Preparing Dicke states in a spin ensemble using phase estimation

At a Glance

Metadata	Details
Publication Date	2021-09-09
Journal	Physical review. A/Physical review, A
Authors	Yang Wang, Barbara M. Terhal
Institutions	QuTech, Delft University of Technology
Citations	18
Analysis	Full AI Review Included

Technical Documentation & Analysis: Dicke State Preparation in NV-Flux Qubit Systems

Reference Paper: Preparing Dicke states in a spin ensemble using phase estimation (arXiv:2104.14310v4)

Executive Summary

This research details an efficient, noise-resilient scheme for preparing highly entangled Dicke states, critical for advancing quantum metrology and sensing applications.

Core Achievement: Demonstration of a Dicke state preparation scheme using standard phase estimation, requiring only O(log₂ N) ancilla qubit measurements.
Physical System: A hybrid quantum architecture utilizing an ensemble of electronic spins in diamond Nitrogen-Vacancy (NV) centers collectively coupled to a single superconducting flux qubit.
Metrology Relevance: The prepared Dicke states $|N, m_z \rangle$ with $m_z \sim O(1)$ can achieve Heisenberg-limited sensitivity (scaling as $O(1/N^2)$) using only global control.
Efficiency: The total preparation time scales efficiently as $O(\log_2 N)$, significantly faster than previous $O(N)$ schemes.
Noise Resilience: The protocol incorporates integrated dynamical decoupling and majority voting (M repeats) to mitigate errors from ancilla qubit dephasing and decay, achieving simulated fidelities exceeding 90%.
Material Requirement: Successful implementation relies on high-quality Single Crystal Diamond (SCD) substrates capable of hosting high-density NV ensembles (up to 10²¹ m^-3) while maintaining long electronic spin coherence times ($T_2 > 50$ ms).

Technical Specifications

The following hard data points were extracted from the analysis of the proposed experimental setup and numerical simulations:

Parameter	Value	Unit	Context
Spin Ensemble Size (N)	Up to 500	Spins	Simulated maximum for K=9 rounds of phase estimation.
Phase Estimation Rounds (K)	[log₂ N] + 1	Rounds	Required for preparation.
Preparation Time Scaling	O(log₂ N)	Time	Total time for controlled rotations.
NV Center Density	10²¹	m^-3	Required for ~1000 NV centers in a 1 µm³ volume below the flux qubit loop.
NV Electronic Spin T₁	> 8	Hours	Energy relaxation time at 25 mK.
NV Electronic Spin T₂	> O(50)	ms	Dephasing time at 77 K (with dynamical decoupling).
Flux Qubit T₁	O(50)	µs	Energy relaxation time (away from flux sweet spot).
Flux Qubit T₂	< O(1)	µs	Dephasing time (away from flux sweet spot).
Magnetic Coupling ($\gamma$)	O(10)	kHz	Estimated coupling strength between flux qubit and NV ensemble.
Target Fidelity	> 90	%	Achieved in simulation (N=500, $\gamma=5$ MHz) with M=5 majority votes.
NV Zero-Field Splitting ($\Delta$)	$\approx 2.88$	GHz	Intrinsic property of NV centers.

Key Methodologies

The Dicke state preparation relies on the standard phase estimation algorithm applied to the collective spin operator $J_z$.

Initialization: The N electronic spins in the NV ensemble are initialized into the product state $| \psi_0 \rangle = (|0\rangle + |1\rangle)^{\otimes N} / \sqrt{2^N}$.
Unitary Operator: The unitary operator for phase estimation is $U = e^{i 2\pi J_z / 2^K}$, where $J_z = \frac{1}{2} \sum_{i=1}^N Z_i$.
Controlled Rotation Implementation: The core operation is the controlled-U$^{2^{K-j}}$ gate, realized via the ZZ-interaction Hamiltonian $H_{\text{coupl}} \approx \frac{1}{2} \gamma Z_f J_z$, where $Z_f$ is the Pauli Z operator on the ancilla flux qubit.
Integrated Dynamical Decoupling (DD): To mitigate dephasing of both the flux qubit and the NV spins, the controlled rotation is performed with integrated echo pulses ($e^{i\pi J_y}$) applied simultaneously to the spins and the flux qubit.
Sequential Measurement: The phase estimation is executed sequentially over $K = [\log_2 N] + 1$ rounds, using only a single ancilla qubit.
Error Mitigation: Each round of phase estimation is repeated $M$ times, and a simple majority vote is performed on the measurement outcomes to suppress errors arising from ancilla decay and dephasing.
Target State Projection: The algorithm non-deterministically projects the ensemble into a random Dicke state $|N, m_z \rangle$, with a probability $O(1/\sqrt{N})$ of obtaining the desired Heisenberg-limited states ($m_z \sim O(1)$).

6CCVD Solutions & Capabilities

The successful replication and extension of this quantum metrology research hinge on the quality and customization of the diamond substrate and its integration with the superconducting circuit. 6CCVD is uniquely positioned to supply the necessary materials and engineering support.

Applicable Materials

The proposed hybrid system requires diamond optimized for NV center performance at cryogenic temperatures (mK range).

Research Requirement	6CCVD Solution	Technical Advantage
High Coherence NV Host	Optical Grade Single Crystal Diamond (SCD)	Ultra-low strain and low nitrogen concentration ensure maximum $T_1$ (> 8 hours) and $T_2$ (> 50 ms) coherence times for the NV electronic spins.
High NV Density	Custom Doping/Implantation Services	We provide SCD substrates optimized for high-density NV creation (native or implanted) up to 10²¹ m^-3, crucial for maximizing the magnetic coupling strength ($\gamma$).
Large Ensemble Size (N=500+)	Large Area SCD/PCD Wafers	We offer SCD plates and PCD wafers up to 125 mm in diameter, allowing for scalable integration of large NV ensembles and multiple flux qubit circuits.

Customization Potential

The experimental setup requires precise integration of the superconducting flux qubit onto the diamond surface.

Precision Substrates: The flux qubit loop (1 µm x 1 µm) demands exceptional surface quality. 6CCVD provides SCD wafers polished to Ra < 1 nm and inch-size PCD polished to Ra < 5 nm, ensuring compatibility with high-fidelity electron beam lithography.
Custom Dimensions: We supply custom-cut SCD plates up to 500 µm thick, tailored to fit specific cryogenic sample holders and dilution refrigerator stages.
Integrated Metalization: The superconducting circuit requires specific metal stacks. 6CCVD offers in-house custom metalization including Au, Pt, Pd, Ti, W, and Cu deposition, enabling the fabrication of high-quality superconducting resonators and flux qubits directly on the diamond surface.

Engineering Support

The paper identifies the weak magnetic coupling ($\gamma$) versus the short flux qubit $T_2$ as the main experimental challenge.

Hybrid System Optimization: 6CCVD’s in-house PhD team specializes in material selection and optimization for Quantum Metrology projects. We assist researchers in tuning diamond growth parameters to achieve the optimal NV depth and density required to maximize the coupling strength ($\gamma$) while preserving NV coherence.
BDD Capabilities: For related quantum sensing applications requiring charge control or integrated electronics, we also offer Boron-Doped Diamond (BDD) films with tunable conductivity.

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

View Original Abstract

<p>We present a Dicke state preparation scheme which uses global control of N spin qubits: our scheme is based on the standard phase estimation algorithm, which estimates the eigenvalue of a unitary operator. The scheme prepares a Dicke state nondeterministically by collectively coupling the spins to an ancilla qubit via a ZZ interaction, using log2N+1 ancilla qubit measurements. The preparation of such Dicke states can be useful if the spins in the ensemble are used for magnetic sensing: we discuss a possible realization using an ensemble of electronic spins located at diamond nitrogen-vacancy centers coupled to a single superconducting flux qubit. We also analyze the effect of noise and limitations in our scheme. </p>

Tech Support

Original Source

DOI: https://doi.org/10.1103/physreva.104.032407