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A practical and efficient approach for Bayesian quantum state estimation

Joseph M. Lukens, Kody J. H. Law, Ajay Jasra, Pavel Lougovski

arXiv:2002.10354v1quant-ph

TL;DR

Bayesian quantum state tomography quantifies uncertainty, but difficult high-dimensional sampling limits its practical use. The paper proposes a self-contained approach using preconditioned Crank–Nicolson sampling and a pseudo-likelihood; examples report faster sampling and increasing computational savings for the pseudo-likelihood. The authors note that the pseudo-likelihood’s practical advantage remains uncertain in some settings.

  • Problem

    Bayesian QST represents state uncertainty, but sampling its complex, high-dimensional distributions impedes practical use.

    Source: §I · p. 1, §II · p. 2

  • Method

    The paper combines a physical density-matrix parameterization, a pseudo-likelihood, and preconditioned Crank–Nicolson posterior sampling.

    Source: §II · p. 2, B. Further Details on Specific Steps · p. 4

  • Results

    pCN gives about 3.5× temporal speedup over slice sampling at the same convergence level, while pseudo-likelihood evaluation reaches about 10× the full-likelihood speed at D = 49.

    Source: §IV · p. 5, §V · p. 7

  • Takeaways & Limitations

    The method offers a numerically efficient route to Bayesian state estimation, with pseudo-likelihood savings emerging as Hilbert-space dimension grows in the simulated example.

    Source: §I · p. 1, §V · p. 7

    The pseudo-likelihood’s practical advantage remains unresolved because its speedup at larger dimensions may be offset by the computational cost of the full likelihood.

    Source: §VI · p. 8

Abstract

from arXiv · show

Bayesian inference is a powerful paradigm for quantum state tomography, treating uncertainty in meaningful and informative ways. Yet the numerical challenges associated with sampling from complex probability distributions hampers Bayesian tomography in practical settings. In this Article, we introduce an improved, self-contained approach for Bayesian quantum state estimation. Leveraging advances in machine learning and statistics, our formulation relies on highly efficient preconditioned Crank--Nicolson sampling and a pseudo-likelihood. We theoretically analyze the computational cost, and provide explicit examples of inference for both actual and simulated datasets, illustrating improved performance with respect to existing approaches.

I. INTRODUCTION

Quantum state tomography estimates a density matrix from repeated measurements, supporting predictions about quantum protocols and analysis of noise. Bayesian QST represents uncertainty as a probability distribution, but sampling its complex, high-dimensional distributions impedes practical use.

I. INTRODUCTION

  • QST estimates a density matrix from repeated measurements, enabling quantitative predictions, noise analysis, and quantum process tomography.Source: §I · p. 1
  • Bayesian QST returns a probability distribution over states, naturally quantifying uncertainty and avoiding unjustifiably optimistic low-rank estimates.It uses available information to minimize an operational divergence.Source: §I · p. 1
  • Sampling complex probability distributions is a numerical obstacle to wider use of Bayesian tomography.Source: §I · p. 1
  • The proposed self-contained Bayesian method combines efficient sampling with experimental and simulated demonstrations to improve on prior Bayesian QST approaches.The introduction reports about 3.5× speedup over slice sampling in the experimental example and a computational scaling advantage for pseudo-likelihood on simulated higher-dimensional data.Source: §I · p. 1

II. BACKGROUND

QST faces exponential growth in the state description, while common estimators trade physicality, uncertainty quantification, and computational practicality. Bayesian inference represents uncertainty explicitly, but evaluating its posterior expectations requires challenging high-dimensional integration.

II. BACKGROUND

  • For n qudits, the Hilbert-space dimension is D = d^n and specifying a density matrix requires D^2 − 1 real numbers, alongside normalization, Hermiticity, and positivity constraints.Source: §II · p. 1
  • Linear inversion uses outcome frequencies as probabilities but least-squares estimates can violate positive semidefiniteness.Source: §II · pp. 1–2
  • Maximum-likelihood estimation enforces physicality through parameterization but gives a point estimate; its ad hoc error-bar procedure simulates additional experiments.Source: §II · p. 2
  • Bayesian QST combines a likelihood and prior into a posterior distribution, from which expectations such as means and standard deviations can be computed.Source: §II · p. 2
  • Posterior expectations require high-dimensional integration, prompting use of MCMC methods including Metropolis–Hastings, SMC, and slice sampling.Slice sampling is general and needs no proposal distribution, but convergence can make these methods intractable for systems of interest.Source: §II · p. 2
  • Adaptive QST uses previous measurements to guide later measurement choices, whereas this work focuses on post-experiment state reconstruction.Source: §II · p. 2

A. Steps

The workflow converts measurement outcomes into a physical density-matrix parameterization and samples a posterior to estimate quantities of interest. It accommodates incomplete measurements through projection and uses a pseudo-likelihood with preconditioned Crank–Nicolson sampling.

A. Steps

  • The workflow begins with measurements and a least-squares estimate, projecting onto observed directions when measurements are tomographically incomplete.Source: A. Steps · p. 2
  • Density-matrix parameters comprise weights and complex vectors, with a prior specified over those parameters and a construction that satisfies physicality conditions.Source: A. Steps · pp. 2–3
  • Figure 1 summarizes the proposed Bayesian QST workflow.Source: Figure 1 · p. 3
  • A pseudo-likelihood scores the proposed state against the least-squares estimate using a projection and Frobenius-norm loss, rather than an explicit experimental model.Source: A. Steps · p. 3
  • Preconditioned Crank–Nicolson Metropolis–Hastings sampling draws from the posterior, after which samples estimate expectation values.Source: A. Steps · p. 3

B. Further Details on Specific Steps

The method combines a projector for incomplete measurements with a pseudo-likelihood and pCN sampling; its prior and density-matrix parameterization are designed for efficient computation.

B. Further Details on Specific Steps

  • The chosen density-matrix parameterization uses 2D2+D real numbers and is reported more efficient to sample and evaluate than the cited Cholesky approach.Constructing ρ from the parameters requires O(D3) operations; the prior's α parameter can favor low- or high-rank states.Source: B. Further Details on Specific Steps · p. 3
  • The pseudo-likelihood assigns a loss between a proposed ρ(x) and the least-squares estimate, rather than deriving from an explicit experimental model.Its scale sets the balance between evidence and prior; the authors use σ2 = 1/N, yielding uncertainties comparable to a full likelihood but slightly larger.Source: B. Further Details on Specific Steps · p. 4
  • For incomplete measurements, the projector retains only the density-matrix elements accessed experimentally; the two-qubit example measures eight of fifteen coefficients.For tomographically complete measurements, PM(ρ) = ρ.Source: B. Further Details on Specific Steps · p. 4
  • For Q ∼ O(D) complete measurements, pseudo-likelihood evaluation costs O(D2), versus O(D4) for the full likelihood; incomplete measurements can raise its cost to O(D4).The incomplete-measurement cost includes explicitly computing the projection.Source: B. Further Details on Specific Steps · pp. 4–5
  • pCN scales the previous sample before adding a random shift, simplifying Metropolis–Hastings acceptance by removing proposal terms that can necessitate smaller stepsizes.Unlike independence sampling, the proposal retains random-walk behavior.Source: B. Further Details on Specific Steps · p. 5

IV. EXAMPLE WITH EXPERIMENTAL DATA

In experimental two-qubit tomography, pCN reaches the same convergence level faster than slice sampling. Full and pseudo-likelihoods yield similar estimates, but the pseudo-likelihood gives no observable sampler-time gain in this small example.

IV. EXAMPLE WITH EXPERIMENTAL DATA

  • In the frequency-bin experiment of Ref., the basis set is tomographically incomplete; fidelity converges to 0.93 ± 0.01, slightly above its 0.92 mean, with singles counts excluded and a different likelihood.Source: §IV · p. 5
  • Sampler times are approximately 18× lower for pCN at the same thinning, although pCN needs larger thinning to reach slice sampling’s convergence level.The paper attributes the timing difference to the slice sampler evaluating π(x) more often.Source: §IV · p. 5
  • A ∼3.5× temporal speedup for pCN over slice sampling reaches the same convergence level; both show an initial log-log slope of −1/2.The comparison fits the first five slice and seven pCN points.Source: §IV · p. 5, Figure 3 · p. 6
  • For incomplete Pauli-X/Z data, projection and full-likelihood probabilities are vectorized into precomputable linear maps, enabling efficient function evaluation and fair likelihood comparisons.The projection map is formed from the measured-basis relationship; the probability map acts on the vectorized density matrix.Source: §IV · p. 6
  • Full and pseudo-likelihoods converge to similar estimates, though the pseudo-likelihood gives slightly lower mean fidelity and higher uncertainty without exceeding the complete model’s predicted fidelity.Both comparisons use α = 1.Source: §IV · p. 6
  • With α = 1/4, both likelihoods give slightly higher fidelity, but full-likelihood chains show more outliers while pseudo-likelihood appears more robust to sampling fluctuations.The sparser prior favors pure states; the authors caution against strong conclusions from the outliers.Source: §IV · p. 6
  • In the D = 4, 16-outcome example, the pseudo-likelihood gives no observable sampler-time speedup because state construction and projection costs dominate; larger-dimensional gains remain prospective.The authors expect the pseudo-likelihood’s efficiency advantage to emerge as dimension increases.Source: §IV · p. 6

V. EXAMPLE WITH SIMULATED DATA

The simulated-data example benchmarks full and pseudo-likelihood timing across increasing dimensions. The gap widened to about 10× at D=49, though dataset limits kept the tests from reaching the O(D^2) asymptotic scaling improvement.

V. EXAMPLE WITH SIMULATED DATA

  • Simulations used entangled two-qudit states mixed with identity noise, with λ = 0.95 and 100D coincidences per basis pair.The parameter λ controls fidelity with respect to the ideal state.Source: §V · p. 7
  • Data cycled through all pairwise combinations of d + 1 mutually unbiased bases, yielding Q = (d + 1)^2 settings with D outcomes each.Source: §V · p. 7
  • At D = 49, full and pseudo-likelihood evaluation times diverged to ∼10×, while remaining comparable at low D.Dataset limits prevented reaching the O(D^2) asymptotic scaling improvement.Source: §V · p. 7
  • Tests covered d ∈ {2, 3, 5, 7} (D ∈ {4, 9, 25, 49}), but dataset limits prevented reaching the O(D^2) asymptotic scaling improvement.Source: §V · p. 7

VI. CONCLUSION

The method may apply to other quantum inference problems, while pseudo-likelihood gains are reported for sufficiently large Hilbert spaces with complete measurements. Their practical importance remains uncertain where data acquisition may outweigh computation.

VI. CONCLUSION

  • The present pCN approach uses Gaussian priors; non-Gaussian priors require a proposal reversible with respect to the prior, while further sampling enhancements remain future work.The authors also identify derivative information, SMC samplers, and finite approximations as directions for further work.Source: §VI · p. 7
  • The pseudo-likelihood appears more robust to sampler initial conditions and offers computational improvements for sufficiently large Hilbert spaces with complete measurements.Source: §VI · p. 7, Figure 5 · p. 8
  • At dimensions where order-of-magnitude speedups occur, acquiring required QST data may outweigh the cost of evaluating the full likelihood, leaving practical benefits uncertain.The full potential of the pseudo-likelihood in this QST problem remains unresolved.Source: §VI · pp. 7–8
  • With minor modifications, the method's key features may apply to other quantum inference problems, including photon loss and linear-optic transformations with dark counts.Source: §VI · p. 7
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