2020/09/08 by Yordan S. Yordanov, Yordanov, Yordan S., Jacob Chevalier-Drori +8 · 1 citation
Computer Science · #Coding theory and cryptography #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #Quantum-Dot Cellular Automata
paper · pdf · doi:10.48550/arxiv.2009.03642
openalex publication_date 2020/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the groups generated by the sets of CP, CNOT and\nSWAP^\α (power-of-SWAP) quantum gate operations acting on n qubits.\nIsomorphisms to standard groups are found, and using techniques from\nrepresentation theory, we are able to determine the invariant subspaces of the\nn-qubit Hilbert space under the action of each group. For the CP operation,\nwe find isomorphism to the direct product of n(n-1)/2 cyclic groups of order\n2, and determine 2n 1-dimensional invariant subspaces corresponding to\nthe computational state-vectors. For the CNOT operation, we find isomorphism\nto the general linear group of an n-dimensional space over a field of 2\nelements, GL(n,2), and determine two 1-dimensional invariant subspaces and\none (2n-2)-dimensional invariant subspace. For the SWAP^\α operation\nwe determine a complex structure of invariant subspaces with varying dimensions\nand occurrences and present a recursive procedure to construct them. As an\nexample of an application for our work, we suggest that these invariant\nsubspaces can be used to construct simple formal verification procedures to\nassess the operation of quantum computers of arbitrary size.\n