Benchmarking non-Clifford gates using only Pauli twirling group

Quantum gate benchmarking is unavoidably influenced by state preparation and measurement errors. Randomized benchmarking addresses this challenge by employing group twirling to regularize the noise channel, then provides a characterization of quantum channels that is robust to these errors through exponential fittings. In practice, local twirling gates are preferred due to their high fidelity and experimental feasibility. However, while existing RB methods leveraging local twirling gates are effective for benchmarking Clifford gates, they face fundamental challenges in benchmarking non-Clifford gates. In this work, we solve this problem by introducing the Pauli Transfer Character Benchmarking. This protocol estimates the Pauli transfer matrix elements for a quantum channel using only local Pauli operations. Building on this protocol, we develop a fidelity benchmarking method for non-Clifford gates $U$ satisfying $U^2=I$. We validate the feasibility of our protocol through numerical simulations applied to Toffoli gates as a concrete example.

💡 Research Summary

The paper addresses a fundamental limitation of current randomized benchmarking (RB) techniques: while RB with local twirling gates (e.g., Pauli or single‑qubit Clifford) works well for Clifford operations, it struggles to benchmark multi‑qubit non‑Clifford gates because the noise channel of such gates cannot be fully twirled by a purely local group. The authors propose a new protocol called Pauli Transfer Character Benchmarking (PTCB) that uses only the Pauli twirling group together with Pauli‑basis state preparation and measurement to estimate selected elements of the Pauli‑transfer matrix (PTM) of an arbitrary quantum channel.

The core idea is to access off‑diagonal PTM elements by inserting a virtual Clifford pair (C) and (C^{\dagger}) around the target gate (U). For any non‑identity Pauli operators (P) and (Q) there exists a Clifford (C) such that (CPC^{\dagger}= \pm Q). By conjugating the noisy implementation (\tilde U = U\Lambda) with (C) and (C^{\dagger}) the product (\tilde U_{PQ}\tilde U_{QP}) appears on the diagonal of a twirled channel. Importantly, the Clifford pair is never physically implemented; instead the two conjugations are combined into a single Pauli operation, so every gate between successive (\tilde U) instances remains a local Pauli.

The protocol proceeds as follows: (1) Choose a Pauli pair ((P,Q)) and find a Clifford (C) satisfying the above relation. (2) Prepare the +1 eigenstate of (Q) (or a product of such eigenstates for multi‑qubit (Q)). (3) Sample a random sequence of Pauli gates (P_0,\dots,P_{2m}) and execute the circuit
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