Demonstration of the Holonomically Controlled Non-Abelian Geometric Phase in a Three-Qubit System of a Nitrogen Vacancy Center

At a Glance

Metadata	Details
Publication Date	2022-11-02
Journal	Entropy
Authors	Shaman Bhattacharyya, Somnath Bhattacharyya
Institutions	University of the Witwatersrand
Citations	3
Analysis	Full AI Review Included

Executive Summary

This study demonstrates the successful simulation and analysis of holonomically controlled non-Abelian geometric phase gates on a three-qubit Nitrogen Vacancy (NV) center system, validating a robust approach for universal quantum computation.

Core Achievement: Simulation of universal holonomic control on a three-qubit, eight-level NV center system using the IBM Quantum Experience (IBM QE) platform.
Fidelity Improvement: Off-resonant holonomic gates achieved a high fidelity of approximately 85%, significantly outperforming on-resonant gates, which reached only ~70% fidelity.
Decoherence Control: The transition between the system’s three dark states was demonstrated, showing that decoherence is minimized when the gate’s dark state aligns with the qubit’s initial state, achieved via a specific π/3 rotation.
Non-Abelian Phase Generation: The concatenation of three separate Abelian rotation paths effectively produced a non-Abelian geometric phase, which is inherently resilient to local noise and errors.
Viability Confirmation: The results support the use of NV centers in diamond as highly viable, room-temperature solid-state qubits for developing universal, fault-tolerant quantum computers.

Technical Specifications

Parameter	Value	Unit	Context
Qubit System Type	Nitrogen Vacancy (NV) Center	N/A	Solid-state spin qubits in diamond
System Size (Simulated)	3	Qubits	Modeled as an eight-level system
Simulation Platform	IBM Q 14 Melbourne	Device	Actual quantum hardware used for digital simulation
Gate Type	Holonomic (Non-Abelian)	N/A	Gates based on geometric phase accumulation
Off-Resonant Gate Fidelity	~85	%	Maximum achieved fidelity for 3-qubit HQC
On-Resonant Gate Fidelity	~70	%	Comparative fidelity, demonstrating superiority of off-resonance
Qubit 1 Rabi Frequency (Ω)	4966	MHz	Frequency used for holonomic gate implementation
Decoherence Control Rotation	π/3 (60)	Radians (Degrees)	Rotation angle used to align the gate’s dark state
Optimal Dark State Alignment Detuning (Δ)	450	MHz	Detuning frequency (Δ₁) where dark state alignment minimizes deviation
Phase Dependence	Detuning Frequency (Δ)	N/A	Geometric phase magnitude varies as a function of Δ

Key Methodologies

The holonomic control of the three-qubit NV center system was demonstrated through a sequence of quantum simulations and calculations on the IBM QE platform:

System Initialization and Modeling: The three-qubit system was modeled using the Greenberger-Horne-Zeilinger (GHZ) frame of reference. Qubits were initialized using identity gates, followed by Rx and Rz gates to set the rotation parameters (θ and φ).
Holonomic Gate Implementation: The unitary holonomic gate (U) was implemented using standard quantum gates (Rx, Ry, Rz) combined with simulated detuning pulses (Δ) and Rabi frequencies (Ω) applied to all three qubits.
Rotation Path Concatenation: Three separate rotation paths—one for each qubit—were defined and concatenated. The first two qubits underwent complete rotations (θ = ±2π), while the third qubit rotated around a circular path (θ ranging between π/6 and -π/6).
Off-Resonance Optimization: The detuning frequencies (Δ₁, Δ₂, Δ₃) were varied to move the gate operation off-resonance. This variation was critical for maximizing the accumulated geometric phase and improving gate fidelity.
Dark State Analysis: The alignment of the gate’s dark state with the initial qubit state was analyzed by observing the return probability (P_r) as a function of detuning frequency and time. The minimum deviation point (450 MHz) confirmed optimal decoherence control.
Fidelity Calculation: Gate performance was quantified using the fidelity formula F(ρ,κ) = Tr[√√ρ κ √ρ], comparing the ideal density matrix (ρ) from the QasmSimulator with the experimental density matrix (κ) obtained from the IBM Q 14 Melbourne device.

Commercial Applications

The development of robust, high-fidelity holonomic gates on solid-state NV centers has direct implications for several advanced technological sectors:

Universal Quantum Computing: Provides a pathway for building fault-tolerant quantum processors, particularly solid-state devices based on diamond, which can operate at room temperature.
Quantum Sensing and Metrology: NV centers are premier quantum sensors. Holonomic control enhances the robustness of these sensors against environmental noise, leading to higher precision magnetometers, thermometers, and gyroscopes.
Fault-Tolerant Quantum Memory: Geometric phases are inherently robust against noise, making holonomic gates ideal for implementing quantum memory and error correction codes (e.g., holonomic surface codes).
Spin-Orbitronics and Topological Materials: The simulation of non-Abelian geometric phases and dark states relates directly to understanding Rashba spin-orbit coupling (RSOC) and vortex structures in topologically protected materials.
Microwave and RF Control Systems: The technique relies on precise control using microwave and radiofrequency pulses, applicable to advanced signal processing and quantum communication hardware.

View Original Abstract

The holonomic approach to controlling (nitrogen-vacancy) NV-center qubits provides an elegant way of theoretically devising universal quantum gates that operate on qubits via calculable microwave pulses. There is, however, a lack of simulated results from the theory of holonomic control of quantum registers with more than two qubits describing the transition between the dark states. Considering this, we have been experimenting with the IBM Quantum Experience technology to determine the capabilities of simulating holonomic control of NV-centers for three qubits describing an eight-level system that produces a non-Abelian geometric phase. The tunability of the geometric phase via the detuning frequency is demonstrated through the high fidelity (~85%) of three-qubit off-resonant holonomic gates over the on-resonant ones. The transition between the dark states shows the alignment of the gate’s dark state with the qubit’s initial state hence decoherence of the multi-qubit system is well-controlled through a π/3 rotation.

Tech Support

Original Source

DOI: https://doi.org/10.3390/e24111593

References

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