On The Stabilizer Formalism And Its Generalization

TL;DR

This paper explores the mathematical structure of the stabilizer formalism in quantum computing, identifying conditions under which generalized stabilizer groups are trivial, and investigates implications for classical simulability of quantum systems.

Contribution

It provides a rigorous analysis of the stabilizer formalism's generalization, establishing conditions for non-trivial Clifford groups and connecting stabilizer set density with group simplicity.

Findings

Density of stabilizing set implies trivial Clifford group

Formalism equivalent to standard stabilization for few qubits

Conjecture that many generalized stabilizer states are standard

Abstract

The standard stabilizer formalism provides a setting to show that quantum computation restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman-Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of $SU(d)$ transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results can be used to construct novel Gottesman-Knill-type results and consequently draw a sharper line between quantum and classical computation.