High fidelity quantum state tomography of electron-$^{14}$N nuclear hybrid spin register in diamond using Rabi oscillations

Technical Summary: Rabi-based Quantum State Tomography (RQST) for Hybrid Spin Registers

Problem Statement
Quantum state tomography (QST) is essential for characterizing quantum states and gates, serving as a cornerstone for scaling quantum computing architectures. Conventional QST methods typically rely on projection-based measurements using specific unitary operators (e.g., $X_{90}$ and $Y_{90}$ pulses) to transfer off-diagonal density matrix elements to observable diagonal terms. However, the authors identify a need for alternative methodologies that can offer high fidelity and robustness, particularly for solid-state spin systems like the Nitrogen-Vacancy (NV) center in diamond. A specific challenge in NV-based systems is the characterization of "dark" nuclear spins, which lack direct optical readout and require entanglement with an electron spin ancilla for measurement.

Methodology: Rabi-based Quantum State Tomography (RQST)
The authors propose a novel characterization method termed Rabi-based Quantum State Tomography (RQST). Unlike conventional methods that apply discrete $\pi/2$ rotations, RQST utilizes a series of Rabi oscillations with varying pulse durations (e.g., $X_5, X_{10}, X_{15}$, etc.) to reconstruct the quantum state. The method is implemented in two variants:

1. Rabi Amplitude QST (RAQST): Reconstructs the state by analyzing the amplitudes of Rabi oscillations around the x- and y-axes ($A_x, A_y$) relative to a reference amplitude ($A_{ref}$) obtained from an eigenstate. While amplitude data narrows the state location to eight points on the Bloch sphere, phase information is required to uniquely determine the coordinates.
2. Rabi Phase QST (RPQST): Reconstructs the state by extracting the phase angles ($\alpha, \beta$) of the Rabi oscillations. This variant is highlighted as being more robust against sample drift (e.g., a defect moving out of focus), which can alter signal amplitudes but has less impact on phase.

The reconstruction process involves fitting the photoluminescence (PL) intensity data to a sinusoidal function ($PL = \sin(\omega t + \phi)$) to extract amplitudes and phases. These parameters are then used to calculate the Cartesian coordinates ($n_x, n_y, n_z$) and polar/azimuthal angles ($\theta, \phi$) of the Bloch vector, allowing for the construction of the experimental density matrix $\rho_{exp}$.

Experimental Implementation
The methodology was validated on the electron-14N nuclear hybrid spin register of a single NV center in diamond at room temperature.

• System: The NV center provides a 9-dimensional Hilbert space (spin-1 electron and spin-1 nitrogen nucleus). The authors focused on a reduced 4-dimensional subspace corresponding to two two-level systems.
• Initialization: A modified spin drive and readout sequence initializes both electron and nuclear spins to the $|0, 0\rangle$ state.
• Readout Strategy:
  • Electron Spin: Directly measured via optically detected magnetic resonance (ODMR).
  • Nuclear Spin: As a "dark" spin with no direct optical signature, the nuclear state was read out via the electron spin using a C-NOT two-qubit entanglement gate. The nuclear spin was manipulated via radio-frequency (RF) driving, and the resulting state population was inferred through the electron spin's optical response.
• Circuit Design: The authors detail specific quantum circuits for initialization and state preparation, utilizing controlled gates to manipulate the electron-nuclear system within specific subspaces (e.g., conditional on the electron spin being in the $|-1\rangle$ state).

Key Results

• High Fidelity: The method demonstrated high-fidelity control and characterization. For the electron spin, an average fidelity of 0.995 was achieved over more than 40 measurements of different states on the Bloch sphere.
• Nuclear Spin Characterization: The methodology was successfully extended to the nuclear spin. By utilizing the electron spin as an ancilla, the authors achieved fidelities of the same order of magnitude as the electron spin measurements.
• Simulation Comparison: Numerical simulations comparing RQST to projection-based QST under uniform white noise indicated that RQST can perform better than conventional methods for certain state angles ($\theta, \phi$), though it performs slightly weaker near the equator of the Bloch sphere.
• Robustness: The RPQST variant was shown to be less susceptible to sample drift compared to amplitude-based methods, while RAQST allows for the detection of such drifts and analysis of state mixing (decoherence).

Significance and Claims
The paper claims to present a validated, efficient method for high-fidelity quantum state characterization that is particularly suitable for room-temperature solid-state spin systems. By replacing standard unitary rotation operators with a Rabi cycle array, the authors provide a flexible tool for state reconstruction. The work is significant for:

1. Scalability: Offering a pathway to characterize qubits in systems where scalability is a bottleneck, such as NV centers.
2. Fault Tolerance: Contributing to the requirements for fault-tolerant quantum computation by enabling precise error correction code verification through high-fidelity state characterization.
3. Hybrid Systems: Demonstrating the capability to characterize "dark" nuclear spins via entanglement with electron spins, a crucial step for utilizing nuclear spins as long-lived quantum memories.

The authors note that while the current work focuses on pure states, the methodology is general enough to be applied to mixed states, though a detailed study of mixed-state QST is reserved for future work. The fidelity of the nuclear spin measurements is noted to inherently depend on the fidelity of the two-qubit gates used for entanglement.