Multi-shot readout error benchmark of the nitrogen-vacancy center’s electronic qubit

At a Glance

Metadata	Details
Publication Date	2025-06-05
Journal	Journal of Applied Physics
Authors	Péter Boross, Domonkos Svastits, Győző Egri, András Pályi
Institutions	Budapest University of Technology and Economics
Analysis	Full AI Review Included

Multi-Shot Readout Error Benchmark for NV Centers: Material Solutions from 6CCVD

This technical documentation analyzes the requirements for minimizing the multi-shot readout error ($\Delta$) in Nitrogen-Vacancy (NV) center electronic qubits, focusing on how 6CCVD’s advanced MPCVD diamond materials and customization services directly address the hardware limitations identified in the research.

Executive Summary

The research introduces $\Delta$ as a quantitative benchmark for evaluating the efficiency of multi-shot readout in room-temperature NV electronic qubits, providing a critical metric for quantum hardware development.

Core Benchmark: The multi-shot readout error scales as $\Delta / \sqrt{N}$, where $N$ is the number of shots, making $\Delta$ the key indicator for readout quality and efficiency.
Error Budget Quantification: The model successfully quantifies the contribution of three major hardware imperfections to $\Delta$: imperfect photon detection efficiency ($\eta$), background photon flux ($\lambda$), and qubit initialization error ($q$).
Hardware Priority: The analysis confirms that maximizing photon detection efficiency ($\eta$) is the most effective strategy for suppressing $\Delta$, demonstrating a reduction from $\Delta \approx 4.6$ (at $\eta=0.1$) to $\Delta \approx 2.1$ (at $\eta=1.0$).
Material Imperfection Link: Imperfect initialization ($q$) and high background flux ($\lambda$) significantly increase $\Delta$, necessitating high-purity, low-defect diamond substrates for optimal performance.
Optimization Tool: $\Delta$ serves as a powerful metric for experimentalists and engineers to prioritize hardware improvements (e.g., material quality, optics, detectors) that yield the highest performance gain.
Fundamental Limit: The benchmark establishes a theoretical lower bound for multi-shot readout error at $\Delta \ge \sqrt{2/3} \approx 0.816$.

Technical Specifications

The following hard data points were extracted from the analysis, characterizing the performance metrics and simulation parameters used in the study.

Parameter	Value	Unit	Context
Readout Error Benchmark ($\Delta$) Lower Bound	$\approx 0.816$	Dimensionless	Fundamental limit ($\sqrt{2/3}$) for multi-shot readout.
Optimal $\Delta$ (Perfect $\eta=1$)	$\approx 2.1$	Dimensionless	Calculated benchmark value assuming perfect detection, no background/initialization error.
$\Delta$ (Low $\eta=0.1$)	$\approx 4.6$	Dimensionless	Benchmark value for 10% photon detection efficiency (common in non-optimized setups).
Required Shots (N) for 1% Error ($\delta=0.01$)	$\approx 2 \times 10^{5}$	Shots	Calculated requirement for $\Delta = 4.6$.
Single-Shot Readout Error ($\epsilon$) Example	$\epsilon=0.29$	Probability	Example error for the simulated photon count histograms.
Optimal Measurement Time ($t_{m,opt}$) Range	$75 - 200$	ns	Range where $\Delta$ dependence on $t_m$ is minimal.
Background Photon Flux ($\lambda$) Example	$1$	MHz	Value used to illustrate background effects (higher than state-of-the-art).
Laser Excitation Rate ($\Gamma_P$)	$630$	MHz	Rate used in the 7-level rate-equation model.
Spin-Flip Rate ($\Gamma_{f1}$)	$80$	MHz	Rate used in the 7-level rate-equation model.

Key Methodologies

The multi-shot readout error benchmark $\Delta$ was derived through a rigorous simulation based on a rate-equation model of the NV center photodynamics.

Qubit System Definition: The study focused on the electronic spin states of the negatively charged NV center in diamond, specifically the $|g, 0\rangle$ ($|0\rangle$) and $|g, -1\rangle$ ($|1\rangle$) sublevels, operating at room temperature.
Photoluminescence Modeling: A 7-level rate-equation model was implemented to describe the photoluminescence (PL) dynamics during optical readout, accounting for radiative and non-radiative transitions between ground, excited, and intermediate singlet states.
Photon-Number Statistics: The Photon Mass Functions (PMFs), $P_0(n)$ and $P_1(n)$, representing the probability of detecting $n$ photons for states $|0\rangle$ and $|1\rangle$, were computed by solving the photon-number-resolved rate equations over a finite measurement duration ($t_m$).
Imperfection Integration: The model incorporated three key imperfections:
- Imperfect Photon Detection ($\eta$): Modeled using a binomial distribution relating emitted photons to detected photons.
- Background Photons ($\lambda$): Modeled using a Poisson process added to the NV-emitted photons.
- Imperfect Initialization ($q$): Modeled as a mixture of the computational basis states.
Benchmark Derivation: The multi-shot readout error benchmark $\Delta$ was calculated using the means ($\mu$) and variances ($\sigma^2$) of the resulting detected photon distributions, averaged over the parameter $z$ (qubit expectation value).
Optimization: $\Delta$ was minimized numerically by optimizing the laser pulse duration ($t_m$) for various values of $\eta$ and $\lambda$.

6CCVD Solutions & Capabilities

The research highlights that minimizing the multi-shot readout error $\Delta$ requires significant hardware optimization, particularly maximizing photon collection efficiency ($\eta$) and minimizing initialization error ($q$). These requirements are directly addressed by 6CCVD’s expertise in high-quality MPCVD diamond synthesis and precision engineering.

Applicable Materials

To replicate and extend this research, especially for achieving the lowest possible $\Delta$ values (approaching 0.816), researchers require diamond optimized for optical and spin coherence.

Material Requirement	6CCVD Recommended Solution	Key Benefit for NV Qubits
High $\eta$ (Photon Collection)	Optical Grade Single Crystal Diamond (SCD)	Ultra-low birefringence and minimal scattering losses, maximizing the fraction of photoluminescence photons collected.
Low $q$ (Initialization Error)	High-Purity SCD (Low Nitrogen/Defects)	Our MPCVD process ensures extremely low concentrations of parasitic defects (e.g., substitutional nitrogen), leading to longer spin coherence times and higher fidelity initialization.
Advanced Device Integration	Custom Thin-Film SCD	SCD wafers available in thicknesses from 0.1µm up to 500µm, ideal for fabricating integrated photonic structures (waveguides, reflectors) that enhance $\eta$.
Alternative Qubit Systems	Boron-Doped Diamond (BDD)	Available for researchers exploring spin-to-charge conversion mechanisms ³⁶ or electrochemical sensing applications.

Customization Potential

The optimization of NV readout often involves integrating the diamond substrate with complex optical and electrical components. 6CCVD provides the necessary precision engineering to facilitate these advanced setups.

Precision Polishing: Achieving high $\eta$ often relies on integrating the NV center with nanostructures (e.g., parabolic reflectors, bullseye gratings). 6CCVD guarantees SCD polishing to Ra < 1nm, minimizing surface roughness that causes scattering losses and hinders lithographic patterning.
Custom Dimensions: While the paper focuses on single NV centers, scaling up requires larger substrates. 6CCVD offers custom plates and wafers up to 125mm (PCD) and large-area SCD substrates, ensuring material availability for prototype development.
Integrated Metalization: The experiment requires precise control over the NV environment (e.g., microwave delivery, electric fields). 6CCVD offers in-house metalization services including deposition of Au, Pt, Pd, Ti, W, and Cu, allowing for custom electrode patterns directly on the diamond surface.
Substrate Thickness Control: We provide SCD substrates up to 10mm thick for bulk experiments, or thin films (0.1µm) necessary for membrane-based photonic integration.

Engineering Support

The research emphasizes that minimizing $\Delta$ requires optimizing material properties and measurement parameters simultaneously.

Material Selection Consultation: 6CCVD’s in-house PhD team specializes in the relationship between MPCVD growth parameters and resulting diamond properties (purity, strain, defect density). We can assist researchers in selecting the optimal SCD grade to minimize initialization error ($q$) and maximize photon collection ($\eta$) for similar NV-based Quantum Sensing and Computing projects.
Global Logistics: We ensure reliable, global delivery of custom diamond solutions, with DDU default shipping and DDP options available, supporting international research timelines.

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

View Original Abstract

The ground-state electronic spin of a negatively charged nitrogen-vacancy center in diamond can be used for room-temperature experiments showing coherent qubit functionality. At room temperature, photoluminescence-based qubit readout has low single-shot fidelity; however, the populations of the qubit’s two basis states can be inferred using multi-shot readout. In this work, we calculate the dependence of the error of a multi-shot inference method on various parameters of the readout process. This multi-shot readout error scales as Δ/N, with N being the number of shots, suggesting to use the coefficient Δ as a simple multi-shot readout error benchmark. Our calculation takes into account background photons, photon loss, and initialization error. Our model enables the identification of the readout error budget, i.e., the role various imperfections play in setting the readout error. Our results enable experimentalists and engineers to focus their efforts on those hardware improvements that yield the highest performance gain for multi-shot readout.

Tech Support

Original Source

DOI: https://doi.org/10.1063/5.0261836

References

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2013 - Nanoscale magnetometry with NV centers in diamond [Crossref]
2012 - Decoherence-protected quantum gates for a hybrid solid-state spin register [Crossref]
2014 - Universal control and error correction in multi-qubit spin registers in diamond [Crossref]