Center for Molecular Modeling - A. De Vos (Alexis)

Block-ZXZ synthesis of an arbitrary quantum circuit

wim — Wed, 16 Nov 2016 14:55:23 +0000

A. De Vos (Alexis), S. De Baerdemacker

Physical Review A

94, 052317

2016

Abstract

Given an arbitrary 2w×2w unitary matrix U, a powerful matrix decomposition can be applied, leading to four different syntheses of a w-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. Führ and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bit unitary gates (w=1) to gates with arbitrary value of w. The synthesized circuit consists of controlled one-qubit gates, such as negator gates and phasor gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled not gates in the case that U is a permutation matrix.

DOI

http://dx.doi.org/10.1103/PhysRevA.94.052317

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robotnik-resubmission-revtex.pdf

The Birkhoff theorem for unitary matrices of arbitrary dimensions

wim — Wed, 16 Nov 2016 14:53:12 +0000

S. De Baerdemacker, A. De Vos (Alexis), L. Chen, L. Yu

Linear Algebra and its Applications

514,151–164

2016

Abstract

It was shown recently that Birkho's theorem for doubly stochastic matrices
can be extended to unitary matrices with equal line sums whenever the dimension
of the matrices is prime. We prove a generalization of the Birkho
theorem for unitary matrices with equal line sums for arbitrary dimension.

DOI

http://dx.doi.org/10.1016/j.laa.2016.10.028

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birkhoff-arbitraryunitary-resubmission.pdf

The Birkhoff theorem for unitary matrices of prime dimension

wim — Wed, 20 Jan 2016 10:21:08 +0000

A. De Vos (Alexis), S. De Baerdemacker

Linear Algebra and its Applications

493 (2016), 455-468

2016

Abstract

The Birkhoff's theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners. We prove that any unitary matrix with equal line sums can also be written as a sum of permutation matrices (with sum of weights equal 1). Furthermore, when the matrix dimension is prime, we prove that the unitary matrix lies inside a convex complex Birkhoff polytope.

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1016/j.laa.2015.12.005

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1-s2.0-S0024379515007156-main.pdf

birkhof-priem-dd20150921-metabstract.pdf

Scaling a Unitary Matrix

wim — Thu, 08 Jan 2015 12:13:03 +0000

A. De Vos (Alexis), S. De Baerdemacker

Open Systems & Information Dynamics

21 (4), 1450013

2014

Abstract

The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called 'scaled matrix' which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all line sums equal to 1. We conjecture that a similar procedure exists, which allows, starting from an arbitrary unitary matrix, to find a scaled matrix which is unitary and has all line sums equal to 1. The existence of such algorithm guarantees a powerful decomposition of an arbitrary quantum circuit.

DOI

http://dx.doi.org/10.1142/S1230161214500139

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sinkhorn-dd20140610-revisionnarefreport-v1.pdf

Matrix Calculus for Classical and Quantum Circuits

wim — Wed, 26 Nov 2014 07:48:46 +0000

A. De Vos (Alexis), S. De Baerdemacker

ACM Journal on Emerging Technologies in Computing Systems (JETC)

11 (2), 9

2014

Abstract

Quantum computation on w qubits is represented by the infinite unitary group U(2(w)); classical reversible computation on w bits is represented by the finite symmetric group S-2w. In order to establish the relationship between classical reversible computing and quantum computing, we introduce two Lie subgroups XU(n) and ZU(n) of the unitary group U(n). The former consists of all unitary n x n matrices with all line sums equal to 1; the latter consists of all unitary diagonal n x n matrices with first entry equal to 1. Such a group structure also reveals the relationship between matrix calculus and diagrammatic zx-calculus of quantum circuits.

DOI

http://dx.doi.org/10.1145/2669370

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14_JECT_11(2)9_DeVos.pdf

The NEGATOR as a Basic Building Block for Quantum Circuits

wim — Tue, 19 Mar 2013 08:17:55 +0000

A. De Vos (Alexis), S. De Baerdemacker

Open Systems & Information Dynamics

20 (1), 1350004

2013

Abstract

Between (classical) reversible computation and quantum computation there exists an intermediate computational world, represented by unitary matrices that have all line sums equal to 1. All of these quantum circuits can be synthesized with the help of merely two building blocks: the NEGATOR and the singly controlled square root of NOT.

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1142/S1230161213500042

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13_open.sys_.inf_.dyna_20(1)1350004_DeVos.pdf

13_open.sys_.inf_.dyna_draft_DeVos.pdf

The roots of the NOT gate

wim — Tue, 27 Nov 2012 11:03:15 +0000

A. De Vos (Alexis), S. De Baerdemacker

International Symposium on Multiple-Valued Logic

Book Series: International Symposium on Multiple-Valued Logic, 167,172

2012

Abstract

The quantum gates called 'k th root of NOT' and 'controlled k th root of NOT' can be applied to synthesize circuits, both classical reversible circuits and quantum circuits. Such circuits, acting on w qubits, fill a (2(w) -1)(2)-dimensional subspace of the (2(w))(2)-dimensional space U(2(w)) of the 2(w) x 2(w) unitary matrices and thus describe computers situated between classical reversible computers and full quantum computers.

DOI

http://dx.doi.org/10.1109/ISMVL.2012.14

Reversible Computation, Quantum Computation, and Computer Architectures in Between

wim — Tue, 27 Nov 2012 10:56:32 +0000

A. De Vos (Alexis), M. Boes, S. De Baerdemacker

Journal of Multiple-Valued Logic and Soft Computing

1 (SI), 67-81

2012

Published while none of the authors were employed at the CMM

Abstract

Thanks to the cosine-sine decomposition of unitary matrices, an arbitrary quantum circuit, acting on w qubits, can be decomposed into 2(w) - 1 elementary quantum gates, called controlled V gates. Thanks to the Birkhoff decomposition of doubly stochastic matrices, an arbitrary (classical) reversible circuit, acting on w bits, can be decomposed into 2(w) - 1 elementary gates, called controlled NOT gates. The question arises under which conditions these two synthesis methods are applicable for intermediate cases, i.e. computers based on some group, which simultaneously is a subgroup of the unitary group U(2(w)) and a supergroup of the symmetric group S(2w). It turns out that many groups either belong to a class that might have a cosine-sine-like decomposition but no Birkhoff-like decomposition and a second class that might have both decompositions. For an arbitrary group, in order to find out to which class it belongs, it suffices to evaluate a function phi(m), deduced either from its order (in case of a finite group) or from its dimension (in case of a Lie group). Here m = 2(w) is the degree of the group.

DOI

http://dx.doi.org/

Decomposition of a Linear Reversible Computer: Digital Versus Analog

wim — Tue, 27 Nov 2012 10:54:03 +0000

A. De Vos (Alexis), S. De Baerdemacker

International Journal of Unconventional Computing

6 (3-4), 239-263

2010

Published while none of the authors were employed at the CMM

Abstract

Linear reversible transformations in the Galois field GF(2) and linear reversible transformations in the field of real numbers show both resemblances and differences. The former constitute a finite group isomorphic to the general linear group GL(w, 2), the latter constitute an infinite, i.e. Lie, group isomorphic to the general linear group GL(w, R) (where w is the logic width of the computation, i.e. respectively the number of bits and the number of real numbers, processed by the computer). Generators of the former group consist of merely control gates; generators of the latter group consist of both control gates and scale gates.

http://www.oldcitypublishing.com/IJUC/IJUCabstracts/IJUC6.3-4abstracts/I...

DOI

http://dx.doi.org/

The Group Zoo of Classical Reversible Computing and Quantum Computing

wim — Thu, 11 Aug 2016 07:58:45 +0000

A. De Vos (Alexis), S. De Baerdemacker

Springer

2016

ISBN

978-3-319-33924

Short content

The unconventional computing is a niche for interdisciplinary science, cross-bred of computer science, physics, mathematics, chemistry, electronic engineering, biology, material science and nanotechnology. The aims of this book are to uncover and exploit principles and mechanisms of information processing in and functional properties of physical, chemical and living systems to develop efficient algorithms, design optimal architectures and manufacture working prototypes of future and emergent computing devices.
This first volume presents theoretical foundations of the future and emergent computing paradigms and architectures. The topics covered are computability, (non-)universality and complexity of computation; physics of computation, analog and quantum computing; reversible and asynchronous devices; cellular automata and other mathematical machines; P-systems and cellular computing; infinity and spatial computation; chemical and reservoir computing.
The book is the encyclopedia, the first ever complete authoritative account, of the theoretical and experimental findings in the unconventional computing written by the world leaders in the field. All chapters are self-contains, no specialist background is required to appreciate ideas, findings, constructs and designs presented. This treatise in unconventional computing appeals to readers from all walks of life, from high-school pupils to university professors, from mathematicians, computers scientists and engineers to chemists and biologists.

www.springer.com/us/book/9783319339238

chapter-unconv-dd20150907-resubmission(1).pdf