Simulating the Electronic Structure of Spin Defects on Quantum Computers
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2022-03-10 |
| Journal | PRX Quantum |
| Authors | Benchen Huang, Marco Govoni, Giulia Galli |
| Institutions | University of Chicago, Argonne National Laboratory |
| Citations | 40 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled âExecutive SummaryâThis research presents a novel hybrid classical/quantum computational protocol for simulating the electronic structure of strongly correlated spin defects in solid-state materials, specifically focusing on the Nitrogen Vacancy (NV-) center in diamond and the Double Vacancy (VV) in 4H-SiC.
- Core Value Proposition: Successfully calculated both ground and excited state energies of complex solid-state spin qubits using a hybrid protocol on a Near-Intermediate Scale Quantum (NISQ) computer (ibmq_casablanca).
- Methodology Integration: Combined classical first-principles Quantum Defect Embedding Theory (QDET) to derive an effective Hamiltonian, followed by Variational Quantum Eigensolver (VQE) for ground states and Quantum Subspace Expansion (QSE) for excited states.
- Error Mitigation Strategy: Proposed and implemented a robust error mitigation scheme combining two techniques:
- Post-Selection: Enforcing constraints on electron occupation to overcome the âunphysical state problemâ inherent to VQE on noisy hardware.
- Zero-Noise Extrapolation (ZNE): Utilizing a novel exponential block replication technique to controllably amplify noise and extrapolate results to the zero-noise limit.
- Performance Achievement: The ZNE technique reduced the ground state energy error relative to the noiseless simulator to approximately 0.002 eV (NV-) and 0.005 eV (VV).
- Excited State Calculation: Reported the first-time calculation of excited state energies for these spin defects using the QSE algorithm on real quantum hardware, demonstrating the methodâs applicability beyond ground state properties.
Technical Specifications
Section titled âTechnical Specificationsâ| Parameter | Value | Unit | Context |
|---|---|---|---|
| Quantum Device Used | ibmq_casablanca | N/A | 7-qubit superconducting quantum computer (NISQ hardware). |
| Qubit Count (NV- Simulation) | 4 | Qubits | Active space minimum model (4e, 3o) mapped via Parity transformation. |
| Qubit Count (VV Simulation) | 6 | Qubits | Active space minimum model (6e, 4o) mapped via Parity transformation. |
| DFT Supercell Size (NV-) | 216 | Atoms | Used for initial Density Functional Theory (DFT) calculation. |
| DFT Supercell Size (VV) | 200 | Atoms | Used for initial DFT calculation. |
| Kinetic Energy Cutoff | 50 | Ry | Used in plane-wave pseudopotential DFT calculations. |
| VQE Ansatz Used | UCCSD | N/A | Unitary Coupled Cluster Singles and Doubles (chemistry-inspired). |
| VQE Optimizer Used | SPSA | N/A | Simultaneous Perturbation Stochastic Approximation (chosen for noise robustness). |
| Ground State Error (NV-, ZNE) | ~0.002 | eV | Error relative to noiseless simulation reference. |
| Ground State Error (VV, ZNE) | ~0.005 | eV | Error relative to noiseless simulation reference. |
| Measurement Repetitions (ZNE Stability) | 50 | Repetitions | Number of times each circuit was measured (8192 shots/circuit) to stabilize the extrapolation. |
| Standard Deviation (Ï) of Measurement | ~3 | meV | Associated with the mean of energy measurements (50 repetitions). |
| Qubit T1 Coherence Time (Q1) | 110 | ”s | Calibration data for the ibmq_casablanca device. |
| CNOT Error Rate (Q1, Q3 pair) | 0.671 x 10-2 | N/A | Lowest error CNOT gate pair used in the experiment. |
Key Methodologies
Section titled âKey MethodologiesâThe simulation workflow utilizes a multi-stage hybrid classical/quantum protocol:
- Mean-Field Electronic Structure (Classical DFT):
- Define a large periodic supercell (216 atoms for NV-, 200 atoms for VV) representing the defect in the solid.
- Compute the electronic structure using hybrid DFT (DDH functionals) and plane-wave pseudopotential methods.
- Effective Hamiltonian Derivation (Classical QDET):
- Identify a localized subset of single-particle orbitals around the defect to define a small âactive spaceâ (minimum model).
- Apply Quantum Defect Embedding Theory (QDET) to derive an effective many-body Hamiltonian (Heff) for the active space, incorporating environmental correlation effects.
- Qubit Hamiltonian Mapping:
- Transform the fermionic Heff into a qubit Hamiltonian (Hq) using the Parity transformation (Jordan-Wigner or Bravyi-Kitaev are alternatives).
- Ground State Calculation (VQE):
- Use the Variational Quantum Eigensolver (VQE) with the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz to variationally minimize the energy expectation value on the quantum computer.
- Noise Mitigation: Post-Selection:
- Implement a post-selection filter on the measurement outcomes to discard states that do not conserve the correct number of electrons, thereby enforcing physical constraints and mitigating the unphysical state problem.
- Noise Mitigation: Zero-Noise Extrapolation (ZNE):
- Apply ZNE using a novel exponential block replication technique to artificially increase the circuit depth by integer multiples (n = 1 to 5 replicas).
- Extrapolate the noisy energy results (E(λ)) back to the zero-noise limit (E*) using a polynomial fit.
- Excited State Calculation (QSE):
- Use the Quantum Subspace Expansion (QSE) algorithm, starting from the optimized VQE ground state as a reference.
- Evaluate the Hamiltonian (HQSE) and overlap (SQSE) matrix elements on the quantum computer, applying both post-selection and ZNE to the measurements.
- Solve the generalized eigenvalue problem (HQSEC = SQSECΔ) classically to obtain excited state energies.
Commercial Applications
Section titled âCommercial ApplicationsâThe ability to accurately simulate the electronic structure and excited states of solid-state defects is critical for advancing several quantum technologies:
| Industry/Application | Relevance to Spin Defect Simulation |
|---|---|
| Quantum Sensing and Metrology | NV- centers in diamond are the benchmark for solid-state quantum sensors. Accurate simulation of their ground and excited state transitions (e.g., 3A2 to 3E) is essential for optimizing magnetometry, thermometry, and gyroscopy devices. |
| Quantum Communication | Spin defects (NV-, VV) serve as robust, long-coherence solid-state qubits and quantum memory nodes. Simulations guide the design of materials with optimal spin-photon interfaces for quantum networks. |
| Solid-State Quantum Computing | The methodology is directly applicable to characterizing and optimizing novel point defects in semiconductors (e.g., SiC, GaN) for use as scalable quantum processors and memory elements. |
| Advanced Materials Discovery | Provides a first-principles quantum simulation framework for strongly correlated systems, accelerating the discovery and design of new materials for energy storage, catalysis, and electronics where multi-reference electronic states are dominant. |
| Semiconductor Manufacturing | Understanding defect properties (like the VV center in SiC) is crucial for controlling material quality and performance in high-power electronics and radiation-hardened devices. |
View Original Abstract
We present calculations of both the ground-and excited-state energies of spin defects in solids carried out on a quantum computer, using a hybrid classical-quantum protocol. We focus on the negatively charged nitrogen-vacancy center in diamond and on the double vacancy in 4H Si C, which are of interest for the realization of quantum technologies. We employ a recently developed first-principles quantum embedding theory to describe point defects embedded in a periodic crystal and to derive an effective Hamiltonian, which is then transformed to a qubit Hamiltonian by means of a parity transformation. We use the variational quantum eigensolver (VQE) and quantum subspace expansion methods to obtain the ground and excited states of spin qubits, respectively, and we propose a promising strategy for noise mitigation. We show that by combining zero-noise extrapolation techniques and constraints on electron occupation to overcome the unphysical-state problem of the VQE algorithm, one can obtain reasonably accurate results on near-term-noisy architectures for ground-and excited-state properties of spin defects.