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Practical Bayesian Tomography

Christopher Granade, Joshua Combes, D. G. Cory

arXiv:1509.03770v2quant-phphysics.data-anstat.AP

TL;DR

Bayesian tomography faces computational, prior-selection, and time-dependence challenges. The paper applies sequential Monte Carlo, constructs experimentally informed priors, and introduces state tracking; it reports that these advances make Bayesian tomography practical for current and future experiments.

  • Problem

    Bayesian tomography faces numerical and implementation barriers, limited useful prior choices, and difficulty incorporating detected drift or diffusion into estimation.

    Source: §I · pp. 1–2

  • Method

    The paper uses sequential Monte Carlo for tomography, constructs priors with experimentally informed means, and applies particle-filtering methods to track evolving states.

    Source: §I · p. 2, §V · p. 7

  • Results

    The authors report that their combined advances make Bayesian quantum tomography practical for current and future experiments.

    Source: §VII · p. 10

  • Takeaways & Limitations

    The approach provides a way to include experimental insight in priors and to characterize stochastically evolving sources without ignoring all previous data.

    Source: §I · p. 2, §V · p. 7

Abstract

from arXiv · show

In recent years, Bayesian methods have been proposed as a solution to a wide range of issues in quantum state and process tomography. State-of-the-art Bayesian tomography solutions suffer from three problems: numerical intractability, a lack of informative prior distributions, and an inability to track time-dependent processes. Here, we address all three problems. First, we use modern statistical methods, as pioneered by Husz\'ar and Houlsby and by Ferrie, to make Bayesian tomography numerically tractable. Our approach allows for practical computation of Bayesian point and region estimators for quantum states and channels. Second, we propose the first priors on quantum states and channels that allow for including useful experimental insight. Finally, we develop a method that allows tracking of time-dependent states and estimates the drift and diffusion processes affecting a state. We provide source code and animated visual examples for our methods.

I. INTRODUCTION

Bayesian tomography can incorporate experimental knowledge, but its practical adoption is limited by computational expense, few useful prior choices, and difficulty incorporating time-varying dynamics. The paper uses sequential Monte Carlo (SMC) to address these challenges with reusable software, experimentally informed priors, and a tracking method.

I. INTRODUCTION

  • Bayesian tomography faces practical adoption barriers because inference can be computationally expensive, implementations are difficult to build and optimize, and reusable software is scarce.The paper identifies reusable code as important for experimental practicality and reproducibility.Source: §I · p. 1
  • Time-varying sources exhibit deterministic drift and stochastic diffusion, but detecting these dynamics does not itself incorporate them into estimation protocols.Some existing approaches are not general-purpose or practical across important applications; one cited proposal requires quantum memory scaling with sample count.Source: §I · p. 2
  • The paper uses sequential Monte Carlo as the unifying algorithmic approach to Bayesian tomography.It also implements the techniques in an open-source Python library integrated with QuTiP and modern instrument-control software.Source: §I · p. 2
  • The proposed prior construction transforms samples from a reference prior using quantum operations drawn from an ensemble specified by an experimental prior estimate.The construction requires sampling rather than an analytic prior form and makes the estimate the new prior's mean.Source: §I · p. 2
  • The paper introduces tracking of drifting and diffusing state evolution by perturbing samples with Gaussian noise whose mean and variance represent deterministic and stochastic evolution.Source: §I · p. 2

II. BAYESIAN TOMOGRAPHY

The section formulates Bayesian tomography from measurement likelihoods and priors, then describes estimators and uncertainty summaries. It uses sequential Monte Carlo to approximate inference and credible regions without assuming a normal posterior.

II. BAYESIAN TOMOGRAPHY

  • For N separately measured, identically and independently prepared systems, the measurement record defines a likelihood through Born-rule outcome probabilities.The setup allows different and adaptive measurements in principle, although the discussion restricts to separate measurements.Source: §II · p. 2
  • Bayesian inference updates a prior using the likelihood, and the Bayesian mean estimator is the posterior expectation of the state.Source: §II · p. 3
  • The Bayesian mean estimator is optimal for strictly proper scoring rules on states and remains approximately optimal under infidelity loss.Its error is characterized by the spread of posterior samples.Source: §II · p. 3
  • Representing states in a Hermitian operator basis makes their coordinates real and identifies each coordinate with an observable's mean value.Posterior covariance describes parameter uncertainty; its associated credible ellipsoid has axes given by the covariance eigenvectors and eigenvalues.Source: §II · pp. 3–4
  • Sequential Monte Carlo approximates the posterior integral with weighted state-hypothesis particles and does not require a normal-posterior assumption.The authors choose SMC over rejection sampling, which can become prohibitively expensive as data accumulates.Source: §II · p. 4
  • SMC supports credible regions formed from particle sets whose total weight reaches the requested probability, as well as covariance-based regions.The paper uses naive 3σ covariance ellipsoids for illustration; channel tomography is included by representing a channel through its Choi state.Source: §II · pp. 2, 4

III. DEFAULT PRIORS: THE SAMPLING OF STATES AND CHANNELS

The section distinguishes default priors from priors that encode experimental insight and explains how each can serve as an SMC input. It also introduces Ginibre matrices as a sampling ingredient.

III. DEFAULT PRIORS: THE SAMPLING OF STATES AND CHANNELS

  • Default priors provide reasonable choices when more detailed information is unavailable and express a notion of uniformity without an estimate beyond the maximally mixed state.Source: §III · p. 4
  • A fiducial prior is an algorithmic input whose mean is maximally mixed, whereas an insightful prior has a non-maximally-mixed mean.Source: §III · p. 4
  • Default-prior sampling algorithms use Ginibre matrices, whose complex entries are sampled from normal distributions.The paper supplies pseudocode for the sampling algorithms.Source: §III · p. 4

A. Priors on States

The section presents default priors for pure and mixed quantum states. It gives a shared sampling expression for Ginibre and Bures ensembles and explains how those samples can support either fiducial or insightful priors.

A. Priors on States

  • For pure states, the canonical default prior is Haar measure, sampled by normalizing a Gaussian-distributed complex vector or applying a Haar-random unitary to a fiducial state.Source: A. Priors on States · p. 5
  • Mixed-state defaults include rank-restricted Ginibre ensembles and the Bures measure, represented by a shared sampling equation.Source: A. Priors on States · p. 5
  • Taking U as identity yields a unitarily invariant Ginibre prior supported on states of rank at most K, while Haar-random U with a square Ginibre matrix yields the Bures measure.Samples can be used as fiducial priors for SMC or transformed into insightful-prior samples.Source: A. Priors on States · p. 5

B. Priors on Channels

The BCSZ measure provides a fiducial prior over quantum channels, generated from Ginibre-random density operators and adjusted to satisfy trace preservation. The resulting channel prior is completely positive, has a depolarizing-channel mean, and is supported at a chosen Kraus rank.

B. Priors on Channels

  • The BCSZ measure samples channel priors supported over channels of a given Kraus rank.The Kraus rank is the minimal number of Kraus operators required to specify a channel.Source: B. Priors on Channels · p. 5
  • To generate a channel sample, BCSZ draws a Ginibre-random density operator, takes its partial trace, and transforms it into a Choi state.Source: B. Priors on Channels · p. 5
  • The resulting channel samples are trace-preserving and completely positive, and the prior mean is the completely depolarizing channel.Source: B. Priors on Channels · p. 5

IV. INSIGHTFUL PRIORS FOR STATES AND CHANNELS

The paper constructs experimentally informed priors by transforming samples from fiducial priors. The target mean can encode experimental design or previous estimates, but the mean alone does not specify a unique prior.

IV. INSIGHTFUL PRIORS FOR STATES AND CHANNELS

  • The basic construction transforms samples from fiducial priors by applying a quantum channel to produce insightful priors.Source: §IV · p. 5
  • A target prior mean can reflect experimental design or previous estimates, but different assumptions can yield priors with the same mean.Source: §IV · p. 5
  • The construction requires support over all feasible states and efficient sampling; the paper notes this lets tomography recover from bad priors given sufficient data.Source: §IV · p. 5
  • Figure 1 illustrates priors with the same mean but different support: one covers only pure states, while the other includes mixed preparations.Source: §IV · p. 5, Figure 1 · p. 6

A. Construction of Insightful Priors

The construction transforms fiducial-prior samples with a generalized amplitude damping channel, choosing its fixed point and damping distribution to obtain a target mean while limiting the transformation.

A. Construction of Insightful Priors

B. Convexity and Robustness of Insightful Priors

The paper constructs insightful priors from fiducial priors while preserving their feasible-state support, and reports that even a badly chosen prior mean can yield a reliable estimate with enough data. The construction also extends to rebit and channel priors.

B. Convexity and Robustness of Insightful Priors

  • When the desired mean lies within the fiducial prior’s support, the insightful prior’s support is the convex closure of the fiducial support; outside it, the support expands.The construction preserves the fiducial prior’s support in either case.Source: B. Convexity and Robustness of… · p. 6
  • Using real Ginibre and BCSZ fiducial priors extends the construction to rebits and quantum channels, respectively.The channel prior can cover all CPTP maps of a given dimension, enabling process tomography through the Choi–Jamiołkowski isomorphism.Source: B. Convexity and Robustness of… · p. 6
  • A fiducial prior also specifies which states are feasible, so restricted support can constrain the resulting insightful prior.For example, a purity-limited fiducial prior can restrict support to the convex hull of its allowed states and the desired mean.Source: B. Convexity and Robustness of… · p. 7
  • After 300 random Pauli measurements, the posterior supports the true state and its mean is approximately the true state, despite a nearly orthogonal prior mean.Recovering from a bad initial prior requires additional data.Source: B. Convexity and Robustness of… · p. 7

V. TOMOGRAPHIC STATE TRACKING

Tomographic state tracking applies particle filtering to evolving quantum sources, alternating particle evolution with Bayesian updates. The method learns drift and diffusion parameters alongside the state, but purely diffusive tracking has a frequency limit.

V. TOMOGRAPHIC STATE TRACKING

  • CONDENSATION alternates Bayesian updates with particle movement to track stochastic processes observed through noisy measurements.Drift models deterministic evolution; diffusion models stochastic evolution, and both parameters can be learned online.Source: §V · pp. 7–8
  • The tracking cycle represents the posterior with particles, perturbs them by a Gaussian, truncates invalid probabilities, then updates on the next datum.The resulting posterior approximation becomes the particle representation for the next diffusion step.Source: Figure 4 · p. 8, §V · p. 8
  • Bayesian updates jointly update the state and its drift and diffusion parameters, allowing those parameters to co-evolve with the posterior.This joint update is described as central to CONDENSATION’s tracking capability.Source: §V · p. 8, Figure 4 · p. 8
  • The state model assumes a frame with deterministic evolution already characterized, but diffusion alone can still track deterministic changes when strong enough to cover them.Figure 5 illustrates deterministic tracking without drift updates.Source: §V · p. 8, Figure 5 · p. 9
  • ftrack = (1/10) × (1/∆t): under purely diffusive tracking, higher-frequency oscillations are not tracked and the estimate remains p(tk) = 1/2.Model selection can test for this limitation so a more appropriate algorithm can be used.Source: §V · p. 9

VI. NUMERICAL EXAMPLES

The examples apply SMC to qutrit state tomography, time-dependent state tracking, and qubit process tomography. They also illustrate how the posterior captures uncertainty and responds to a poorly aligned prior.

VI. NUMERICAL EXAMPLES

  • In the qutrit state-tomography example, performance is reported as risk, defined as expected quadratic loss over repetitions of the algorithm; “true” states are drawn from the prior matching the insightful and unbiased prior.Source: §VI · p. 9
  • In well over half of 1,200 trials, QInfer flagged likely failure when the qutrit prior mean was nearly orthogonal to the true state.The risk comparison uses quadratic loss and includes default, insightful, biased, and nearly orthogonal priors.Source: §VI · p. 9, Figure 6 · p. 10
  • The state-space method is demonstrated in an animation of tracking quantum states over time.The animation is available online, with a snapshot in Video 1.Source: §VI · p. 9
  • In this example, uncertainty in the final posterior was dominated by the identity- and Hadamard-channel contributions; the posterior covariance’s principal components are quantum maps describing directions of maximal uncertainty.Source: §VI · pp. 9–10

VII. CONCLUSIONS

The conclusion presents practical Bayesian tomography contributions, while the appendices detail process-tomography representation, SMC, prior sampling, coin priors, and a tracking-frequency estimate.

VII. CONCLUSIONS

  • The paper contributes an experimental-insight prior for states and channels, software for approximate Bayesian tomography, and a method to track time-dependent states.The authors conclude that these advances make Bayesian quantum tomography practical for current and future experiments.Source: §VII · p. 10

Appendix A: Choi-Jamiołkowski Isomorphism and QPT

  • The Choi-Jamiołkowski representation turns process-tomography preparation and measurement into measurements on a normalized Choi state, making QPT a special case of state tomography with an appropriate fiducial prior.The formulation covers both in-place and ancilla-assisted QPT.Source: Appendix A · p. 15

Appendix B: Review of Sequential Monte Carlo

  • Sequential Monte Carlo approximates a posterior with weighted particles, updates their weights using each new datum’s likelihood, and resamples when weight concentration reduces the effective particle count.Resampling perturbs particle locations and resets weights uniformly to restore numerical stability.Source: Appendix B · pp. 15–16

Appendix C: Sampling Default Priors

  • Appendix C reviews algorithms for sampling default priors, including Ginibre states, Bures states, Haar unitaries, and uniform CPTP maps.The listed algorithms specify the input dimensions and the sampled object or distribution.Source: Appendix C · pp. 16–17

Appendix D: GADFLI Prior for Coins

  • The coin example treats bias as a Bloch-sphere axis and constructs GAD-prior samples by transforming fiducial samples; its illustrated prior is symmetric about pµ = 1/2, though that value is disallowed.The coin likelihood assigns heads probability p, and the prior condition is used to generate the illustrated priors.Source: Appendix D · pp. 17–18, Figure 10 · p. 18

Appendix E: An estimate of the tracking frequency

  • The tracking-frequency estimate uses a 95% confidence interval and error tolerance ϵ to derive the required sample count, then applies a Nyquist-Shannon bound to the measurement interval.The appendix frames this as an order-of-magnitude estimate for smooth tracking; higher frequencies may be tracked, but evolution is likely discontinuous.Source: Appendix E · p. 19

Appendix F: QInfer Tomography Tutorial

The tutorial walks through QInfer tomography setup, from choosing a basis and defining models and priors to updating a posterior and examining its covariance.

Appendix F: QInfer Tomography Tutorial

  • The tutorial defines a tomography basis, using the single-qubit Pauli basis as its example; basis choice is largely arbitrary but may suit some experiments better than others.Basis objects convert between QuTiP’s Qobj representation and QInfer’s model parameters.Source: Appendix F · p. 19
  • QInfer models package the likelihood, experimental parameters, and metadata; TomographyModel describes single-shot experiments, while BinomialModel handles batches.The tutorial’s experiment records store a measurement vector and the number of repetitions.Source: Appendix F · p. 20
  • The tutorial constructs an insightful prior by combining a fiducial prior with a desired mean, then uses SMC updates to obtain a posterior from simulated data.It demonstrates a rebit prior and a sequence of 50 experiments.Source: Appendix F · pp. 20–23
  • The posterior covariance superoperator summarizes uncertainty, and its Choi matrix is visualized to show covariance across observables.The tutorial also demonstrates reading the final estimated state as a QuTiP Qobj.Source: Appendix F · p. 24

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