Benchmarking machine learning algorithms for adaptive quantum phase estimation with noisy intermediate-scale quantum sensors

At a Glance

Metadata	Details
Publication Date	2021-06-03
Journal	EPJ Quantum Technology
Authors	Nelson Filipe Costa, Yasser Omar, Aidar Sultanov, Gheorghe Sorin Paraoanu, Nelson Filipe Costa
Institutions	Art Research Centre of the Slovak Academy of Sciences, Instituto Politécnico de Lisboa
Citations	17
Analysis	Full AI Review Included

Technical Documentation & Analysis: Adaptive Quantum Phase Estimation using MPCVD Diamond

This document analyzes the research paper “Benchmarking machine learning algorithms for adaptive quantum phase estimation with noisy intermediate-scale quantum sensors” and outlines how 6CCVD’s specialized MPCVD diamond materials and engineering services are critical for replicating, extending, and industrializing the demonstrated noise-robust quantum sensing protocols, particularly those utilizing Nitrogen-Vacancy (NV) centers in diamond.

Executive Summary

Core Achievement: Machine learning (ML) algorithms—Differential Evolution (DE) and Particle Swarm Optimization (PSO)—are successfully benchmarked for adaptive Quantum Phase Estimation (QPE) to enhance precision in noisy quantum systems.
Noise Robustness: ML-based adaptive protocols demonstrate superior robustness compared to traditional non-adaptive methods, especially when subjected to high levels of Gaussian noise, Random Telegraph Noise (RTN), and quantum decoherence (low visibility $v$).
Application Relevance: The findings are directly applicable to solid-state quantum metrology platforms, including superconducting qubits and, critically, Nitrogen-Vacancy (NV) centers in diamond.
Performance Metric: The algorithms successfully drive the measurement precision close to the Standard Quantum Limit (SQL) in the Noisy Intermediate-Scale Quantum (NISQ) regime ($N \in [5, 25]$ qubits).
Material Requirement: Achieving the long decoherence times ($T_2$) necessary for high visibility ($v$) in NV center experiments requires ultra-high purity, low-defect Single Crystal Diamond (SCD) substrates, a core specialization of 6CCVD.

Technical Specifications

The following hard data points were extracted from the analysis, focusing on parameters relevant to quantum sensor performance and noise mitigation.

Parameter	Value	Unit	Context
Qubit Count Range ($N$)	5 to 25	Qubits	Range tested for ML algorithm benchmarking
Holevo Variance Range ($\ln(V_H)$)	-1.2 to -1.7	N/A	Achieved precision band for $N \in [10, 25]$
Gaussian Noise Standard Deviation ($\sigma$)	0.2, 0.4, 0.8	N/A	Applied to controllable phase shifter $\theta$
Random Telegraph Noise Offset ($\lambda$)	0.2, 0.4, 0.8	N/A	Tested with fixed probability $\eta = 0.4$
Quantum Decoherence Visibility ($v$)	0.9, 0.8, 0.6	N/A	Reduced visibility due to $T_2$ limitations
NV Center Zero-Field Splitting ($D$)	2.87	GHz	Energy separation of $m_s = 0$ and $m_s = \pm 1$ states
NV Center $T_1$ Decay Times	Tens of	Milliseconds	Long relaxation times in diamond
NV Center $T_2$ Decoherence Times	Microseconds	Microseconds	Extendable to hundreds of µs via decoupling
Optimal DE Parameters ($F, C$)	0.7, 0.8	N/A	Amplification and Crossover constants
Optimal PSO Parameters ($\alpha, \beta, w, V_{max}$)	0.8, 0.8, 0.8, 0.2	N/A	Convergence and speed regulators

Key Methodologies

The experimental approach relies on adaptive feedback policies optimized by machine learning to maximize the precision of phase estimation in Ramsey interferometry setups.

Adaptive Phase Control: The core protocol uses a controllable phase shifter ($\theta_m$) that is dynamically adjusted based on the outcome ($\zeta_{m-1}$) of the previous measurement step, following a Markovian feedback rule: $\theta_m = \theta_{m-1} + (-1)^{\zeta_m} x_m$.
Performance Metric: The effectiveness of the adaptive policy ($x_m$) is quantified by minimizing the Holevo Variance ($V_H$), which serves as the cost function for the ML algorithms.
Machine Learning Optimization: Two reinforcement learning algorithms, Differential Evolution (DE) and Particle Swarm Optimization (PSO), are employed for direct search optimization across the high-dimensional policy space.
Noise Modeling: The robustness of the optimized policies is tested against three critical noise sources relevant to solid-state qubits:
- Gaussian Noise (GSN): Affects the controllable phase shifter $\theta_m$.
- Random Telegraph Noise (RTN): Models discrete, random offsets ($\lambda$) in the phase shifter value, highly relevant for solid-state systems.
- Quantum Decoherence: Modeled as a reduction in interference visibility ($v = \exp(-\tau/T_2)$), directly impacting the signal-to-noise ratio.
Computational Scaling: The simulation complexity scales polynomially with the number of qubits $N$ (approximately $O(N^3)$), necessitating constraints on the number of generations ($G=100$) for convergence testing.

6CCVD Solutions & Capabilities

The research highlights the critical need for robust quantum sensors, particularly NV centers in diamond, which require materials engineered to minimize decoherence and noise coupling. 6CCVD is uniquely positioned to supply the necessary MPCVD diamond substrates and customization services to advance this research from simulation to experimental realization.

Applicable Materials

To replicate and extend the noise-robust quantum phase estimation protocols using NV centers, researchers require the highest quality diamond materials:

Optical Grade Single Crystal Diamond (SCD): Essential for maximizing the NV center decoherence time ($T_2$). Our SCD is grown via MPCVD with ultra-low nitrogen and defect concentrations, ensuring high visibility ($v \approx 1$) in the ideal case and providing the longest possible $T_2$ times to mitigate the effects of quantum decoherence (as discussed in Section 4.3.3).
Custom Nitrogen-Doped SCD: For optimal NV center creation, 6CCVD offers precise control over nitrogen incorporation during the growth process, allowing engineers to tune the NV density for specific magnetometry or sensing applications.

Customization Potential

The implementation of Ramsey interferometry and qubit control (Eq. 13) often requires specialized geometries and integrated electronics. 6CCVD provides comprehensive customization capabilities:

Research Requirement	6CCVD Customization Service	Technical Specification
Large-Scale Sensor Arrays	Custom Dimensions & Plates	Plates/wafers up to 125mm (PCD)
Thin Film Integration	Custom Thickness Control	SCD/PCD layers from 0.1µm to 500µm
High-Fidelity Qubit Readout	Ultra-Smooth Polishing	SCD: Ra < 1nm; Inch-size PCD: Ra < 5nm
Microwave Control Lines	In-House Metalization	Deposition of Au, Pt, Pd, Ti, W, Cu
Substrate Integration	Thick Substrates	Diamond substrates available up to 10mm

Engineering Support

The paper demonstrates that the performance tradeoff between increased sensitivity and increased noise coupling is a central challenge in quantum metrology. 6CCVD’s in-house PhD team specializes in material science for quantum applications and can assist researchers in optimizing this tradeoff.

Noise Mitigation Consultation: We provide expert guidance on material selection and surface preparation to minimize specific noise sources (Gaussian, RTN, 1/f noise) that limit $T_2$ in NV center systems.
Material Optimization for Adaptive Sensing: Our team assists with material specifications (e.g., nitrogen concentration, surface termination) tailored for similar adaptive quantum phase estimation projects, ensuring the diamond substrate meets the stringent requirements for high-fidelity, noise-robust quantum control.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.