Preparing Dicke states in a spin ensemble using phase estimation
At a Glance
Section titled āAt a Glanceā| 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 |
Executive Summary
Section titled āExecutive SummaryāThe research proposes a robust method for preparing highly entangled Dicke states, essential for achieving Heisenberg-limited quantum metrology.
- Core Value Proposition: Deterministic preparation of Dicke states (|N, mz ∼ O(1)>) using global control, enabling metrology sensitivity scaling as 1/N2 (Heisenberg limit), surpassing the standard quantum limit (1/√N).
- Methodology: Utilizes sequential Quantum Phase Estimation (QPE) in a hybrid system consisting of an ensemble of Nitrogen-Vacancy (NV) electronic spins coupled to a Superconducting Flux Qubit (ancilla).
- Efficiency: The scheme requires only O(log2 N) ancilla qubit measurements, significantly fewer than methods requiring individual spin control.
- System Implementation: The controlled rotations are realized via a ZZ-coupling (Hcoupl ∼ Zf Jz) between the flux qubit (Zf) and the collective spin operator (Jz) of the NV ensemble.
- Performance Trade-off: Preparation success probability scales as O(1/√N), necessitating O(√N) preparation attempts on average to obtain the target state.
- Noise Resilience: Numerical simulations confirm the scheme is robust against ancilla timing inaccuracies (up to 10 ns deviation) and decoherence, provided strong coupling (Ī³ ∼ 5 MHz) and majority voting (M ≥ 5 repeats per round) are employed.
- Key Challenge: The required coupling strength (O(MHz)) is significantly higher than current experimental estimates (O(10) kHz) for the proposed NV-flux qubit setup, especially given the short coherence time (T2 < 1 Āµs) of the flux qubit when operated away from its sweet spot.
Technical Specifications
Section titled āTechnical Specificationsā| Parameter | Value | Unit | Context |
|---|---|---|---|
| Spin Ensemble Size (N) | 500 | Spins | Used in numerical simulations (K=6 rounds). |
| Spin Ensemble Size (N) | ∼ 106 | Spins | Corresponds to K=20 rounds of QPE. |
| NV Center Density | 1021 | m-3 | Required to achieve ∼ 1000 NV centers in a 1 Āµm3 volume below the flux qubit loop. |
| NV Electronic Spin T1 | > 1 | hour | At 25 mK operating temperature. |
| NV Electronic Spin T2 | > O(50) | ms | With dynamical decoupling (DD) at 77 K. |
| NV Initialization Time | O(100) | Āµs | Duration for resonant optical excitations. |
| Flux Qubit T1 | O(50) | Āµs | Assumed value for simulation. |
| Flux Qubit Tφ (Pure Dephasing) | Tφ = 2 | Āµs | Assumed value for simulation (away from sweet spot). |
| Flux Qubit T2 | < O(1) | Āµs | Typical value when operated away from the flux sweet spot. |
| Flux Qubit Single-Qubit Gate Time | O(1) | ns | Typical duration for single-qubit gates. |
| Magnetic Coupling (Ī³) | O(10) | kHz | Estimated strength for NV-flux qubit ZZ-coupling. |
| Required Coupling (Ī³) | ≥ 5 | MHz | Required in simulations to overcome Tφ = 2 Āµs dephasing. |
| External Magnetic Field | O(100) | Gauss | Used to split the m = Ā±1 levels of the NV S=1 spin. |
| NV Zero-Field Splitting (Ī) | ∼ 2.88 | GHz | Intrinsic splitting of the NV electronic spin. |
| Timing Inaccuracy (Ļ) Tolerance | ≤ 10 | ns | Standard deviation of time deviation tolerated in controlled rotations (maintaining > 90% fidelity). |
| QPE Rounds (K) | ∼ log2 N | Rounds | Number of sequential measurements required. |
Key Methodologies
Section titled āKey MethodologiesāThe Dicke state preparation relies on sequential Quantum Phase Estimation (QPE) applied to the collective spin operator Jz, using a hybrid NV-Flux Qubit system.
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System Setup and Initialization:
- A hybrid system is established: N NV-center electronic spins (acting as qubits, using |m=0> and |m=-1> states) are collectively coupled to a Superconducting Flux Qubit (ancilla).
- The NV ensemble is initialized into the product state |ψ0> = (|0> + |1>)/√2)⊗N, which is a uniform superposition of Dicke states centered around mz = 0.
- The ancilla flux qubit is initialized to |0>.
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Controlled Unitary Operation:
- The QPE requires controlled rotations Uj = exp(iĻ2j-1(Jz - Aj-1)), where Aj-1 is determined by previous measurement outcomes.
- The controlled rotation U2K-j is implemented via the ZZ-coupling Hamiltonian Hcoupl = (Ī³/2) Zf Jz, where Zf is the Pauli Z operator of the flux qubit.
- To mitigate noise from other NV types, integrated dynamical decoupling (echo pulses πJy) is applied simultaneously to the NV spins and the flux qubit during the controlled rotation time tj.
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Sequential Phase Estimation Rounds (K rounds):
- Each round j involves: a. Applying a Hadamard gate (H) to the ancilla. b. Applying the controlled rotation U2K-j for time tj = π / (2j-1 γ). c. Rotating the ancilla around the Z-axis (Rz(ν)) based on previous measurement results. d. Applying a final Hadamard gate (H) to the ancilla. e. Measuring the ancilla qubit (readout bj = 0 or 1).
- The total evolution time T for all K rounds is bounded by T < 2π/γ.
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Noise Mitigation via Repetition and Voting:
- Each QPE round is repeated M times (M ≥ 5 in simulations).
- A simple majority vote is performed on the M measurement outcomes to determine the bit bj, suppressing ancilla phase flip errors (Tφ) and decay (T1).
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State Projection:
- The sequential measurements project the initial product state into a superposition of Dicke states. After K rounds, the state is projected to |N, mz> where mz is determined by the measured bits, maximizing the probability of obtaining the desired Heisenberg-limited state (|N, mz ∼ O(1)>).
Commercial Applications
Section titled āCommercial ApplicationsāThis technology is foundational for advanced quantum hardware and sensing applications, leveraging the unique properties of entangled spin ensembles.
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Quantum Metrology and Sensing:
- Heisenberg-Limited Sensors: Enables the creation of sensors (e.g., magnetometers, gyroscopes) that operate at the fundamental quantum limit (1/N2 scaling), vastly improving precision over classical sensors.
- Magnetic Field Sensing: Direct application for high-sensitivity magnetic field detection using NV ensembles, relevant for medical imaging (MRI) and fundamental physics research.
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Hybrid Quantum Computing and Memory:
- Quantum Transduction: The NV-flux qubit coupling provides a mechanism for transferring quantum information between fast superconducting circuits (processing) and long-coherence spin systems (memory).
- Quantum Memory: Dicke states prepared in NV ensembles can serve as robust, long-lived quantum memory elements (NV T1 > 1 hour).
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Quantum Error Correction (QEC):
- The preparation scheme can be adapted to create specific superpositions of Dicke states that form logical codewords for permutation-invariant quantum error correction codes, enhancing robustness against collective noise.
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Solid-State Quantum Hardware Development:
- The methodology provides a blueprint for controlling and entangling large ensembles of solid-state qubits, a critical step toward scalable quantum processors.
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>