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Quantum Multiple-Valued Decision Diagrams Containing Skipped Variables.

David Y. Feinstein, Mitchell Aaron Thornton

Published Jan 1, 2015• 2 citations
OpenAlex

Abstract

Quantum multiple-valued decision diagrams (QMDD) are data structures that enable efficient representation and manipulation of unitary matrices representing reversible and quantum circuits. QMDDs are of the form of a rooted, directed, acyclic graph with initial and terminal vertices where each vertex is annotated with a variable name representing a circuit line. Directed paths from the intial to a terminal vertex can be represented as a sequence of variable names in the order in which they appear in the path. The existence of QMDD paths that do not contain all variable names, or “skipped variables”, has direct ramifications in the formula-tion of synthesis and other algorithms for reversible and quantum logic. Skipped variables are generally rare and tend to appear in quantum circuits that are intended to operate using superimposed values on the control lines. We have found that a unitary matrix representing a circuit whose QMDD contains skipped variables is likely to exhibit a specific anomaly when decomposed into a cascade of unitary matrices using the Reck-Zeilinger-Bernstein-Bertani algorithm.

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Quantum multiple-valued decision diagrams (QMDD) are data structures that enable efficient representation and manipulation of unitary matrices representing reversible and quantum circuits. QMDDs are of the form of a rooted, directed, acyclic graph with initial and terminal vertices where each vertex is annotated with a variable name representing a circuit line. Directed paths from the intial to a terminal vertex can be represented as a sequence of variable names in the order in which they appear in the path. The existence of QMDD paths that do not contain all variable names, or “skipped variables”, has direct ramifications in the formula-tion of synthesis and other algorithms for reversible and quantum logic. Skipped variables are generally rare and tend to appear in quantum circuits that are intended to operate using superimposed values on the control lines. We have found that a unitary matrix representing a circuit whose QMDD contains skipped variables is likely to exhibit a specific anomaly when decomposed into a cascade of unitary matrices using the Reck-Zeilinger-Bernstein-Bertani algorithm.
The data structure referred to as quantum multiple-valued decision diagrams (QMDD) is used to efficiently represent the unitary matrices describing reversible and quantum circuits. This paper investigates the conditions that cause skipped variables to appear in the QMDD of some binary and ternary quantum circuits. We have found that a unitary matrix that produces a skipped variable in a QMDD is likely to cause a specific anomaly when it is decomposed into a cascade of two-level unitary matrices by the Beck-Zeilinger-Bernstein-Bertani algorithm.
In this paper, we present a novel structure, QuantumMultiple- valued Decision Diagrams (QMDD), specifically designed to represent and manipulate the matrices encountered in the specification of reversible and quantum gates and circuits, both binary and multiple-valued. QMDD use many common decision diagram techniques, ideas introduced in QuIDDPro and novel techniques introduced here. Building the QMDD for the matrices for individual gates and the subsequent construction of the QMDD for the matrix describing a circuit are discussed. A prototype C implementation is described and experimental results are given that show the new structure is a promising and compact representation for reversible and quantum logic circuits.
In previous research, a novel structure was discussed for representing the matrices that can be built from an n-variable r-valued reversible/quantum circuit. This structure, called a QMDD, takes on a form similar to that of a reduced-ordered-binary-decision-diagram (ROBDD). It is known that the order of variables used for developing an ROBDD from a binary logic circuit is relevant to the size and structure of that ROBDD. This paper determines what effect, if any, variable order has on the QMDD structure and proposes a simple heuristic for choosing a 'good' variable order.
In [1], a novel structure was discussed for representing the matrices that can be built from an n-variable r-valued reversible/quantum circuit. This structure, called a QMDD, takes on a form similar to that of a reduced-orderedbinary-decision-diagram (ROBDD). It is known that the order of variables used for developing an ROBDD from a binary logic circuit is relevant to the size and structure of that ROBDD [3]. This paper determines what effect, if any, variable order has on the QMDD structure and proposes a simple heuristic for choosing a ‘good’ variable order.
This paper describes new metrics for the data structure referred to as quantum multiple-valued decision diagrams (QMDD) which are used to represent the matrices describing reversible and quantum gates and circuits. These metrics provide information about QMDD that allows for improvement of minimization techniques. We explore metrics related to the frequency of edges with non-zero weight for the entire QMDD data structure and their histograms with respect to each variable. We observe some unique regularity particular to the methodology of the QMDD. We develop new heuristics for QMDD dynamic variable ordering (DVO) that are guided by the proposed metrics. Experimental results show the effectiveness of the proposed techniques.