Experimental characterization of the Toffoli gate via channel spectrum benchmarking

TL;DR

This work addresses the challenge of benchmarking quantum gate fidelity when spectral degeneracies and off-diagonal noise hinder standard CSB analyses. It extends CSB with a 6-term noisy-eigenvalue model and introduces FEI, an interval fidelity estimate that remains informative under degeneracy. Numerical simulations on a three-qubit Toffoli gate show that FEI midpoints closely approximate true fidelity and outperform the original 4-term approach, while experiments on a trapped-ion platform compare qubit and qutrit Toffoli implementations and reveal leakage in the qutrit path. The results demonstrate a SPAM-robust, degeneracy-tolerant benchmarking framework with potential applicability across platforms and gate sets.

Abstract

Channel spectrum benchmarking (CSB) provides a robust framework for characterizing quantum gate fidelities while remaining insensitive to state preparation and measurement (SPAM) errors. Yet, current CSB implementations encounter fundamental challenges when reconstructing noisy eigenvalues, particularly in the presence of spectral degeneracies and off-diagonal noise components in the target gate's eigenbasis. These issues become especially pronounced in the strong noise regime for gates with fidelities around $90\%$. To address these limitations, we introduce an extended CSB model together with a fidelity estimate interval (FEI) -- an interval-valued estimate of the target gate fidelity. Numerical simulation demonstrates that FEI remains sufficiently narrow, with its midpoint reliably approximating the true fidelity. We further validate the protocol on a trapped-ion quantum processor by benchmarking two implementations of the three-qubit Toffoli gate. The results reveal a clear advantage of the qutrit-based implementation over its qubit-based counterpart.

Figures (8)

• Figure 1: Schematic quantum circuit for the CSB protocol. Quantum gates are represented by black rectangles. Red circles indicate the circuit locations where we model the presence of inherent quantum noise.
• Figure 2: Typical $\hat{P}^{ab}(L)$ dependencies from CSB simulations. (a) State $(|101\rangle + |010\rangle)/\sqrt{2}$ shows smooth decay; (b) State $(|100\rangle + |11-\rangle)/\sqrt{2}$ exhibits oscillations. Orange lines: simulated data; blue curves: fitted 6-term models \ref{['eq:g_model']}.
• Figure 3: Noisy eigenvalues of the Toffoli gate in the complex plane. Orange: theoretical values; blue: reconstructed estimates obtained by fitting the 6-term model \ref{['eq:g_model']}; green: estimates after applying the filtering procedure proposed in this work.
• Figure 4: Results of the algorithm proposed in gu2023benchmarking. (a) The dependency $\hat{P}^{ab}(L)$ for state $(|101\rangle + |11-\rangle)/\sqrt{2}$ (orange line: simulated data; blue curve: 4-term model \ref{['eq:P(L)_small_form']} fitted with MP method). (b) Reconstructed noisy eigenvalues of the Toffoli gate in the complex plane (orange: theoretical values; blue: raw estimates obtained using the approach from gu2023benchmarking).
• Figure 5: Comparison of fidelity estimation methods for the Toffoli gate. The distribution shows fidelity estimates obtained using: our approach with 6-term model, numerical optimization, filtering, and FEI construction (blue); the same method but using Eq. \ref{['eq:fid_degenerate']} for point estimation (green); the theoretical value (black); and the MP method with 4-term model and Eq. \ref{['eq:fid_degenerate']} (red).