[Paper Review] Sequential Bayesian experiment design for adaptive Ramsey sequence measurements

R. D. McMichael, Sergey Dushenko+1 more|arXiv (Cornell University)|May 5, 2021

Quantum Information and Cryptography71 references22 citations

TL;DR

This paper proposes a sequential Bayesian experiment design for adaptive Ramsey sequence measurements in nitrogen-vacancy (NV) centers, optimizing the phase accumulation time τ to improve measurement efficiency. By continuously updating beliefs about unknown parameters using Bayesian inference and selecting τ values that maximize information gain, the method reduces measurement time by factors of 2 and 4 compared to adaptive heuristic and random protocols, respectively, with concurrent workflows eliminating computational overhead.

ABSTRACT

The Ramsey sequence is a canonical example of a quantum phase measurement for a spin qubit. In Ramsey measurements, the measurement efficiency can be optimized through careful selection of settings for the phase accumulation time setting, $ au$. This paper implements a sequential Bayesian experiment design protocol for the phase accumulation time in low-fidelity Ramsey measurements, and performance is compared to both a previously reported adaptive heuristic protocol and random setting choices. A workflow allowing measurements and design calculations to run concurrently largely eliminates computation time from measurement overhead. When precession frequency is the lone parameter to estimate, the Bayesian design is faster by factors of 2 and 4 relative to the adaptive heuristic and random protocols respectively.

Motivation & Objective

• To improve measurement efficiency in low-fidelity, averaged-readout Ramsey sequences for NV centers.
• To address the challenge of inefficient parameter estimation due to limited photon collection and classical readout noise.
• To develop and evaluate a sequential Bayesian experiment design that adaptively selects the phase accumulation time τ to maximize information gain.
• To compare the Bayesian approach against an adaptive heuristic and random protocols in terms of speed and accuracy.
• To demonstrate that concurrent measurement and design computation eliminates computational overhead, enabling real-time optimization.

Proposed method

• Uses sequential Bayesian inference to update posterior distributions over unknown parameters (a, c, ω₀, T₂) after each measurement epoch.
• Employs decision theory to select the next τ value that maximizes expected information gain, based on current posterior uncertainty.
• Models signal and background photon counts using Poisson distributions with time-dependent rates λₛ(t) and λ_b(t), incorporating drift via moving averages.
• Derives a likelihood function P(ns|ms, nb, mb, θ) that integrates over background count rate λ_b using a conjugate gamma prior (ν = -1) for robustness.
• Implements a concurrent workflow where measurement and design computation run in parallel, effectively eliminating computation time from measurement overhead.
• Uses a ratio model R(θ) = a * [1 + c/2 * (1 + cos(ω₀τ)) * exp(-(τ/T₂)²)] to relate signal and background counts to the unknown parameters.

Experimental results

Research questions

• RQ1 How does sequential Bayesian experiment design improve measurement efficiency in low-fidelity Ramsey sequences compared to heuristic and random protocols?
• RQ2 What is the impact of concurrent measurement and design computation on effective measurement time and overhead?
• RQ3 How does background count averaging affect the accuracy and convergence of the Bayesian inference protocol?
• RQ4 To what extent does the Bayesian method reduce uncertainty in ω₀ estimation compared to alternative protocols?
• RQ5 Can the Bayesian approach achieve Heisenberg scaling in uncertainty (σ ∝ 1/T) under realistic experimental constraints?

Key findings

• The sequential Bayesian design reduces measurement time by a factor of 2 compared to the adaptive heuristic protocol when estimating the precession frequency ω₀.
• The Bayesian method achieves a 4× improvement in measurement speed over the random protocol in the same parameter estimation task.
• Concurrent execution of measurements and design calculations effectively eliminates computation time from the measurement overhead, enabling real-time optimization.
• Background count averaging improves likelihood sharpness and parameter convergence, but gains saturate when nb ≳10 × ns, indicating diminishing returns beyond this point.
• The Bayesian protocol converges to the true parameter values within a few standard deviations, while incorrect prior choices (e.g., ν = 0) lead to biased estimates.
• The method supports Heisenberg scaling of uncertainty (σ ∝ 1/T) under constant repeats per epoch, indicating optimal scaling behavior in theory.