls the probability that any estimate ˆPi differs from its mean by more than ε.Source: Appendix D · p. 14
- Theorem 4 gives a sufficient sample-complexity guarantee for kernel-estimator infidelity at most ε, under an a priori O(1) upper bound on the coupling magnitudes.The guarantee concerns a k-body Hamiltonian supported on m specified Pauli operators and depends on the spectral gap Δ of the noisy K matrix.Source: Appendix D · p. 14
- The full reconstruction analysis combines estimates at quadrature points to approximate K and bounds the resulting kernel-estimator error.The analysis distinguishes quadrature error from sampling error and optimizes the averaging time for fixed other parameters.Source: Appendix D · p. 15