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Acta Cybernetica

Storing the Quantum Fourier Operator in the QuIDD Data Structure

Katalin Friedl, László Kabódi

Published Jan 1, 2017• 1 citation
Publisher

Abstract

Quantum algorithms can be simulated using classical computers, but the typical time complexity of the simulation is exponential. There are some data structures which can speed up this simulation to make it possible to test these algorithms on classical computers using more than a few qubits. One of them is QuIDD by Viamontes et al., which is an extension of the Algebraic Decision Diagram. In this paper, we examine the matrix of Fourier operator and its QuIDD representation. To utilize the structure of the operator we propose two orderings (reversed column variables and even-odd order), both resulting in smaller data structure than the standard one. After that, we propose a new method of storing the Fourier operator, using a weighted decision diagram that further reduces its size. It should be the topic of subsequent research whether the basic operations can be performed efficiently on this weighted structure.

DOI
10.14232/actacyb.23.2.2017.5

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Quantum algorithms can be simulated using classical computers, but the typical time complexity of the simulation is exponential. There are some data structures which can speed up this simulation to make it possible to test these algorithms on classical computers using more than a few qubits. One of them is QuIDD by Viamontes et al., which is an extension of the Algebraic Decision Diagram. In this paper, we examine the matrix of Fourier operator and its QuIDD representation. To utilize the structure of the operator we propose two orderings (reversed column variables and even-odd order), both resulting in smaller data structure than the standard one. After that, we propose a new method of storing the Fourier operator, using a weighted decision diagram that further reduces its size. It should be the topic of subsequent research whether the basic operations can be performed efficiently on this weighted structure.
Simulating quantum algorithms is a hard problem on classical computers, it usually needs exponential time and space. Viamontes et al. proposed a new data structure the Quantum Information Decision Diagram (QuIDD) to overcome this problem and implemented it in the QuIDDPro software. Using this structure several algorithms can be simulated on classical computers with polynomial time and space. In this paper we suggest further improvement and analyse in detail its behavior on Grover’s search algorithm.
Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and vectors modeling qubit states grow exponentially with an increase in the number of qubits. However, by using a new data structure called the Quantum Information Decision Diagram (QuIDD) that exploits the structure of quantum operators, many of these matrices and vectors can be represented in a form that grows polynomially. Using QuIDDs, we implemented a general-purpose quantum computing simulator in C++ called QuIDDPro and tested it on Grover's algorithm. Our QuIDD technique asymptotically outperforms other known simulation techniques.
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.
Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and vectors modeling qubit states grow exponentially with the number of qubits. It has been shown experimentally that the QuIDD (Quantum Information Decision Diagram) datastructure greatly facilitates simulations using memory and runtime that are polynomial in the number of qubits. In this paper, we present a complexity analysis which formally describes this class of matrices and vectors. We also present an improved implementation of QuIDDs which can simulate Grover's algorithm for quantum search with the asymptotic runtime complexity of an ideal quantum computer up to negligible overhead.
Classical representations of quantum states and operations as vectors and matrices are plagued by an exponential growth in memory and runtime requirements for increasing system sizes. Based on their use in classical computing, an alternative data structure known as Decision Diagrams (DDs) has been proposed, which, in many cases, provides both a more compact representation and more efficient computation. In the classical realm, decades of research have been conducted on DDs and numerous variations tailored for specific applications exist. However, DDs for quantum computing are just in their infancy and there is still room for tailoring them to this new technology. In particular, existing representations of DDs require extending all operations in a quantum circuit to the full system size through extension by nodes representing identity matrices. In this work, we make an important step forward for quantum DDs by stripping these identity structures from quantum operations. This significantly reduces the number of nodes required to represent them as well as eases the pressure on key building blocks of their implementation. As a result, we obtain a structure that is more natural for quantum computing and significantly speeds up computations—with a runtime improvement of up to 70 × compared to the state-of-the-art.
Quantum computing promises substantial speedups by exploiting quantum mechanical phenomena such as superposition and entanglement. Corresponding design methods require efficient means of representation and manipulation of quantum functionality. In the classical domain, decision diagrams have been successfully employed as a powerful alternative to straightforward means such as truth tables. This motivated extensive research on whether decision diagrams provide similar potential in the quantum domain-resulting in new types of decision diagrams capable of substantially reducing the complexity of representing quantum states and functionality. From an implementation perspective, many concepts and techniques from the classical domain can be re-used in order to implement decision diagrams packages for the quantum realm. However, new problems-namely how to efficiently handle complex numbers-arise. In this work, we propose a solution to overcome these problems. Experimental evaluations confirm that this yields improvements of orders of magnitude in the runtime needed to create and to utilize these decision diagrams. The resulting implementation is publicly available as a quantum DD package at http://iic.jku.at/eda/research/quantum_dd.