This paper presents a decision diagram structure intended for the simulation and verification of reversible and quantum circuits. The structure is designed to efficiently represent the matrices describing reversible and quantum gate and circuit behaviour and takes advantage of key properties of those matrices. Algorithms are described for efficiently building the decision diagram representation directly from a gate without constructing the actual matrix and for performing matrix multiplication which is the fundamental operation to determine the function realized by a cascade of gates. Experimental results show the methods presented are applicable to large circuits at the state of the art in reversible and quantum synthesis and design.
This paper presents a decision diagram structure intended for the simulation and verification of reversible and quantum circuits. The structure is designed to efficiently represent the matrices describing reversible and quantum gate and circuit behaviour and takes advantage of key properties of those matrices. Algorithms are described for efficiently building the decision diagram representation directly from a gate without constructing the actual matrix and for performing matrix multiplication which is the fundamental operation to determine the function realized by a cascade of gates. Experimental results show the methods presented are applicable to large circuits at the state of the art in reversible and quantum synthesis and design.
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.
The problem of synthesis and optimisation of logical reversible and quantum circuits from functional descriptions represented as decision diagrams is considered. It is one of the key problems being solved with the aim of creating quantum computing technology and quantum computers. A new method of stepwise transformation of the initial functional specification to a quantum circuit is proposed, which provides for the following project states: reduced ordered binary decision diagram, if-decision diagram, functional if-decision diagram, reversible circuit and quantum circuit. The novelty of the method consists in extending the Shannon and Davio expansions of a Boolean function on a single variable to the expansions of the same Boolean function on another function with obtaining decomposition products that are represented by incompletely defined Boolean functions. Uncertainty in the decomposition products gives remarkable opportunities for minimising the graph representation of the specified function. Instead of two outgoing branches of the binary diagram vertex, three outgoing branches of the if-diagram vertex are generated, which increase the level of parallelism in reversible and quantum circuits. For each transformation step, appropriate mapping rules are proposed that reduce the number of lines, gates and the depth of the reversible and quantum circuit. The comparison of new results with the results given by the known method of mapping the vertices of binary decision diagram into cascades of reversible and quantum gates shows a significant improvement in the quality of quantum circuits that are synthesised by the proposed method.
Quantum-mechanical phenomena are playing an increasing role in information processing as transistor sizes approach the nanometer level, while the securest forms of communication rely on quantum data encoding. When they involve a finite number of basis states, these phenomena can be modeled as quantum circuits, the quantum analogue of conventional or classical logic circuits. Simulation of quantum circuits can therefore be used as a tool to evaluate issues in the design of quantum information processors. Unfortunately, simulating such phenomena efficiently is exceedingly difficult. The matrices representing quantum operators (gates) and vectors modeling quantum states grow exponentially with the number of quantum bits. The information represented by quantum states and operators often exhibits structure that can be exploited when simulating certain classes of quantum circuits. We study the development of simulation methods that run on classical computers and take advantage of such repetitions and redundancies. In particular, we define a new data structure for simulating quantum circuits called the quantum information decision diagram (QuIDD). A QuIDD is a compressed graph representation of a vector or matrix and permits computations to be performed directly on the compressed data. We develop a comprehensive set of algorithms for operating on QuIDDs in both the state-vector and density-matrix formats, and evaluate their complexity. These algorithms have been implemented in a general-purpose simulator program for quantum-mechanical applications called QuIDDPro. Through extensive experiments conducted on representative quantum simulation applications, including Grover's search algorithm, error characterization, and reversible circuits, we demonstrate that QuIDDPro is faster than other existing quantum-mechanical simulators such as the National Institute of Standards and Technology's QCSim program, and is far more memory-efficient. Using QuIDDPro, we explore the advantages of quantum computation over classical computation, simulate quantum errors and error correction, and study the impact of numerical precision on the fidelity of simulations. We also develop several novel algorithms for testing quantum circuit equivalence and compare them empirically. The QuIDDPro software is equipped with a user-friendly interface and is distributed with numerous example scripts. It has been used as a laboratory supplement for quantum computing courses at several universities.
Recently much attentions have been paid to quantum circuit design to prepare for the future ldquoquantum computation era.rdquo Like the conventional logic synthesis, it should be important to verify and analyze the functionalities of generated quantum circuits. For that purpose, we propose an efficient verification method for quantum circuits under a practical restriction. Thanks to the restriction, we can introduce an efficient verification scheme based on decision diagrams called Decision Diagrams for Matrix Functions (DDMFs). Then, we show analytically the advantages of our approach based on DDMFs over the previous verification techniques. In order to introduce DDMFs, we also introduce new concepts, quantum functions and matrix functions, which may also be interesting and useful on their own for designing quantum circuits.
While thousands of experimental physicists and chemists are currently trying to build scalable quantum computers, it appears that simulation of quantum computation will be at least as critical as circuit simulation in classical VLSI design. However, since the work of Richard Feynman in the early 1980s little progress was made in practical quantum simulation. Most researchers focused on polynomial-time simulation of restricted types of quantum circuits that fall short of the full power of quantum computation. Simulating quantum computing devices and useful quantum algorithms on classical hardware now requires excessive computational resources, making many important simulation tasks infeasible. In this work we propose a new technique for gate-level simulation of quantum circuits which greatly reduces the difficulty and cost of such simulations. The proposed technique is implemented in a simulation tool called the Quantum Information Decision Diagram (QuIDD) and evaluated by simulating Grover's quantum search algorithm. The back-end of our package, QuIDD Pro, is based on Binary Decision Diagrams, well-known for their ability to efficiently represent many seemingly intractable combinatorial structures. This reliance on a well-established area of research allows us to take advantage of existing software for BDD manipulation and achieve unparalleled empirical results for quantum simulation.
Abstract Reversible circuit synthesis methods based on decision diagrams achieve low quantum costs but do not account for quantum bit (qubit) limits for the application of reversible logic in quantum computing. Here, a synthesis method using sub-graphs of shared functional decision diagrams (SFDDs) is proposed for reducing the number of lines when synthesizing reversible circuits. An SFDD is partitioned into sub-graphs by exploiting the longest dominant-active paths, and the sub-graphs are mapped to reversible gate cascades. To further reduce the number of lines, template root matching is presented for reusing circuit lines. Experimental results indicate that the proposed method achieves the known minimum number of lines in many cases and has good scalability. Although the proposed method increases the quantum cost over a prior method based on functional decision diagrams, it significantly reduces the number of lines in most cases. Compared with the one-pass method using quantum multiple-valued decision diagrams, the proposed method reduces the quantum cost without increasing the number of lines in many cases. When compared with the lookup table-based method using a direct mapping flow, the method reduces the number of lines in a few cases. Thus, the method aids in the physical realization of a quantum circuit.