[Paper Review] Quantum process tomography of a single solid state qubit

Mark Howard, Jason Twamley+4 more|ArXiv.org|Mar 16, 2005

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper presents the first experimental demonstration of quantum process tomography (QPT) on a single solid-state qubit using the nitrogen-vacancy (NV) center in diamond. By applying microwave pulses to prepare a basis of input states and using optical readout via ODMR, the authors reconstruct the process matrix χ and its physically valid counterpart 𝔀, revealing increasing deviations due to decoherence over time (20–80 ns), with fidelity metrics showing growing discrepancies between experimental and ideal processes.

ABSTRACT

We present an example of quantum process tomography performed on a single solid state qubit. The qubit used is two energy levels of the triplet state in the Nitrogen-Vacancy defect in Diamond. Quantum process tomography is applied to a qubit which has been allowed to decohere for three different time periods. In each case the process is found in terms of the $χ$ matrix representation and the affine map representation. The discrepancy between experimentally estimated process and the closest physically valid process is noted.

Motivation & Objective

• To demonstrate quantum process tomography (QPT) on a single solid-state qubit, a critical step toward characterizing and debugging quantum devices.
• To identify and quantify decoherence effects in a single NV-center qubit by analyzing the evolution of the process matrix χ over time.
• To compare experimentally reconstructed processes with the closest physically valid processes (via χ̃) to assess validity and fidelity.
• To validate the use of QPT in solid-state systems using high-fidelity single-shot readout via optically detected magnetic resonance (ODMR).

Proposed method

• Prepared four input states (|0⟩, |1⟩, |+⟩, |+i⟩) on the NV center's spin triplet state using microwave pulses.
• Applied controlled unitary operations via microwave Rabi oscillations to calibrate π and π/2 pulses.
• Performed quantum state tomography on output states after each process using ODMR for single-shot readout.
• Reconstructed the process matrix χ using the χ-matrix formalism from input-output state pairs via the inverse of the β matrix.
• Transformed the process into a density matrix via the Jamiolkowski isomorphism to compute fidelity and distance metrics.
• Computed the nearest physically valid process 𝔀 by projecting χ onto the set of completely positive, trace-preserving maps, and quantified discrepancies using matrix norms and trace distance.

Experimental results

Research questions

• RQ1 Can quantum process tomography be successfully applied to a single solid-state qubit with high-fidelity readout?
• RQ2 How does decoherence over time (20–80 ns) affect the physical validity and fidelity of the reconstructed quantum process?
• RQ3 To what extent do experimentally estimated process matrices deviate from physically valid processes, and how can this be quantified?
• RQ4 Can fidelity metrics such as trace distance and Bures metric be reliably used when the process is unphysical, or must alternative measures be employed?

Key findings

• The experiment successfully performed QPT on a single NV-center qubit, marking the first such demonstration in a solid-state system.
• The process matrix χ showed increasing deviation from physicality with longer decoherence times, with the largest discrepancy observed at 80 ns.
• The trace distance between the ideal process and the reconstructed process (D_pro) increased from 0.056 at 20 ns to 0.096 at 80 ns, indicating growing error.
• Matrix norms (e.g., ∥X∥_1) rose from 0.101 at 20 ns to 0.175 at 80 ns, quantifying the growing deviation between χ and the physically valid 𝔀.
• The Bloch sphere representations revealed that dephasing (Z-axis collapse) was the dominant decoherence channel, while amplitude damping was not resolved on these timescales.
• Fidelity-based metrics such as Bures and C metrics became unreliable when the process was unphysical, necessitating the use of norm-based measures for error quantification.