Coherent and non-unitary errors in ZZ-generated gates

TL;DR

This paper compares the error characteristics of continuous-angle controlled phase (CP) and fixed-angle controlled Z (CZ) gates in variational quantum algorithms, highlighting how non-unitary errors impact their fidelities and suggesting simpler implementations for low-error regimes.

Contribution

It provides a detailed error analysis of CP and CZ gates under coherent and incoherent noise, proposing that CZ with virtual Z decomposition can reduce calibration complexity in low-error conditions.

Findings

CP and CZ achieve similar fidelities below 0.03% incoherent error and 0.8% coherent error.

Above 2% coherent error, CZ gate fidelity is highly dependent on the variational parameter b3.

CZ with virtual Z decomposition can simplify calibration in low-error quantum devices.

Abstract

Variational algorithms such as the Quantum Approximate Optimization Algorithm have attracted attention due to their potential for solving problems using near-term quantum computers. The $ZZ$ interaction typically generates the primitive two-qubit gate in such algorithms applied for a time, typically a variational parameter, $\gamma$. Different compilation techniques exist with respect to the implementation of two-qubit gates. Due to the importance of the $ZZ$-gate, we present an error analysis comparing the continuous-angle controlled phase gate (CP) against the fixed angle controlled $Z$-gate (CZ). We analyze both techniques under the influence of coherent over-rotation and depolarizing noise. We show that CP and CZ compilation techniques achieve comparable $ZZ$-gate fidelities if the incoherent error is below $0.03 \, \%$ and the coherent error is below $0.8 \, \%$. Thus, we argue that for small coherent and incoherent error a non-parameterized two-qubit gate such as CZ in combination with virtual $Z$ decomposition for single-qubit gates could lead to a significant reduction in the calibration required and, therefore, a less error-prone quantum device. We show that above a coherent error of $0.04 \pi$ ($2 \, \%$), the CZ gate fidelity depends significantly on $\gamma$.