# Quantifying magic for multi-qubit operations

## TL;DR

This paper develops a framework to quantify the non-stabiliserness of quantum operations, introducing new monotones and algorithms that improve classical simulation efficiency for multi-qubit quantum circuits.

## Contribution

It extends the concept of magic monotones from states to channels, introduces channel robustness and magic capacity, and provides more efficient simulation algorithms.

## Key findings

- New monotones for quantum channels are defined.
- Simulation algorithms outperform previous methods in certain cases.
- Techniques for easier calculation of monotones are proposed.

## Abstract

The development of a framework for quantifying "non-stabiliserness" of quantum operations is motivated by the magic state model of fault-tolerant quantum computation, and by the need to estimate classical simulation cost for noisy intermediate-scale quantum (NISQ) devices. The robustness of magic was recently proposed as a well-behaved magic monotone for multi-qubit states and quantifies the simulation overhead of circuits composed of Clifford+T gates, or circuits using other gates from the Clifford hierarchy. Here we present a general theory of the "non-stabiliserness" of quantum operations rather than states, which are useful for classical simulation of more general circuits. We introduce two magic monotones, called channel robustness and magic capacity, which are well-defined for general n-qubit channels and treat all stabiliser-preserving CPTP maps as free operations. We present two complementary Monte Carlo-type classical simulation algorithms with sample complexity given by these quantities and provide examples of channels where the complexity of our algorithms is exponentially better than previous known simulators. We present additional techniques that ease the difficulty of calculating our monotones for special classes of channels.