Optimal control of fast and high-fidelity quantum gates with electron and nuclear spins of a nitrogen-vacancy center in diamond
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2015-05-15 |
| Journal | Physical Review A |
| Authors | Yi Chou, ShangâYu Huang, HsiâSheng Goan |
| Institutions | National Center for Theoretical Sciences, National Taiwan University |
| Citations | 40 |
| Analysis | Full AI Review Included |
Technical Documentation and Analysis: High-Fidelity Quantum Gates using Optimal Control in MPCVD Diamond NV Centers
Section titled âTechnical Documentation and Analysis: High-Fidelity Quantum Gates using Optimal Control in MPCVD Diamond NV CentersâExecutive Summary
Section titled âExecutive SummaryâThis document analyzes the application of Quantum Optimal Control (QOC) theory, specifically the Krotov method, to achieve high-speed and high-fidelity quantum gate operations using the Nitrogen Vacancy (NV-) center electron and nuclear spins in diamond. This research sets critical benchmarks for Fault-Tolerant Quantum Computation (FTQC) utilizing robust MPCVD diamond materials.
- Ultra-High Fidelity: Achieved quantum gate infidelities ($K$) as low as $3.9 \times 10^{-5}$ for single-qubit operations (X Gate) and $4.3 \times 10^{-4}$ for the two-qubit CNOT gate.
- FTQC Compliance: The calculated worst-case gate errors ($K < 6.0 \times 10^{-4}$) are significantly below the $10^{-3}$ threshold required for current models of Fault-Tolerant Quantum Computation (FTQC).
- Nano-second Speeds: Demonstrated extremely fast gate operation times, achieving single-qubit (Z, X) operations in 10 ns (0.01 ”s) and the critical two-qubit CNOT gate in 50 ns (0.05 ”s).
- Robustness via QOC: The Krotov optimization method successfully designs complex control pulses that suppress decoherence effects from nearby noise qubits ($^{15}$N, $^{13}$C), leakage states ($m_s = +1$), and non-Markovian distant spin baths.
- Material Foundation: The entire system relies on high-purity, isotopically engineered Single Crystal Diamond (SCD) to ensure long spin coherence times ($\tau_c$ up to 25 ”s).
- Scaling Potential: The ability to execute universal quantum gates (including CNOT) in a single, optimally designed pulse sequence, rather than serial decomposition, accelerates potential scale-up of diamond quantum registers.
Technical Specifications
Section titled âTechnical SpecificationsâThe following parameters summarize the core operational conditions and achieved performance metrics derived from the QOC application on the NV center spin register.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Core Quantum System | NV- center | N/A | Hybrid spin register in diamond ($^{15}$N, $^{13}$C, electron spin). |
| Static Magnetic Field ($B_z$) | 500 - 1000 | G | Used for Zeeman splitting and defining quantization axis. |
| Electron Spin ZFS ($\Delta$) | $2.87 \cdot 2\pi$ | GHz | Zero-field splitting between $m_s = 0$ and $m_s = \pm 1$. |
| $^{13}$C Hyperfine Coupling ($A^{\text{C}}$) | $127 \cdot 2\pi$ | MHz | Strong coupling between NV electron and nearest neighbor $^{13}$C. |
| X Gate Operation Time (T) | 0.01 (10) | ”s (ns) | Target speed for high-fidelity single-qubit operation. |
| Optimal X Gate Error ($K$) | $3.9 \times 10^{-5}$ | N/A | Lowest error achieved with QOC (includes $^{15}$N and $^{13}$C noise). |
| CNOT Gate Operation Time (T) | 0.05 (50) | ”s (ns) | Demonstrated ultra-fast two-qubit control. |
| Optimal CNOT Gate Error ($K$) | $4.3 \times 10^{-4}$ | N/A | High fidelity achieved even with $^{15}$N noise and spin bath effects. |
| Spin Bath Correlation Time ($\tau_c$) | 25 | ”s | Input parameter for non-Markovian decoherence modeling. |
| Required FTQC Error Threshold ($P_{\text{th}}$) | < $6.0 \times 10^{-4}$ | N/A | Error achieved for CNOT gate meets surface code requirements. |
Key Methodologies
Section titled âKey MethodologiesâThe research utilizes an advanced implementation of Quantum Optimal Control (QOC) theory to synthesize robust, short-duration control pulses.
- Hybrid Spin Register Definition:
- Qubits: NV electron spin (control) and proximal $^{13}$C nuclear spin (target).
- Noise/Ancilla: $^{15}$N nuclear spin and the $m_s = +1$ state (treated as a leakage state).
- Hamiltonian Specification:
- The total system Hamiltonian $H$ incorporates static magnetic field effects ($H_0$), time-dependent control fields ($H_{cx}(t), H_{cy}(t)$), and the effect of the environment ($H_{eb}$).
- Hyperfine interactions ($A^{\text{C}}, A^{\text{N}}$) govern coupling between the NV electron spin and the $^{13}$C and $^{15}$N nuclear spins, respectively.
- Decoherence Treatment (Open System Model):
- The distant spin bath is modeled as a classical random field $B(t)$ acting on the electron spin $S_z$, causing pure dephasing.
- The dynamics are governed by a Time-Local Non-Markovian Master Equation which is coupled with a differential equation for the dissipator $D(t)$. This ensures accurate modeling of memory effects in the decoherence.
- Optimal Control Algorithm (Krotov Method):
- The QOC algorithm minimizes the Infidelity function ($K$) relative to the ideal target gate ($G$), ensuring the resulting propagator ($U$) achieves maximum overlap.
- The algorithm iteratively updates the control fields $B_x(t)$ and $B_y(t)$, employing a feedback mechanism to ensure monotonic increase of the objective function (fidelity $F$) without requiring expensive line searches.
- Robust Gate Synthesis:
- Unlike conventional $\pi$-pulse techniques, which suffer fidelity loss due to leakage at high power, QOC designs complex, arbitrarily shaped microwave/RF pulses that specifically suppress unwanted off-resonance transitions and crosstalk, enabling extremely fast, high-fidelity gate realization (e.g., CNOT in a single 50 ns run).
6CCVD Solutions & Capabilities: Enabling Advanced NV Quantum Research
Section titled â6CCVD Solutions & Capabilities: Enabling Advanced NV Quantum ResearchâThis research underscores the absolute requirement for premium, high-purity, and isotopically engineered Single Crystal Diamond (SCD) materials. 6CCVD is an expert MPCVD diamond manufacturer, uniquely positioned to supply the foundational materials needed to replicate and advance this cutting-edge quantum control research.
| Research Requirement | 6CCVD Material/Capability | Value Proposition |
|---|---|---|
| Foundational Material Quality | Optical Grade Single Crystal Diamond (SCD) | Critical for achieving the long coherence times ($\tau_c$ up to 25 ”s) necessary for QOC gates. 6CCVD provides SCD with exceptionally low native nitrogen and metallic impurity background, minimizing unwanted decoherence sources. |
| Isotope Engineering | Custom Doping via MPCVD Growth | The experiment requires precise incorporation of target isotopes ($^{15}$N for the NV center, and proximal $^{13}$C spins). 6CCVD utilizes customized MPCVD gas recipes to control NV concentration and isotopic purity, allowing researchers to define specific spin registers accurately. |
| Integrated Control Fields | In-House Custom Metalization Services | The implementation of QOC requires complex, time-dependent $B_x(t)$ and $B_y(t)$ control fields. 6CCVD offers in-house deposition and patterning of standard and refractory metals (Au, Pt, Pd, Ti, W, Cu) for lithographically defining the microwave and RF strip lines necessary to generate the optimal control pulses directly on the diamond surface. |
| Scalability and Integration | Custom Dimensions and Thickness Control | 6CCVD delivers large-area SCD substrates and wafers up to 125mm (PCD), enabling future scale-up of diamond quantum chip architectures. We control SCD thickness from 0.1 ”m up to 500 ”m, allowing precise positioning of NV layers relative to surface control structures. |
| Surface Precision | Ultra-Low Roughness Polishing | High-fidelity gates require smooth surfaces to minimize surface noise interactions. 6CCVD achieves superior polishing results: Ra < 1 nm for Single Crystal Diamond, ensuring optimal performance for near-surface NV centers and high optical coherence. |
| Boron Doped Diamond (BDD) | BDD Substrates and Films | While this paper focuses on NV-, BDD is vital for related electrochemistry and sensing applications. We offer high-quality BDD films with tunable conductivity. |
Engineering Support: 6CCVDâs in-house PhD team provides specialized consultation services, assisting engineers and scientists with material specification, isotopic selection, and structural design for similar hybrid spin quantum computation projects. We ensure the raw diamond material meets the rigorous demands of nano-second scale optimal control experiments.
For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.
View Original Abstract
A negatively charged nitrogen vacancy (NV) center in diamond has been recognized as a good solid-state qubit. A system consisting of the electronic spin of the NV center and hyperfine-coupled nitrogen and additionally nearby carbon nuclear spins can form a quantum register of several qubits for quantum information processing or as a node in a quantum repeater. Several impressive experiments on the hybrid electron and nuclear spin register have been reported, but fidelities achieved so far are not yet at or below the thresholds required for fault-tolerant quantum computation (FTQC). Using quantum optimal control theory based on the Krotov method, we show here that fast and high-fidelity single-qubit and two-qubit gates in the universal quantum gate set for FTQC, taking into account the effects of the leakage state, nearby noise qubits and distant bath spins, can be achieved with errors less than those required by the threshold theorem of FTQC.