Invariants of the local Clifford group (9 pages)

M. Van den Nest

Maarten Van den Nest, Jeroen Dehaene, Bart De Moor

We study the algebra of complex polynomials which remain invariant under the action of the local Clifford group under conjugation. Within this algebra, we consider the linear spaces of homogeneous polynomials degree by degree and construct bases for these vector spaces for each degree, thereby obtaining a generating set of polynomial invariants. Our approach is based on the description of Clifford operators in terms of linear operations over GF(2). Such a study of polynomial invariants of the local Clifford group is mainly of importance in quantum coding theory, in particular in the classification of binary quantum codes. Some applications in entanglement theory and quantum computing are briefly discussed as well.

Ingemar Bengtsson

Maarten Van den Nest, Jeroen Dehaene, Bart De Moor

The classification of stabilizer states under local Clifford (LC) equivalence is of particular importance in quantum error-correction and measurement-based quantum computation. Two stabilizer states are called LC equivalent if there exists a local Clifford operation which maps the first state to the second. We present a finite set of invariants which completely characterizes the LC equivalence class of any stabilizer state. Our invariants have simple descriptions within the binary framework in which stabilizer states are usually described.

Q. Ten

We do not claim to solve the general graph automorphism problem. Whileit is a known strategy to attempt to solve the Graph isomorphism and autom-rphism problems on a quantum computer by treating them as a HSP oversome symmetric group this paper takes a different approach by using the graphformlism presented in [1] so we present a intrisically quantum algorithm for ex-ploring the local-Clifford (LC) equivalence neighborhood of a graph using thegraph-state data structure introduced by [1]. By leveraging efficient local com-plementation, edge complementation, and ancilla-driven set-complementationoperations, we construct a quantum superposition over the LC-orbit of a graphand introduce an oracle that implements structural invariants such as automor-phisms and fixed-point subgraphs. We show that amplitude amplification yieldsa quadratic speedup for detecting LC-invariant features compared to classicalrandomized search. Our method provides a provably correct quantum proce-dure for structural graph analysis within the graph-state formalism. The localCliford orbit is a massive, exponentially large space that cannot be searched ingeneral classically with polynomial time complexity

M. Van den Nest, Jeroen Dehaene, Bart De Moor