# Constructive quantum scaling of unitary matrices

Constructive quantum scaling of unitary matrices In this work, we present a method of decomposition of arbitrary unitary matrix $$U\in \mathbf {U}(2^k)$$ U ∈ U ( 2 k ) into a product of single-qubit negator and controlled- $$\sqrt{\text{ NOT }}$$ NOT gates. Since the product results with negator matrix, which can be treated as complex analogue of bistochastic matrix, our method can be seen as complex analogue of Sinkhorn–Knopp algorithm, where diagonal matrices are replaced by adding and removing an one-qubit ancilla. The decomposition can be found constructively, and resulting circuit consists of $$O(4^k)$$ O ( 4 k ) entangling gates, which is proved to be optimal. An example of such transformation is presented. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Constructive quantum scaling of unitary matrices

, Volume 15 (12) – Oct 12, 2016
10 pages

/lp/springer_journal/constructive-quantum-scaling-of-unitary-matrices-HCzAwQlnvm
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-016-1448-z
Publisher site
See Article on Publisher Site

### Abstract

In this work, we present a method of decomposition of arbitrary unitary matrix $$U\in \mathbf {U}(2^k)$$ U ∈ U ( 2 k ) into a product of single-qubit negator and controlled- $$\sqrt{\text{ NOT }}$$ NOT gates. Since the product results with negator matrix, which can be treated as complex analogue of bistochastic matrix, our method can be seen as complex analogue of Sinkhorn–Knopp algorithm, where diagonal matrices are replaced by adding and removing an one-qubit ancilla. The decomposition can be found constructively, and resulting circuit consists of $$O(4^k)$$ O ( 4 k ) entangling gates, which is proved to be optimal. An example of such transformation is presented.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Oct 12, 2016

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