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Journal of Physics Conference Series

QUANTUM: A WolframMathematicaadd-on for Dirac Bra-Ket Notation, Non-Commutative Algebra, and Simulation of Quantum Computing Circuits

J L Gómez Muñoz, Francisco Delgado

Published Mar 1, 2016• 8 citations
Publisher

Abstract

This paper introduces QUANTUM, a free library of commands of Wolfram Mathematica that can be used to perform calculations directly in Dirac braket and operator notation. Its development started several years ago, in order to study quantum random walks. Later, many other features were included, like operator and commutator algebra, simulation and graphing of quantum computing circuits, generation and solution of Heisenberg equations of motion, among others. To the best of our knowledge, QUANTUM remains a unique tool in its use of Dirac notation, because it is used both in the input and output of the calculations. This work depicts its usage and features in Quantum Computing and Quantum Hamilton Dynamics.

DOI
10.1088/1742-6596/698/1/012019

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This paper introduces QUANTUM, a free library of commands of Wolfram Mathematica that can be used to perform calculations directly in Dirac braket and operator notation. Its development started several years ago, in order to study quantum random walks. Later, many other features were included, like operator and commutator algebra, simulation and graphing of quantum computing circuits, generation and solution of Heisenberg equations of motion, among others. To the best of our knowledge, QUANTUM remains a unique tool in its use of Dirac notation, because it is used both in the input and output of the calculations. This work depicts its usage and features in Quantum Computing and Quantum Hamilton Dynamics.
For many practical applications, Quantum computers offer major speed-ups over traditional machines. The production of quantum chips has recently made tremendous strides, with the quantity of Qubits increasing and fidelity increasing. The smallest unit of information is the binary digit i.e. bit in general computing. In Quantum Computing the term Qubit (Quantum Bit) is used for the same purpose. In order to differentiate between a classical bit and a Qubit, Bra-Ket or Dirac notation is used. So, the Qubits are represented as |0〉 and |1〉 and are often read as Ket 0 for zero state and Ket 1 for one state. In this paper mainly programming and application sides of Quantum Computing are focused. For that, Qiskit module and its tools are used in Python programming language. General circuit optimization scheme is given. The Quantum circuits with the concept of Quantum Gates implemented are stronger compared to the optimizing algorithm and with faster execution time. In addition, for prime factorization, access to IBM's Quantum computer is carried out successfully.
We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.
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.
In several problems of quantum many body physics, one is required to handle complex expressions originating in the non-commutative nature of quantum operators. Their manipulation requires precise ordering and application of simplification rules. This can be cumbersome, tedious and error prone, and often a challenge to the most expert researcher. In this paper we present a software package DiracQ to facilitate such manipulations. The package DiracQ consists of functions based upon and extending considerably the symbolic capabilities of Mathematica. With DiracQ, one can proceed with large scale algebraic manipulations of expressions containing combinations of ordinary numbers or symbols (the c-numbers) and arbitrary sets of non-commuting variables (the q-numbers) with user defined properties. The DiracQ package is user extendable and comes encoded with the algebraic properties of several standard operators in popular usage. These include Fermionic and Bosonic creation and annihilation operators, spin operators, and canonical position and momentum operators. An example book is provided with some suggestive calculations of large-scale algebraic manipulations.
Quantum computing (QC) is an exciting new technology that completes the text with an application that relies heavily on the mathematics of linear algebra. There is more to explain than the conversion of common matrix notation into Dirac bra-ket notation. There is also the bizarre world of quantum physics that is inspiring a range of quantum information technologies. Qubits and their entangled superposition states are introduced to show that the “quantum advantage” stems from qubits storing exponentially more information that can be processed at exponentially faster rates. The process of preparing, modifying, and measuring qubit states is explained. Linear algebra skills are exercised by representing QC coding through the application of unitary operators to qubit states. QC algorithms are designed by diagramming quantum circuits, casting those circuits into sequences of operators applied to qubit states, and converting the operators into matrix forms. These are the tools of QC. Several algorithms that demonstrate a range of computational tasks from simple arithmetic through quantum Fourier analysis complete the chapter.
One of the first challenges that new learners of Quantum Computing face is learning how to use and interpret quantum circuits represented in circuit, Dirac, and matrix notation. It is necessary for learners to understand the different notations used in Quantum Computing in order to ensure that they have a robust comprehension of the field and can make use of a variety of resources. Depending on their technical background, some learners may struggle with certain notations more than others. We present QNotation, a visual tool that helps learners explore the aforementioned notations. Circuit, Dirac, and matrix notation. This allows them to be able to identify the differences and similarities between the different notations. QNotation was built to be used at multiple stages throughout the Quantum Computer learning journey. While one may start using QNotation to help them become familiar with the different notations used in Quantum Computing, the tool can also be used as a visual quantum circuit debugging tool. In addition to QNotation being a tool that can be used by learners, it can also be used by instructors to support the teachings of quantum computing concepts and building blocks, including entanglement and Quantum Fourier Transform.