Characterization of decohering quantum systems: Machine learning approach

Markku P. V. Stenberg; Oliver Köhn; Frank K. Wilhelm

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Characterization of decohering quantum systems: Machine learning approach

Markku P. V. Stenberg, Oliver Köhn, Frank K. Wilhelm

arXiv:1510.05655v1Oct 19, 2015quant-ph

TL;DR

Decoherence limits the accuracy and efficiency of adaptive characterization for coupled quantum systems, especially when only a few Rabi oscillations occur before relaxation. The paper uses Bayesian inference and machine-learning optimization to design measurement policies for uncertain parameters and experimental imperfections. The resulting policies make characterization more efficient under decoherence, including extreme initial uncertainty and readout imperfections.

Problem
Decoherence makes adaptive characterization of coupled quantum systems difficult and can severely limit accuracy when relaxation permits only a few Rabi oscillations.
Source: §I · p. 1, §IV · p. 3
Method
The paper combines Bayesian inference with machine-learning search for policies tailored to relaxation, initial parameter uncertainty, and experimental imperfections.
Source: §I · p. 1, §V · p. 3, §VI · p. 5
Results
Policies optimized through particle swarm optimization significantly outperform the other considered policies, with median error not saturating over the evaluated measurement-shot interval.
Source: §VI · p. 5
Takeaways & Limitations
Machine-learning-designed measurement policies can make characterization of decohering quantum systems vastly more efficient despite experimental imperfections.
Source: §IX · p. 7

Abstract

from arXiv · show

Adaptive data collection and analysis, where data are being fed back to update the measurement settings, can greatly increase speed, precision, and reliability of the characterization of quantum systems. However, decoherence tends to make adaptive characterization difficult. As an example, we consider two coupled discrete quantum systems. When one of the systems can be controlled and measured, the standard method to characterize another, with an unknown frequency $\omega_{\rm r}$, is swap spectroscopy. Here, adapting measurements can provide estimates whose error decreases exponentially in the number of measurement shots rather than as a power law in conventional swap spectroscopy. However, when the decoherence time is so short that an excitation oscillating between the two systems can only undergo less than a few tens of vacuum Rabi oscillations, this approach can be marred by a severe limit on accuracy unless carefully designed. We adopt machine learning techniques to search for efficient policies for the characterization of decohering quantum systems. We find, for instance, that when the system undergoes more than 2 Rabi oscillations during its relaxation time $T_1$, $O(10^3)$ measurement shots are sufficient to reduce the squared error of the Bayesian initial prior of the unknown frequency $\omega_{\rm r}$ by a factor $O(10^4)$ or larger. We also develop policies optimized for extreme initial parameter uncertainty and for the presence of imperfections in the readout.

I. INTRODUCTION

Decoherence remains a major source of quantum-information errors and makes accurate device characterization more difficult. The paper studies adaptive policies tailored to relaxation, readout errors, and initial parameter uncertainty using machine learning.

Decoherence remains a major error source and increases the resources required for fault-tolerant quantum computation, while also complicating characterization.Source: §I · p. 1
Adaptive measurements update settings using collected data and can outperform nonadaptive schemes, but short relaxation times can limit their numerical accuracy.Source: §I · p. 1
The paper uses machine learning to develop policies tailored to energy relaxation, readout error, and initial parameter uncertainty.The policies characterize an unknown resonator frequency ωr coupled to a controllable qubit with unknown coupling strength g.Source: §I · p. 1

II. EXPERIMENTAL RESOURCES

The paper considers swap spectroscopy for a qubit coupled to another mode, using qubit frequency and waiting time as adaptive controls. This retains modest experimental resources while enabling adaptive characterization.

The unknown mode frequency ωr and coupling strength g are inferred from probability patterns over the control surface (ωq, t), with Δω = ωq − ωr and T1 denoting detuning and relaxation time.Source: §II · p. 2
Swap spectroscopy prepares an excited qubit, evolves the coupled system for time t at fixed ωq, and measures the qubit in the σz basis.The excitation undergoes vacuum Rabi oscillations, while relaxation produces an increasing envelope in the ground-state occupation probability.Source: §II · p. 2
Unlike conventional spectroscopy, which repeats measurements at fixed settings many times, the considered setup adaptively updates (ωq, t) while keeping these as the only experimental controls.Source: §II · p. 2

III. BAYESIAN INFERENCE

The inference scheme uses Bayesian updating to refine uncertainty about g and ωr after each measurement. The posterior becomes the prior for the next setting, enabling iterative learning implemented with sequential Monte Carlo.

Bayes’ theorem updates the joint posterior P(g, ωr|d) from the likelihood of observed data and the prior distribution over the unknown parameters.The normalization averages the likelihood over all possible parameter values.Source: §III · p. 2
The posterior after one measurement serves as the prior for the next measurement, making iterative parameter learning possible.Source: §III · p. 2
After sufficient data collection, the estimates (g̃, ω̃r) are obtained from the posterior mean.The procedure can use sequential Monte Carlo because of its computational efficiency.Source: §III · p. 2

IV. MANUALLY OPTIMIZED POLICY

The manually optimized policy chooses measurement settings from current posterior statistics, but its performance deteriorates for ωr estimation when relaxation permits fewer than a few tens of Rabi cycles. This motivates policies optimized for decoherence.

IV. MANUALLY OPTIMIZED POLICY

The policy uses posterior means and standard deviations to focus ωq near the relevant frequency range and choose waiting times from current uncertainty scales.It also uses the estimated coupling mean μg because μg determines the expected likelihood width on the frequency axis.Source: §IV · pp. 2–3
Figure 1 contrasts adaptive Bayesian inference with the particle swarm optimization algorithm used to search policy parameters.Source: Figure 1 · p. 3
When fewer than a few tens of Rabi cycles occur before relaxation, the manually optimized policy becomes unsatisfactory for estimating ωr.This limitation is especially relevant for characterizing spurious TLSs because their decoherence times range from nanoseconds to microseconds.Source: §IV · p. 3

V. PARTICLE SWARM OPTIMIZATION

The paper uses particle swarm optimization to search policy parameters for adaptive quantum characterization. Candidate policies are evaluated on simulated parameter samples using median squared estimation error, while particles exchange neighborhood information to improve their positions.

V. PARTICLE SWARM OPTIMIZATION

Manual optimization is impractical because suitable policy parameters depend on relaxation time, readout error, and initial parameter uncertainty.The particle swarm algorithm improves policies from past experience and is adopted as the paper’s machine-learning approach.Source: §V · p. 3
Particle swarm optimization searches four-parameter candidate policies by representing each as a position in an Nd-dimensional policy space with Nd = 4.The initial swarm contains 60 particles placed randomly within a region expected to contain the optimum.Source: §V · p. 3
Each policy’s fitness is the negative median squared error of the estimated ωr after 200 adaptive measurement updates.The negative sign makes higher fitness correspond to better policies; repeated measurements are also evaluated after MrNu total shots.Source: §V · p. 3
Fitness evaluation applies each policy to K = 2000 simulated samples with randomly chosen true parameters and matching initial priors.Unless otherwise specified, the initial standard deviations are σg,0 = 0.25µg,0 and σω,0 = 2µg,0, with performance summarized by medians over 10 000 simulated samples.Source: §V · pp. 3–4
Particles exchange information through a fixed ring topology, retaining the best position found within each particle’s neighborhood.Each particle has two neighbors, including wraparound between the first and last particles.Source: §V · p. 4
Particle positions are updated iteratively using attraction coefficients, random variables, and an inertia weight, with bounded velocities and iterations continuing until fitness saturates.The selected parameters are β1 = 0.5, β2 = 1.0, and w = 0.7.Source: §V · p. 4

VI. POLICIES OPTIMIZED THROUGH PSO

Particle-swarm optimization searches policies adapted to relaxation, readout, and uncertainty conditions, improving frequency estimation in decohering systems. The optimized policies outperform manual alternatives, avoid error saturation, and show that uniform waiting-time sampling can be near-optimal.

VI. POLICIES OPTIMIZED THROUGH PSO

Strong relaxation can make adaptive estimates saturate because long waiting times allow Rabi oscillations to decay and posteriors to converge to wrong values.The manual policy’s error improves initially but saturates after hundreds of shots, while numerical error and uninformative measurements limit accuracy.Source: §VI · p. 4
The optimized policies impose a maximum waiting time and repeat each setting 10 times before posterior updates to limit relaxation and accumulated numerical error.The fitness function is evaluated after 2000 measurement shots, with Mr = 10 repeated measurements per setting.Source: §VI · pp. 4–5
The optimized policy P(20) mach,u significantly outperforms the compared policies, with median error continuing to decrease over the measured shot range.Figure 2 compares normalized median squared error and waiting times for the policies at T1µg,0 = 20π.Source: §VI · pp. 4–5, Figure 2 · p. 4
Policies are optimized separately for different relaxation times and initial uncertainties, including cases where nR ranges from 20 to 2.Figure 4 evaluates normalized median squared error for σω,0 = 2µg,0, 10µg,0, and 20µg,0.Source: §VI · pp. 5–6, Figure 4 · p. 6
The optimized waiting-time cutoff is advantageous near 1.37T1–1.55T1, and the policy search spans an 8-dimensional space for P(nR) mach,u.The tests cover nR = 20, 12, 8, and 2 Rabi cycles during relaxation.Source: §VI · p. 6, Figure 4 · p. 6
Uniform Pt(t) produces approximately the same error as optimized probability-density shapes, making it a near-optimal choice across the tested relaxation times.Optimized densities peak near 1.31T1–1.49T1, while the median error is insensitive to shape when the mean has the correct scale.Source: §VI · pp. 5–6, Figure 3 · p. 5

VII. EXTREME INITIAL UNCERTAINTY

The policies are tested under extreme initial uncertainty in the unknown frequency, with separate optimization for relaxation time and prior width. Their adaptive frequency targeting makes the median error decrease more steeply as the initial uncertainty grows.

VII. EXTREME INITIAL UNCERTAINTY

The policies are optimized separately for normalized relaxation times with nR = 20, 12, 8, and 2 and initial frequency uncertainties σω,0 = 2µg,0, 10µg,0, and 20µg,0.Source: §VII · p. 6
For large initial frequency uncertainty, the qubit frequency follows an adaptive rule and then focuses near the relevant frequency scale after the first detection exceeds Dth.This reduces the uncertainty to O(µg).Source: §VII · p. 6
The median error exhibits an increasingly steep drop as the initial frequency uncertainty σω,0 increases.This behavior is reported for the policies optimized for the corresponding relaxation time and prior uncertainty.Source: §VII · p. 6

VIII. EXPERIMENTAL IMPERFECTIONS

The readout-imperfection tests use a conservative error probability and examine policies whose presumed relaxation time differs from the true value. Estimates continue improving even when the true relaxation time is substantially shorter than assumed.

VIII. EXPERIMENTAL IMPERFECTIONS

The readout-imperfection analysis assumes a conservative misidentification probability Pe = 0.1.The tested policies are optimized for normalized relaxation times 20π, 12π, 8π, and 2π.Source: §VIII · p. 7
When the true relaxation time is longer than presumed, the resulting estimates are more accurate than those obtained with the presumed value.The shorter-than-assumed case is treated as the more challenging mismatch.Source: §VIII · p. 7
Improving estimates persist when the true relaxation time is only ∼50% of the value presumed by the policy.Shorter-than-assumed relaxation can cause signal loss when most waiting times exceed the true T1.Source: §VIII · p. 7
The results suggest treating T1 as an additional unknown could improve robustness to relatively large relaxation-time errors.This suggestion follows from the observed improvement despite a shorter true relaxation time.Source: §VIII · p. 7

IX. CONCLUSION

The conclusion identifies decoherence as a continuing barrier to efficient characterization and uses machine learning to design adaptive policies that retain efficiency under relaxation and experimental imperfections.

IX. CONCLUSION

Decoherence limits both device performance and the efficient characterization of quantum information processing systems.The paper frames decoherence as an ubiquitous source of errors that remains consequential despite improved coherence times.Source: §IX · p. 7
Short relaxation limits adaptive speed-up because exponentially increasing waiting times eventually exceed the interval over which useful Rabi oscillations survive.In the absence of relaxation, the same waiting-time strategy can exponentially speed estimation.Source: §IX · p. 7
Machine-learning-designed policies make measurements more informative and system characterization more efficient in the presence of experimental imperfections.The conclusion presents this as the paper's delivered result for decohering systems.Source: §IX · p. 7

APPENDIX

The appendix records the optimized policy parameterizations and points to the equations defining their parameters and update rules.

APPENDIX

Candidate policies P(nR),re are parameterized by a vector including continuous and integer policy parameters.The listed parameter space is R6 × N2.Source: APPENDIX · p. 7
Candidate policies P(nR),re use a 15-component vector of values Pt(t2) through Pt(t16).This parameterization is identified for the policy family associated with the referenced figures.Source: APPENDIX · p. 7
Table I lists the parameters determining the optimized policies, whose meanings are defined by Eqs. (11) and (12).The table serves as a parameter reference for the optimized policy families.Source: Table I · p. 7