10 algorithms for classical specifications of Stabilizer states and Clifford gates

[Fe-r-oz]

Hi, Stefan!

Please refer to the brief summary of a recent paper on stabilizer formalism tools, posted on 3 Jan, 2025. Thank you!

This paper Fast algorithms for classical specifications of stabiliser states and Clifford gates introduces ten efficient algorithms for handling classical representations of stabilizer theory elements. Specifically, it examines three ways to describe $n$-qubit stabilizer states:

• [SV]: As a complex vector of amplitudes.
• [SQ]: As a combination of a quadratic form, a linear map, and an affine subspace of $\mathbb{Z}_2^n$.
• [SP]: As a check matrix, which is a concise list of Pauli gate generators for the stabilizer group.

It also explores two methods for representing $n$-qubit Clifford gates:

• [CU]: As a unitary matrix.
• [CT]: As a tableau, consisting of $2n$ Pauli gates that represent the conjugation images of basic Pauli gates.

It provides algorithms for converting between these different representations and for verifying whether a given state vector corresponds to a stabilizer state or a unitary operation qualifies as a Clifford gate.

Algorithms (6 for conversions between representations of stabilizer states, and 4 for conversion between Clifford gates)

• [SV] <-> [SQ]
• [SV] <-> [SP]
• [SQ] <-> [SP]
• [CU] <-> [CT]

PR #461 was inspired from the aforementioned paper (posted on 3 Jan, 2025) that provided 10 'new' algorithms for stabilizer formalism tools. This referred to an earlier paper that had the new algorithm for random stabilizer state generation.

I think we might have most of functionality mentioned in this paper to some extent already, we can benchmark these 10 algorithms against stim and qiskit as this paper provided a lot of benchmarks if that sounds good?

Please refer to Appendix B: Timed Benchmarks for the benchmarks.

These feature (if some of the algorithms need to be implemented) may not be a high priority for now.

Python/C++ implementation: https://github.com/ndesilva/stabiliser-tools