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Matrix Decompositions and Quantum Circuit Design

Published

Author(s)

Stephen Bullock

Abstract

The n-qubit concurrence canonical decomposition (CCD) is a generalization of the two-qubit canonical decomposition SU(4)=[SU(2) (x) SU(2)] ? [SU(2) (x) SU(2)], where ? is the commutative group which phases the maximally entangled Bell basis. A prequel manuscript creates the CCD as a particular example of the G=KAK metadecomposition theorem of Lie theory. We hence denote it by SU(2n)=KAK. If Cn(|?)= &| is the concurrence entanglement monotone, then computations in the K group are symmetries of a related bilinear form and so do not change the concurrence. Hence for a quantum computation v=k1 a k2, analysis of a in e A allows one to study one aspect of the entanglement dynamics of the evolution v, i.e. the concurrence dynamics. Note that analysis of such an a in e A is simpler than the generic case, since A is a commutative group whose dimension is exponentially less than that of SU(N). In this manuscript, we accomplish three main goals. First, we expand upon the treatment of the odd-qubit case of the sequel, in that we (i) present an algorithm to compute the CCD in case n=2p-1 and (ii) characterize the maximal odd-qubit concurrence capacity in terms of convex hulls. Second, we interpret the CCD in terms of a time-reversal symmetry operator, namely the quantum bit flip |?> ? (-i sy) (x)n | ?*>. In this context, the CCD allows one to write any unitary evolution as a two-term product of a time-reversal symmetric and anti-symmetric evolution; no Trotterization is required. Finally, we use these constructions to study time-reversal symmetric Hamiltonians. In particular, we show that any | ?> in the ground state of such an H must either develop a Kramer's degeneracy or be maximally entangled in the sense that Cn(| ?>)=1. Many time-reversal symmetric Hamiltonians are known to be nondegenerate and so produce maximally concurrent ground states.
Proceedings Title
Invited talk to Laboratoire d'Informatique Theorique et Quantique
Conference Location
University of Montreal,

Citation

Bullock, S. (2004), Matrix Decompositions and Quantum Circuit Design, Invited talk to Laboratoire d'Informatique Theorique et Quantique, University of Montreal, (Accessed May 26, 2024)

Issues

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Created December 1, 2004, Updated February 19, 2017