Mitigating Coherent Errors through a Decoherence-Resistant Variational Framework employing Stabilizer State

TL;DR

Coherent errors from small gate miscalibrations degrade stabilizer-state preparation by breaking stabilizers of $|\psi_S\rangle$. The paper introduces Variational Coherent Error Mitigation (VCEM), which transpiles the stabilizer circuit into native gates and optimizes parameters $\vec{\theta}$ to minimize $C(\vec{\theta})=-\sum_{i=1}^n \langle \psi_S(\vec{\theta}+\vec{\epsilon})|\hat{S}_i|\psi_S(\vec{\theta}+\vec{\epsilon})\rangle$, achieving the global minimum $C(\vec{\theta}_{opt})=-n$ at $\vec{\theta}_{opt} = -\vec{\epsilon}$. The authors show that coherent errors can be cancelled at the circuit level and that the optimization landscape remains well-behaved (convex near the minimum) even in the presence of modest incoherent noise, with $\tilde{C}$ related to $C$ by factors like $(1-p)$ or more generally $\tilde{C} = \tilde{C}^{(\mathcal{P})} - \Delta$, and provide a stationarity theorem and an $O(\epsilon^2 n)$ bound on deviations. They validate VCEM via numerical experiments on graph states and GHZ states, demonstrating rapid convergence and robustness under Pauli noise, supporting pre-compensation of coherent errors prior to standard incoherent QEM.

Abstract

Stabilizer states are a central resource in quantum information processing, underpinning a wide range of applications. While they can be efficiently generated via Clifford circuits, the presence of coherent errors, such as small-angle miscalibrations in native gate implementations, can significantly impact their quality. In this work, we introduce Variational Coherent Error Mitigation (VCEM), a method that employs the stabilizer formalism to suppress coherent errors through variational optimization of native gates parameters. VCEM demonstrates robust performance, remaining largely unaffected by incoherent noise, enabling pre-compensation of coherent errors prior to the application of standard incoherent error mitigation techniques. We demonstrate the effectiveness and robustness of VCEM through numerical simulations.

Figures (11)

• Figure 1: Parametrized circuit to evaluate the VCEM cost function.
• Figure 2: Parametrized circuit to evaluate the modified VCEM cost function with a Pauli noise channel after $\mathcal{U}_S$.
• Figure 3: Sketch of how the landscape of the cost function could be modified by the Pauli incoherent noise $\mathcal{P}$ acting at the end of the circuit. The value of $\vec{\theta}$ in black is the initial choice of the parameters.
• Figure 4: The action of global depolarizing maps after each momentum is equal to a single global depolarizing map after $\mathcal{U}_S$.
• Figure 5: The action of Pauli maps after each moment is equivalent to an effective Pauli map after $\mathcal{U}_S$ plus a reminder $\mathcal{R}$.

Theorems & Definitions (20)

• Theorem 1: Stationarity of the solution
• Theorem 2: Upper bound on $\Delta \tilde{C}$
• Definition 1: Pauli map
• Lemma 1: Action of Pauli maps on Pauli strings
• proof
• Lemma 2: Pauli and unitary Clifford maps
• proof
• Lemma 3
• proof
• Lemma 4