# Quantifying the magic of quantum channels

## TL;DR

This paper develops a resource theory for quantifying the non-stabilizerness or 'magic' of quantum channels, introduces measures like mana and thauma, and proposes classical simulation algorithms for noisy quantum circuits.

## Contribution

It introduces a novel resource theory for quantum channels' magic, along with computable measures and improved simulation algorithms for noisy quantum circuits.

## Key findings

- Mana and thauma provide bounds on distillable magic.
- Classical simulation algorithms outperform existing methods.
- Thresholds for non-stabilizerness under noise are identified.

## Abstract

To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum "magic" or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension $d$, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.