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Physics of Particles and Nuclei Letters

A mathematica program for constructing quantum circuits and computing their unitary matrices

Vladimir P. Gerdt, Robert Kragler, Alexander N. Prokopenya

Published Nov 19, 2009• 12 citations
Publisher

Abstract

In this paper we briefly describe a Mathematica program for simulation of quantum circuits and illustrate some of its facilities by simple examples. Unlike other Mathematica-based quantum simulators, our program provides a user-friendly graphical interface for generating quantum circuits and computing the circuit unitary matrices. In addition to standard linear algebra-based tools, our program implements special computer-algebra technique for constructing the multivariate polynomial system that, for a circuit composed from the Toffoli and Hadamard gates, uniquely defines the circuit matrix.

DOI
10.1134/s1547477109070061

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In this paper we briefly describe a Mathematica program for simulation of quantum circuits and illustrate some of its facilities by simple examples. Unlike other Mathematica-based quantum simulators, our program provides a user-friendly graphical interface for generating quantum circuits and computing the circuit unitary matrices. In addition to standard linear algebra-based tools, our program implements special computer-algebra technique for constructing the multivariate polynomial system that, for a circuit composed from the Toffoli and Hadamard gates, uniquely defines the circuit matrix.
An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z_2 arising when applying the Feynman's sum-over-paths approach to quantum circuits. The matrix elements determined by a circuit can be computed by counting the number of common roots in Z_2 for the polynomial system associated with the circuit. To determine the number of solutions in Z_2 for the output polynomial system, one can use the Groebner bases method and the relevant algorithms for computing Groebner bases.
A correct simulation of a quantum circuit on a classical computer is more important because of their future use.The main purpose of this work is to illustrate a full adder circuit by using a standard Mathematica add-on package.The circuit can be constructed by using CNOT-based quantum gates.The program provides a curriculum unit, to generate the basic elements that make up quantum circuit.This paper shows effective computational design by using analogy of classical circuits.We presented an explicit example to show efficiency of the 4 qubit full adder circuit on classical computer.The method given in this paper can be used to design various quantum circuits.
Recently quantum computing has achieved significant results both in theoretical and experimental areas and quantum algorithms have drawn the attention of many scientists. At the present there exist several experiments that realized some parts of certain important quantum algorithms. While those experiments work with only up to seven qubits, quantum simulators allow scientists to design algorithms with many more qubits. The existing simulators have one or more of the following restrictions: either they require powerful computers and additional computation packages such as Matlab or Mathematica or they don't provide an easy method for interacting between the classical and quantum parts of an algorithm. More importantly, none of them can simulate a generic 64-qubit circuit. It is important at this stage of quantum computing development to investigate technological improvements in simulating quantum circuit design. In this thesis we developed a specialized data structure, QSV, for quantum simulators along with new method for calculation of the new state vector after application of a quantum gate in a quantum simulator. We also developed a technique that speeds up the simulation time with our QSV vectors. With those results we built a quantum simulator that overcomes many shortcomings in other simulators and it has many advanced features. Our simulator can simulate many 64-qubit circuits and it can directly simulate multiple qubit gates as well as allows users to define gates with up to 10 qubits. The simulation time is much less in comparison with other simulators we tested. The simulator defines specifications so that users can easily design and test quantum circuits. With the simulator we also developed some new quantum circuits and verified many designs and constructions.
A quantum computer simulator is presented. This simulator is an engineering work and no deep understanding of quantum mechanics is required from the user. The simulator is based on the circuit model of quantum computation in which quantum gates act on quantum registers which comprise a number of quantum bits (qubits). The inputs to the simulator are the initial states of the qubits that form a quantum register and the quantum gates applied at each computation step. The inputs are entered through a graphical user interface. The outputs of the simulator are the matrices that represent the quantum register state at each quantum computation step and graphical outputs that show the probability of measuring each one of the possible quantum register base states and the phase of each state at each computation step. The well-known Deutsch's algorithm and the quantum Fourier transform, which is the base of many quantum algorithms, are presented using this simulator. Furthermore, the generation and variation of entanglement during quantum computations can be calculated using this simulator. The quantum computer simulator is a useful tool for the study of quantum computer circuits, quantum computing, and the development of new quantum algorithms.
A model of quantum computation based on unitary matrix operations was introduced by Feynman and Deutsch. It has been asked whether the power of this model exceeds that of classical Turing machines. We show here that a significant class of these quantum computations can be simulated classically in polynomial time. In particular we show that two-bit operations characterized by 4 × 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time. This contrasts with the known universality of two-bit operations and demonstrates that efficient quantum computation of restricted classes is reconcilable with the Polynomial Time Turing Hypothesis. Therefore, it is possible that, The techniques introduced bring the quantum computational model within the realm of algebraic complexity theory. In a manner consistent with one view of quantum physics, the wave function is simulated deterministically, and randomization arises only in the course of making measurements. The results generalize the quantum model in that they do not require the matrices to be unitary. In a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read-twice Boolean formulae. All our results are based on the use of gates that are defined in terms of their graph matching properties.