Invariant subspaces of twoqubit quantum gates and their application in the verification of quantum computers
Abstract
We investigate the groups generated by the sets of $CP$, $CNOT$ and $SWAP^\alpha$ (powerofSWAP) 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 $n(n1)/2$ cyclic groups of order $2$, and determine $2^n$ $1$dimensional invariant subspaces corresponding to the computational statevectors. 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 $(2^n2)$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.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 DOI:
 10.48550/arXiv.2009.03642
 arXiv:
 arXiv:2009.03642
 Bibcode:
 2020arXiv200903642Y
 Keywords:

 Quantum Physics