Invariant subspaces of two-qubit quantum gates and their application in the verification of quantum computers

TL;DR

This paper analyzes the invariant subspaces generated by key two-qubit quantum gates and demonstrates their potential use in verifying quantum computer operations.

Contribution

It characterizes the invariant subspaces of groups generated by $CP$, $CNOT$, and $SWAP^eta$ gates, providing a foundation for quantum computer verification methods.

Findings

Invariant subspaces for $CP$, $CNOT$, and $SWAP^eta$ gates are explicitly determined.

Isomorphisms to standard groups are established for these gate operations.

A recursive method to construct invariant subspaces of $SWAP^eta$ is presented.

Abstract

We investigate the groups generated by the sets of $CP$, $CNOT$ and $SWAP^\alpha$ (power-of-SWAP) quantum gate operations acting on $n$ qubits. Isomorphisms to standard groups are found, and using techniques from representation theory, we are able to determine the invariant subspaces of the $n-$qubit Hilbert space under the action of each group. For the $CP$ operation, we find isomorphism to the direct product of $n(n-1)/2$ cyclic groups of order $2$, and determine $2^n$ $1$-dimensional invariant subspaces corresponding to the computational state-vectors. For the $CNOT$ operation, we find isomorphism to the general linear group of an $n$-dimensional space over a field of $2$ elements, $GL(n,2)$, and determine two $1$-dimensional invariant subspaces and one $(2^n-2)$-dimensional invariant subspace. For the $SWAP^\alpha$ operation we determine a complex structure of invariant subspaces with varying dimensions and occurrences and present a recursive procedure to construct them. As an example of an application for our work, we suggest that these invariant subspaces can be used to construct simple formal verification procedures to assess the operation of quantum computers of arbitrary size.