World s First Universal Quantum Computation Simulator. Quantum Computer Simulator. For Windows/Macintosh. User Guide

Transcription

1 World s First Universal Quantum Computation Simulator Quantum Computer Simulator For Windows/Macintosh User Guide

2 Published By SENKO Corporation River Steel Bldg. 9-30, Kitasaiwai 2-chome, Nishi-ku, Yokohama, JAPAN Copyright 1999 SENKO Corporation. All rights reserved. Quantum Computer Simulator and QCS are trademarks of SENKO Corporation. Macintosh and TrueType are trademarks of Apple Computer Inc. Adobe, Adobe Type Manager, and Adobe Acrobat are trademarks of Adobe Systems, Inc. Windows is a trademark of Microsoft Corporation. Mathematica is a trademark of Wolfram Research, Inc. All other company and product names are trademarks and/or registered trademarks of their respective owners. First Printing: September 1999

3 Contents Contents 0. PREFACE INTRODUCTION Why is a quantum computer to be studied? What is a quantum computer? Summary INSTALLATION Windows Version Macintosh Version TUTORIAL Essential Knowledge and Reference Books Features of Quantum Computer Simulator Gates Matrices Qubits Selecting Lines Selecting Gates Selecting Qubits Unitary Gates Swap Gates Custom Gates Linking with Mathematica Composite Gates Measurement Gates Exponential Gates DFT (Discrete Fourier Transform) Gates Inserting and Deleting Qubits and Gates Editing by Drag and Drop, Copy, Cut, and Paste Creating Circuits Executing Simulation Quantum Computer Simulator User Guide i

4 Contents 3.21 Settings Other Features DESCRIPTION OF FUNCTIONS Windows Version Macintosh Version SPECIFICATIONS ii Quantum Computer Simulator User Guide

5 PREFACE 0. PREFACE This manual describes basics of quantum computation, operation of Quantum Computer Simulator (hereinafter sometimes called QCS or Simulator), tutorials, and other information. Operating the Simulator in steps with the manual enables you to create various quantum algorithms. Sample quantum algorithms attached may help you to create more sophisticated algorithms. It is our great pleasure that this Quantum Computer Simulator will help you to understand and study a quantum computer which must be realized in the near future. Quantum Computer Simulator User Guide 1

6 INTRODUCTION 1. INTRODUCTION 1.1 Why is a quantum computer to be studied? Computers are used everywhere, day and night in our current society. These are essential to our ordinary, comfortable life. Recently we have encountered a number of problems when using such essential computers in various opportunities. One of them is security in communication between computers in a network. It is a very serious problem if a secret document exchanged between two computers can be read by third party computers. To avoid such problem, cryptographic systems are noticed and studied widely. Now among various cryptographic systems, RSA public key cryptosystems are used commonly. These systems are based on factorization of a large integer, which is hardly done or may take many years to be solved even by current supercomputers. Namely, present cryptography is done by focusing on a weakpoint of computers. Those systems are very commonly used in communication between computers since it is very easy to create them. In 1994, however, P. Shor in the Bell Laboratory of AT&T has discovered that such factorization may be computed much faster with a computer model called a quantum Turing machine, which is the basis of a quantum computer. This discovery has given a kind of highlight to a quantum computer, which may allow computer codebreakers to successfully attack even the most impenetrable cryptographic systems in a practically short time. Unlike classical (current) computers processing digital information of 0 and 1, quantum computers process superposition of 0 and 1 (e.g. 0 at some percentage, and also 1 at some percentage). Thus the latter are completely different from the former. There is another reason why the emergence of quantum computers is expected. It is very difficult to solve large-number factorization with classical computers. Could faster computers, then, solve such factorization easily? High speed computers depend on high speed CPUs. Making CPUs faster will need more larger-scale integration of CPUs, which can be considered equivalent to the more higher density of transistors in CPUs. Those transistors will, however, reach fundamental physical limits when they begin to approach the size of atoms, where they behave with quantum mechanics. CPUs for quantum computers will be comprised of basic elements such as electrons and photons, so they could be much smaller that those for current computers. The size of controllers that control those small elements will depend on the development of science and technologies Most scientists and researchers in both corporate and university laboratories, however, acknowledged that there is much work left to do to build commercially or scientifically useful quantum computers. What on earth is, then, a quantum computer? 2 Quantum Computer Simulator User Guide

7 INTRODUCTION 1.2 What is a quantum computer? A quantum computer has various features quite different from classical (current) computers, thereby making it possible to perform computations that could not be done by classical computers. To learn well about quantum computers, we have to make a mathematical approach as well as comparison between these two different types of computers Expression of bits Bit expression in quantum computers is completely different from in classical computers. Bits are minimum units to handle data in computers, being expressed in binary notation of 0 and 1 in case of classical computers. In quantum computers, however, bits are expressed not only as 0 and 1 but also as superposition of 0 and 1. We can obtain probability variables between 0 and 1 from superposition of 0 and 1. A bit in classical computers corresponds to a qubit in quantum computers. This qubit in quantum computers has two basic states, states 0 and 1, which are equivalent to those in classical computers. In classical computers, for instance, 0 is a state of 0 volt plus basic voltage and 1, a state of 5 volts plus basic voltage. (This basic voltage 5V is supposed for TTL logic gates. The basic voltage for widely used CMOS logic gates is approximately 3.5V.) Movements of electrons are most widely accepted in the study of quantum computers to physically express qubits. Electrons have internal motions corresponding to spins in classical dynamics. These have two eigen values, called an upward and downward spins. To express qubits, it is very convenient if the upward spin is deemed as state 0 and the downward one as state 1. A qubit has a state where it is not certain that the qubit is 0 or 1 until observed at an instance. This state is called a state of superposition. Electrons have such state where it is not certain until observed that their spins are upward or downward. As such, electron spins are considered as best suitable, among various quantum phenomena to express qubits in quantum computers. Thus these computers are called quantum computers because they utilize quantum phenomena. The state of superposition is identifiable as state 1 or 0 only after observed. Before observation, it may be 0 or 1 at a certain probability. For example, it is 0 at a probability of 30% and 1 at a probability of 70%. In quantum computers, states 0 and 1 are strictly expressed in matrices. States 0 and 1 are expressed simply 0 and 1 respectively Classical computers: 0 or 1... bit Quantum computers: Superposition of 0 and 1... qubit Quantum Computer Simulator User Guide 3

8 INTRODUCTION 1 0 : ground state : excited state 1 Superposition x is expressed in the equation below when 0 and 1 are matrices above. In this equation, a and b are complex numbers, called probability amplitudes. The squares of absolute values of probability amplitudes of 0 and 1 are respective probabilities of 0 and 1 when x is observed. Probabilities of 0 and 1 are summed to 1. x x = a 0 + b 1 where a and b are complex numbers. 2 2 a probability of 0, b probability of 1 The above is bit expression about one qubit. In case of two or more qubits, they are expressed as blow Classical computers Example) (2 bits) (8 bits) Quantum computers Example) (2 qubits) (8 qubits) Qubits can be expressed with the tensor product of matrices of single qubits. The number of elements of a matrix is 2 raised power to the number of qubits when a column of qubits is calculated using the tensor product. This indicates that columns of bits in quantum computers are different from those in classical computers. 4 Quantum Computer Simulator User Guide

9 INTRODUCTION Quantum Computer Simulator User Guide = = L M From the above, the following are given: = = (2 qubits) columns (8 qubits) columns (16 qubits) In the case of n qubits, a matrix of 2 n is given. Tensor product... = b b a b b a b b a a b b a a arbitrary x x x x L L M = =

10 INTRODUCTION Circuits We then describe differences between circuits in classical and quantum computers resulting from differences in bit columns mentioned in the preceding paragraph. Electronic circuits in classical computers are often expressed with the following main three elements AND OR NOT 0 1 It is important that one or two bits are output when two bits are input. This means that the original two bits cannot be restored from the result of AND or OR circuits. That is, electronic circuits in classical computers are not reversible. Complicated circuits can be created with these basic elements. Those circuits are actually built in classical computers. Complicated processing systems: Combination of basic elements AND, OR and NOT How are those circuits in quantum computers? There are such circuits as AND, OR and NOT in quantum computers as well. Those are, however, not always smallest units in quantum computers, where smaller elements CN (Controlled-NOT) are proposed to represent AND, OR and NOT used in classical computers. 6 Quantum Computer Simulator User Guide

11 INTRODUCTION Basic element of quantum computer: CN controlled-not CONTROL CONNECTION 0 0 σ x 1 1 TARGET CN is expressed by a quadratic unitary matrix: CN inverts a acting qubit only when the state of a qubit on the control side is 1. It acts as shown in the following table: IN OUT CONTROL TARGET CONTROL TARGET It is known that similar circuits to those in classical computers can be expressed with this element CN. In quantum computers circuits can be expressed with matrices as well as qubit columns. CN is expressed with a quadratic square matrix. As seen from a circuit diagram of CN, the size of a side of a matrix is for two qubits, namely 2 2, as it acts to two qubits. It means a simple calculation of a matrix that a circuit acts to an input qubit column. The result is an output qubit. In the above CN, it is given as follows: Quantum Computer Simulator User Guide 7

12 INTRODUCTION Where an input qubit is = = 0 0 As an output qubit is also 01, = Output Input The above indicates that the number of output qubits equals to that of input qubits. Thus, unlike classical computers, a quantum computer always outputs qubits in the same number as of input qubits. Furthermore, it is reversible that when the output result is input in the same circuit, the exactly same input qubits are output. This reversibility is obvious from the above matrix. a 1a2a3a4 b1b2b3b 4 Operator or gate a 1 b 1 a 2 σ x b 2 a 3 b 3 a 4 σ x σ x b 4 In quantum computers, we can design various circuits using not only AND, OR and NOT created with CN but also varied matrices. Those circuits will give excellent performance that cannot be done with classical computers. Matrices for circuits in quantum computers, however, must satisfy the following conditions: Unitary matrices with the respective numbers of column and row elements to be 2 n where n is an arbitrary positive integer Elements operating to qubits are called unitary operators. 8 Quantum Computer Simulator User Guide

13 INTRODUCTION Physical Implementations of Quantum Computers What is quantum computer hardware alike? In a sense, quantum computers make user of electron spins as mentioned before. It requires very high technology to control such small particles. Various technologies are being studied at many institutes throughout the world. In this paper, we do not discuss about those high technologies, but explain of simple images only. It is supposed that an electron exists in a space. This electron is equivalent to a qubit. The electron always spins either upward or downward. State 0 or 1 is identified with the upward or downward spin as follows: qubit 0 Upward spin of electron qubit 1 Downward spin of electron Some operation to change the direction of spin is done to such electron. This operation must be performed by a unitary operator. The operation is considered being done, for example, by light or electromagnetics. Electrone Action (light, etc.) For example, a NOT operation is given to an electron. Initial State Result Electrone x = a 0 + b 1 x = b 0 + a 1 Probability of upward spin: Probability of downward spin: 2 a b 2 Element: NOT Probability of upward spin: Probability of downward spin: b a 2 2 In two or more qubits, it is supposed that some electrons fly in parallel. This gives us an impression that building an actual quantum computer is very difficult because we must make accurate operations onto electrons when they are caught. An actual quantum computer could be very fast and small. Quantum Computer Simulator User Guide 9

14 INTRODUCTION 1.3 Summary We have described the basics of quantum computers as above. Quantum computers are considered one of the feasible next-generation computers, but are not superior in all aspects. The following is a summary of quantum computers in both advantages and disadvantages. Advantages will be: Processing Better algorithms than in classical computers can be created to solve some problems. Example) Factorization Algorithms Algorithms are more widely applicable than in classical computers. They are so reversible that input data are obtainable from output data Example) Addition Subtraction Exponential calculation Logarithm Hardware Classical computers: Electronic circuits Example) Production process Size Current (1998) million mil/side Future ( ) Size of atom? Approx. 0.2 billion mil/side Quantum computers: Electrons Very fast, small Disadvantages will be: Implementing an actual quantum computer seems very difficult. The computer equipment itself might be very large. A coherent state may break, resulting in a wrong solution. 10 Quantum Computer Simulator User Guide

15 INTRODUCTION Coherent State where qubits 0 and 1 coexist in a certain probability Those advantages and disadvantages mentioned above are only for examples, and there will be many more. It will not be better to replace classical computers with quantum computers in all fields. The latter may not be good in cost performance for simple computation or control of electric devices. The former should be used for those purposes. Both should, therefore, be used for different purposes. A quantum computer may be added to a classical computer in the form of QPU (quantum processing unit). It will depend on the development of quantum computers, of which physical implementation will be sooner than expected since many scientist and researchers in the world have been making every effort for such implementation. Quantum Computer Simulator User Guide 11

16 INTRODUCTION Quantum computer II Qubit (Superposition of 0 and 1) Classical computer II Bit (0 or 1) Hardware Electron spin Fast, small Size, price of equipment Very high technology needed Electronic circuit Fairy fast Accumulated technology Limited size Software Excellent algorithms Reversibility Error by decoherence Accumulated technology Some algorithms not good 12 Quantum Computer Simulator User Guide

17 INSTALLATION 2. INSTALLATION 2.1 Windows Version Install the Quantum Computer Simulator in the following procedure: 1. Insert the Quantum Computer Simulator CD-ROM in the CR-ROM drive, and the SETUP.EXE is automatically activated to start set-up. (If it is not done so, doubleclick the SETUP.EXE file.) 2. Proceed with installation as instructed in the QC Simulator Setup dialogs. Use the Browse... button in the Choose Destination Location page if you want to change the drive or folder where the QCS is to be installed. We recommend you to use the Typical setup type for installation if you install the QCS for the first time. Adobe Acrobat Reader is necessary to read the PDF manual in the Documents folder contained in the QC Simulator folder. Install the Reader using its installer provided in the QC Simulator CD-ROM if you do not have it in your computer. Quantum Computer Simulator User Guide 13

18 INSTALLATION 2.2 Macintosh Version Install the Quantum Computer Simulator in the following procedure: 1. Insert the Quantum Computer Simulator CD-ROM in the CD-ROM drive and double-click the QC Simulator Installer icon. 2. Click the Install button in the QC Simulator Installer dialog box. Use the Switch Disk button or the drive popup menu if you want to change the drive or folder where the QCS is to be installed. We recommend you to use Easy Install for installation if you install the QCS for the first time. Adobe Acrobat Reader is necessary to read the PDF manual in the Documents folder contained in the QC Simulator folder. Install the Reader using its installer provided in the QC Simulator CD-ROM if you do not have it in your computer. 14 Quantum Computer Simulator User Guide

19 3. TUTORIAL This chapter explains how to create basic circuits (algorithms) using the Quantum Computer Simulator, and we learn the features and operations of the Simulator. Essential knowledge and reference books Features of Quantum Computer Simulator Gates Matrices Qubits Selection of gates Selection of matrix Selection of qubit Basic gates Swap gates Custom gates Linking with Mathematica Composite gates Inserting and deleting qubits and gates Editing by drag and drop, copy, cut, and pa Creating algorithms File input and output Executing simulation Settings Other features Quantum Computer Simulator User Guide 15

20 3.1 Essential Knowledge and Reference Books Knowledge of basic linear algebra and understanding of descriptions in Chapter 1 are enough to create quantum algorithms and understand their operations. The reference books listed below will be recommended to learn and understand quantum computers in more details. 1) Explorations in Quantum Computing, Colin P. Williamsand Scott H. Clearwater, Springer-Verlag (ISBN X) 2) Feynman Lectures on Computation, Richard P. Feynman, AddisonWesley (ISBN ) Research and study of quantum computers are getting more and more active day by day. We recommend you to access the following web site for better understanding of the latest situation: The Quantum Computer Simulator performs operations effectively using Mathematica for input and output of matrices. For operations of Mathematica, please refer to the Mathematica 3.0 Reference Guide included in the Mathematica package. 16 Quantum Computer Simulator User Guide

21 3.2 Features of Quantum Computer Simulator The Quantum Computer Simulator is a simulator of a quantum computer which is considered as one of the next-generation computers. Programming languages will be invented for quantum computers in future. Until then, however, algorithms with matrices native to quantum computers will be mainly used. It is the Quantum Computer Simulator that is programmed to create and verify quantum circuits without considering those matrices so much. The major features are, basically, creation and verification of quantum computer circuits, but the Simulator has the following features in addition: Number of qubits: Upto 32, subject to memory Types of input qubits: Pure states 0 and 1 only Expressions of arbitrary probability amplitude: Number of input gates: Unlimited, subject to memory Types of input gates: 1. Arbitrary quadratic unitary matrices 2. Controlled quadratic unitary matrices 3. Swap gates 4. Custom gates 5. Composite gates 6. Measurement gates 7. Exponential gates 8. DFT (discrete Fourier transform) gates - Input and output of files (file extension "qc") for each quantum algorithm - Quantum Computer Simulator User Guide 17

22 File outputs are compatible for both Windows and Macintosh. Each quantum algorithm can be output to a printer. Any number of gates can be cut, copied and pasted from the quantum algorithm window. Images copied from the quantum algorithm window can be output in bit map or PICT. Drag and drop is possible inside and outside the quantum algorithm window. Any names (one character and suffix each) can be attached to unitary and custom gates. The state of probability amplitude during simulation can be observed in the Graph and Numeric windows. Enlargement and reduction are possible at any positions in the Graph window, using the zoom cursor. Results can be sorted by either state or probability in the Numeric window. Import and export of unitary matrices from and to Mathematica 18 Quantum Computer Simulator User Guide

23 3.3 Gates A unit of a command for quantum computation is called a gate in the Quantum Computer Simulator. An algorithm running on the Simulator is simply a set of gates even if it is huge. Each of those gates contains a matrix. We can create various algorithms editing matrices. It may take a considerable time or is a waste of time to prepare those matrices by inputting manually. To avoid such time, the Simulator has the following ready-to-use gates built in as templates Note: This tutorial describes simulation in the Windows version. The exactly same simulation can be done in the Macintosh version except some minor differences displayed in the screen. Such differences are described each time when they appear. Bear those differences in mind to perform tutorial simulation in the case you use the Macintosh version. Types of gates Basic gate Swap gate Custom gate Composite gate Measurement gates Exponential gates DFT (Discrete Fourier Transform) gates Five frequently used unitary gates, including NOT gate, are pre-defined for your convenience. Swap gates are used to replace the low order of qubits with the high order. Custom gates is used to operate a unitary transformation on more than two qubits. Mathematica will help you to create more complicated matrices. A composite gate is a set of various gates. It is very convenient since complicated gates can be combined easily to only one composite gate. The portion of the algorithm not filled by any gates are considered to be equivalent to an identity matrix. This classification of gates provides very easy-to-understand, easy-to-use interface. Quantum Computer Simulator User Guide 19

24 3.4 Matrices All the gates appear on an algorithm have coresponding matrix representation. For instance, NOT gate appears on the screen as a symbol below and is represented in a matrix form as follows: Controlled-NOT gate appears on the screen as a symbol below: and has a matrix representation as follows: Quantum algorithm is considered to be a successive unitary transformation. 20 Quantum Computer Simulator User Guide

25 3.5 Qubits Qubits are of bit columns to be input into quantum circuits. The values can be set to 0 or 1, and no other numeric values can be used. Small numbers indicate lower qubits and large ones, upper qubits > > Quantum Computer Simulator User Guide 21

26 3.6 Selecting Lines Gates must be selected before they are moved by drag and drop or edited. To select a single gate, do the following operation on the gate: Windows version: Left click on a gate Macintosh version: Click on a gate To select two or more gates, do the following: Windows version: (1) Left click on a gate at one end (2) Shift left click on a gate at the other end. Macintosh version: (1) Click on a gate at one end (2) Shift key + click on a gate at the other end. 22 Quantum Computer Simulator User Guide

27 3.7 Selecting Gates Different shapes will be displayed with the types of gates, but the way of selection is the same. To select a gate, do the following: Windows version: Left click on a gate Macintosh version: Click on a gate To select a unitary gate: To select the point of a unitary gate s control line: To select a custom or composite gate: Quantum Computer Simulator User Guide 23

28 3.8 Selecting Qubits You must select a qubit as well before editing. To select a qubit, do the following: Windows version: Left click on a qubit Macintosh version: Click on a qubit To change the selected qubit by key operation: Qubit 0: Press key 0. Qubit 1: Press key 1. This key operation is common to the Windows and Macintosh versions. 24 Quantum Computer Simulator User Guide

29 3.9 Unitary Gates A unitary gate is composed of a unitary (quadratic) matrix and a control line attached to it. Do the following to create it: Windows version: Macintosh version: Control key + left click on a qubit where a gate is to be placed Control key + click on a qubit where a gate is to be placed You can also place a gate by the Insert Matrix command in the Edit menu. A gate for NOT is placed in the default setting. This gate can be changed easily to any of the predefined matrices by the following mouse operation: Windows version: Left double-click on a gate to be edited Macintosh version: Double-click on a gate to be edited Note: In the Macintosh version, a gate can also be edited using the Property... command ( +I) in the Edit menu after selecting the gate. Quantum Computer Simulator User Guide 25

30 The following dialog box appears: Note: In the Macintosh version, the dialog shown below appears. The functions are the same as in the Windows version except for the locations of buttons and others. The above dialog box has the following gates predefined: Selecting the Custom matrix at the bottom will allow you to use an arbitrary quadratic unitary 26 Quantum Computer Simulator User Guide

31 matrix. A real and imaginary numbers can be set in each of the columns and rows. When you close the dialog box, an entered custom matrix is checked for correctness. If it is not correct, the original matrix is set. To set a custom matrix more easily, this Simulator has a function to define a matrix by linking with Mathematica. For details, refer to 3.11, Custom gates and 3.12, Linking with Mathematica. You can name the matrix freely by entering a character in each of the Name and Suffix columns. For example, A is entered in the Name column and x, in the Suffix column. Press the OK button and the following appears: You may not enter a suffix. Without a suffix entered, the matrix is displayed as shown below. When it is difficult to identify matrices in a circuit of many gates, this naming function will make those matrices more easily identifiable in the circuit. A control line can be attached to a matrix of basic gate. Control lines control matrices connected to them. When a qubit at a control point on a control Quantum Computer Simulator User Guide 27

32 line is 0, the matrix does not act. When the qubit is 1, on the contrary, the matrix does act. This control line can be set only for basic gates that contain matrices, in the following manner. Windows version: Macintosh version: Control key + left click at a qubit where a control point is to be placed Control key + click at a qubit where a control point is to be placed Then the matrix is connected to the control point automatically. Any number of control points can be set on empty qubits. Control points can also be set by the Insert Matrix command in the Edit menu in the same manner as for placing matrices. 28 Quantum Computer Simulator User Guide

33 3.10 Swap Gates A swap gate replaces an upper qubit with a lower qubit. There are various usages of this gate, such as selecting an input qubit when using a custom gate. Input qubit To create a swap gate, first select a position to insert it, then use the Swap Gate command in the Edit menu to place it at the selected position. Do the following operation to swap gates: Windows version: Macintosh version: Drag and move the point of a qubit to be exchanged. Drag and move the point of a qubit to be exchanged while pressing the control key. When you click a notch at the lefthand side on a swap gate, a notch at the righthand side becomes an anchor point. When you click a notch at the righthand side, a notch at the lefthand side becomes an anchor point. Quantum Computer Simulator User Guide 29

34 3.11 Custom Gates Only custom gates can handle any types of matrices. Use custom gates for larger unitary matrices than quadratic ones. To create a custom gate, first select a position to insert it, then use the Custom Gate command in the Edit menu to place it at the selected position. The following dialog box appears. 30 Quantum Computer Simulator User Guide

35 Note: In the Macintosh version, the dialog shown below appears. The functions are the same as in the Windows version except for the locations of buttons and others. Press the Create button, and the dialog box shown below appears to specify the size of a matrix. Then the following matrix information appears to specify the name and elements of the matrix. An identity matrix is specified at the default setting Quantum Computer Simulator User Guide 31

36 Press the OK button, and the created matrix is automatically checked whether it is unitary. If unitary, a custom gate is placed at the selected position. If you want to edit the created matrix, take the following steps to do it: Windows version: Left double-click on a custom gate to be edited Macintosh version: Double-click on a custom gate to be edited Note: In the Macintosh version, a gate can also be edited using the Property... command ( +I) in the Edit menu after selecting the gate. And the dialog box appears again for you to edit the matrix as shown below. 32 Quantum Computer Simulator User Guide

37 3.12 Linking with Mathematica When preparing a custom gate of which matrix has a large number of qubits, many elements have to be entered as explained in the preceding section. To save time of such entry, this Simulator is provided with a function that can import matrices from Mathematica (version 3.0 or later), which must have been installed in your computer. Note: Mathematica 3.0 or later must have been properly installed for linking with Mathematica. If not, the Import/Export button is displayed in gray to disable the linking function. If this happens, open the system folder and check to see if the MathLinkLibraries file is contained in the extensions folder. If not, reinstall Mathematica properly. This MathLinkLibraries file contains a library to allow the kernel part of Mathematica to be used from outside. The Quantum Computer Simulator makes use of this library to communicate with Mathematica. Press the Import/Export button in the above dialog box, and the following dialog appears to link with Mathemacia. The Simulator needs the QCS.nb file to link with Mathematica. (It is the Mathematica Connection.nb file in the Macintosh version.) If the QCS.nb file is not found, a dialog to specify a file will appear. Specify the QCS.nb file in the dialog. Note: In the Macintosh version, a progress bar appears showing connection in the window instead of a connection status dialog. Quantum Computer Simulator User Guide 33

38 When connection is successful, Mathematica is activated to open the QCS.nb file and the dialog shown below asks you for initialization. Press the Yes button to start initialization Then the result of linking is shown at the LinkResult part in the Notebook. 34 Quantum Computer Simulator User Guide

39 If linking is not successful, we recommend you to restart Mathematica and retry linking. After successful linking, enter a matrix at the cell in the Notebook as shown below and evaluate it. Then the elements of the matrix are automatically transferred to the Custom matrix dialog box in the QCS to check the matrix whether it is unitary and square. If the matrix contains imaginary numbers, capital letter I must be entered. Quantum Computer Simulator User Guide 35

40 If the matrix is not unitary or square, an error message will appear as shown below. Check and correct the matrix and retry import. Other than custom gates, unitary gates can also be linked with Mathematica. For unitary gates, quadratic unitary matrices only can be imported from Mathematica. In addition to the above import function, the Simulator can export matrices to Mathematica from the Custom matrix or Unitary matrix dialog box. After linking, evaluate the In cell in the Notebook and the matrix exported from the dialog box in the QCS will be displayed below the cell as shown: 36 Quantum Computer Simulator User Guide

41 Mathematica will be a great help for you to create complicated large matrices to be exported to the QCS. Quantum Computer Simulator User Guide 37

42 3.13 Composite Gates A composite gate is a set of gates. It can be composed of unitary, custom, swap, and composite gates. When you design a complicated circuit of many gates, composite gates will be very helpful to make your circuit much simpler. It is very easy to create a composite gate. First select gates to be a set. Then, use the Create command of Composite Gate in the Edit menu, and a composite gate will be created automatically. Note: In the Macintosh version, use the Group as Composite command in the Edit menu to create a composite gate. If a swap gate is included in a set of gates, only the swapped part becomes the matrix part of the composite gate. 38 Quantum Computer Simulator User Guide

43 Expanding a composite gate is very easy as well. First select a composite gate to be expanded. Then, use the Expand command of Composite Gate in the Edit menu. Note: In the Macintosh version, use the De-Group Composite command in the Edit menu to expand a composite gate. The composite gate is automatically decomposed to original gates as shown below. Quantum Computer Simulator User Guide 39

44 The matrix of a composite gate can be named in the same manner as for unitary and custom gates. Windows version: Left double-click on a composite matrix to be edited Macintosh version: Double-click on a composite matrix to be edited Note: In the Macintosh version, a gate can also be edited using the Property... command ( +I) in the Edit menu after selecting the gate. The following dialog box appears. Enter a name and suffix. The result will be as shown below. 40 Quantum Computer Simulator User Guide

45 3.14 Measurement Gates Measurement gates are used to observe qubits. Quantum computation features linear multiple operation on superposition of qubits. In actual physics, however, we cannot measure such superposition. We can only measure a state resulting after a superposition of states is shrinked at a certain probability. When, for instance, the state of a system of four qubits is as shown in the expression: State of quantum system = ,,, ,,, the state is measured to shrink to 0000,,, or 1111,,, at a probability of 0.5. State of four-qubit system Shrink to either one at a probability of ,,, ,,, 0000,,, 1111,,, To create a measurement gate, select the Put Measurement Gate command from the Edit menu. Measurement gates are expressed smaller in width than other gates. A default measurement gate is set to measure a qubit. To increase the number of qubits to be measured, double-click a measurement gate or select the Property... command from the Edit menu. The dialog shown below will appear. Enter a desired number. Quantum Computer Simulator User Guide 41

46 The Simulator chooses at random one of the qubits to be measured from those of which probability amplitude is not 0, and normalizes it to simulate a measurement gate. Application of a measurement gate will give different results each time of simulation. 42 Quantum Computer Simulator User Guide

47 3.15 Exponential Gates Exponential computation means operations of exponents containing residue, being used for Shor s factorization algorithm. b Specifically, the Simulator computes a on mod N input b and outputs the result to a pair of registers in which 0 must be input. This is because the operation below is not irreversible. In quantum computation, all gates must be expressed by products of unitary matrices. To secure this invertibility, operations to perform exponential computation must be in the following form:- b b a a mod N b b, 0 a b, a mod N Choose the Put Exponential Gate command from the Edit menu to create an exponential gate (a gate for exponential computation). b Input parameter is output as it is. The upper shows an input parameter. The result of computation is output in the lower. Input values in the lower must be all 0 b a mod N> is output. Each exponential gate is divided to the upper and lower parts on the screen. A newly created exponential gate has an input parameter of 1 as the number of qubits a and N the default values for a and N. These values are not generally suitable for execution of computation as needed. Quantum Computer Simulator User Guide 43

48 To change those values, double-click a created exponential gate or select the gate and choose the Property command from the Edit menu. The following dialog will appear. Specify a size of input parameter b. Specify a value of a. Specify a value of N. 44 Quantum Computer Simulator User Guide

49 3.16 DFT (Discrete Fourier Transform) Gates Discrete Fourier transform is an operation most popular in quantum computation. The Simulator performs the following computation against input parameter a In the above formula, q has the following relation with the number of qubits n: a q 1 a 1 2 e q q n = 2 c= 0 π ia c/ q To create a DFT (discrete Fourier transform) gate, choose the Put DFT Gate command from the Edit menu. This gate has its name DFT displayed at the center on the screen, as shown below. c The number of input qubits is set to 1 as default. To change this number, double-click a created DFT gate or select the gate and choose the Property command from the Edit menu. The dialog shown below will appear. Specify the number of qubits for an operation required. Quantum Computer Simulator User Guide 45

50 Specify the number of qubits. 46 Quantum Computer Simulator User Guide

51 3.17 Inserting and Deleting Qubits and Gates Qubits and gates can be inserted or deleted freely after opening a new circuit. To insert a gate: Windows version: Macintosh version: Select the position of a gate to be inserted and press the Insert key. Select the position of a gate to be inserted and press the command key and + key. To delete a gate: Windows version: Macintosh version: Select a gate to be deleted and press the Delete key. Select a gate to be deleted and press the delete or clear key. To insert a qubit: Windows version: Macintosh version: Select the position of a qubit to be inserted and press the Insert key. Select the position of a qubit to be inserted and press the command key and + key. To delete a qubit: Windows version: Macintosh version: Select a qubit to be deleted and press the delete key. Select a qubit to be deleted and press the delete or clear key. There are some points to be noted in insertion and deletion of a qubit when various gates exist on the qubit. In case of deletion, a dialog box will appear asking you to make sure. Recheck the circuit before deleting a qubit. Listed below is a summary of deletion. Quantum Computer Simulator User Guide 47

52 Unitary gate (Matrix) Insertion: A qubit is inserted before the selected qubit. Inserting position Deletion: The selected qubit is deleted. Deleting position Unitary gate (Control) Insertion: A qubit is inserted before the selected qubit Inserting position Deletion: The selected qubit is deleted. Deleting position 48 Quantum Computer Simulator User Guide

53 Swap gate Insertion: A qubit is inserted before the selected qubit. Inserting position Deletion: The selected qubit is deleted. Any qubits that lose the swapping qubit restore to original. Deleting position Custom and composite gates Insertion: Qubits below the qubit to be inserted move upward only when that qubit is inserted before the lowest qubit position of the matrix. Inserting position Deletion: The selected qubit is deleted together with any gates on it. Deleting position Quantum Computer Simulator User Guide 49

54 3.18 Editing by Drag and Drop, Copy, Cut, and Paste A circuit can be designed and created easily by editing with a mouse. Gates can be moved by drag and drop. Gates can also be moved or copied by cut and paste or copy and paste or by key operation. Windows version: Macintosh version: Drag and drop a gate to be moved using the left button. (The gate is copied by dragging and dropping it while pressing the control key.) Drag and drop a gate to be moved. (The gate is copied by dragging and dropping it while pressing the option key.) Note: Note: In the Macintosh version, dragging a gate between windows will be deemed as copying even if the option key is not pressed. In the Windows version, this drag and drop operation is just simple. But in the Macintosh version, the operation with a gate selected is different from with both a gate and line selected. When a gate is selected, the position of the gate can be adjusted only vertically. To change the position of a gate on a quantum algorithm or drag it to another window, the line containing the gate must be selected beforehand. If the number of qubits to be moved differs from that at the destination, a dialog shown below appears. When qubits to be moved are fewer than those at the destination: Press the Yes button, and the qubits are increased to match with those at the destination. 50 Quantum Computer Simulator User Guide

55 When qubits to be moved are more than those at the destination: Press the Yes button, and the qubits are decreased to match with those at the destination. If there are any matrices over qubits to be deleted, such matrices are deleted as well. In case of swap gates, they are initialized to the default. Any control points on qubits to be deleted are also deleted as shown below. Quantum Computer Simulator User Guide 51

56 Unitary gates (Matrices) Unitary gates (Controls) Custom and composite gates In the Windows version only,. matrices can also be moved horizontally or between circuits. 52 Quantum Computer Simulator User Guide

57 Cut (Copy) and paste The cut (copy) and paste function is effective only for gates. This function is one of the standard system functions and is done by usual cut (copy) and paste operation in the Windows or Macintosh system. Windows version: (1) Select a gate to be moved. (2) Press the Ctrl and X keys (or the Ctrl and C keys for copying). (3) Select a destination gate. (4) Press the Ctrl and P keys. Macintosh version: (1) Select a gate to be moved. (2) Press the command and X keys (or the command and C keys for copying). (3) Select a destination gate. (4) Press the command and V keys. Quantum Computer Simulator User Guide 53

58 3.19 Creating Circuits We have described basic editing functions in the above sections, and now explain of creating simple circuits with such editing functions. We take for example basic logic circuits used in classical computers. As typical logic circuits, there are an AND and OR circuits, which work as shown: AND circuit: OR circuit: Input 2 Input 2 Input Input Let s create quantum circuits that work as an AND and OR circuits. An AND circuit is expressed as shown: x y 0 σ x x y x y Start the Simulator and decrease the number of qubits to 3 since three qubits are required as shown in the above circuit. 54 Quantum Computer Simulator User Guide

59 Place a NOT matrix at the first gate position: An AND circuit is created based on the above NOT matrix. Then, move the NOT matrix to the third qubit position as shown: Set two controls: Quantum Computer Simulator User Guide 55

60 A basic AND circuit is now created. The output and input of this circuit are as shown: x x y y 0 x y When the values of the first and second qubits are changed, the result will be displayed at the third qubit. To change the input and output positions, use a swap gate. The figure below is an example where swap gates are used. Let s create an OR circuit while keeping the AND circuit as it is. An OR circuit is expressed as below, which is quite similar to the AND circuit. x σ x x y σ x y 1 σ x x y Symbol σχ in the above figure represents a NOT symbol in logic operations. Open a new circuit by selecting the New command in the File menu to create an OR circuit since the Simulator allows you to create two or more circuits at the same time. 56 Quantum Computer Simulator User Guide

61 You will notice that the AND and OR circuits have a common part shown in the figure below. σ x We utilize this common part by copying it into an OR circuit. Copy the common part in the already created AND circuit and paste it in the OR circuit by drag and drop while pressing the control key, as shown in the figure below. Insert two NOT matrices as shown: Enter 1 as the input value of the third qubit as shown: Quantum Computer Simulator User Guide 57

62 The OR circuit is completed now. Its output and input are as shown in the figure below. x y x y 1 x y These AND and OR circuits created as above can be utilized in various circuits. We recommend you to make composite gates of them for use in creating new circuits. 58 Quantum Computer Simulator User Guide

63 3.20 Executing Simulation This section describes how to simulate the quantum circuits created in section 3.16 and how to evaluate the result of simulation. The Quantum Computer Simulator is provided with two probability distribution windows to show the result of simulation. Graphic window The Graphic window shows distribution of probabilities visually. The vertical axis is for the probability of observation of each state and the horizontal axis, for states. The graph can be enlarged upto 32 times bigger. Windows version To enlarge: To reduce: Left click on the graph Right click on the graph Macintosh version To enlarge: To reduce: Click on the graph Click on the graph while pressing the option key Quantum Computer Simulator User Guide 59

64 Numeric window The Numeric window shows the state probability of each qubit. Any state of probability 0 is not displayed. The numeric values displayed in the example figure below are the state and probability (real and imaginary number parts of complex number) from the left-hand side. The results displayed in the Numeric window can be sorted by state of qubit and probability respectively. Select By State or By Probability of the Sort command in the Simulator menu as shown below to start sorting. Sort by State (example) For example, the result of sort by state is displayed as shown in the figure below. 60 Quantum Computer Simulator User Guide

65 Note: The Macintosh version differs in the display of probability portions from the Windows version. These are displyed exponentially in the Windows version while they are not in the Macintosh version as shown in the example below. Example) This will be displayed in the following manner: Windows version: 6.25e-002 Macintosh version: Sort by Probability (example) For example, the result of sort by probability is displayed as shown in the figure below. The two figures above show that the qubit of the highest probability is closest to the answer. Simulation This Simulator has three simulation modes which are a full simulation, step simulation, and to-cursor simulation. Full simulation Open the And.qc file created in section 3.16 and enter input values, for example, 1 for x and 0 for y as shown below. Quantum Computer Simulator User Guide 61

66 x = 1 y = 0 Then, select the Run command in the Simulator menu as shown in the figure below to start simulation. After simulation, the Simulator will give the result that the probability of state 001 is 1 as shown below. INPUT OUTPUT (3) Result (2)y (1)x (3) Result x (2)y (1)x The answer is 0 at the third qubit since AND of x and y is expected to be Quantum Computer Simulator User Guide

67 x = 1 y = 1 How about a case of x=1 and y=1 then? The result will be expected to be 1 at the third qubit. Execute simulation in the same manner as above after entering 1 for x and 1 for y. The result will be displayed as shown in the figure below. The above figure shows that the answer is 1 at the third qubit. The result of simulation can be saved as a text file by the Save Result command in the File menu as shown below. In this save operation, the result is output and saved in the form as specified for sort and notation in the Numeric window. Examples of output result: Quantum Computer Simulator User Guide 63

68 Step simulation Open the Or.qc file created in section 3.16 and execute stepped simulation by selecting the Gate command of the Step command in the Simulator menu as shown below. When the Gate command is conducted, simulation is executed on a gate and suspended. Conduct the Gate command again, and simulation is executed on the next gate as illustrated in the figure below. 64 Quantum Computer Simulator User Guide

69 Quantum Computer Simulator User Guide 65

70 To-cursor simulation Select the Gate command of the Step command in the Simulator menu, simulation is executed upto the cursor position and stops there as illustrated below. In the step or to-cursor simulation, the result of simulation can be saved while simulation is suspended. 66 Quantum Computer Simulator User Guide

71 3.21 Settings Select the Preferences command in the File menu to open the Preferences dialog box as shown in the figure below and perform settings as you like referring to the descriptions of setting items mentioned below. Note that there are some differences between the Windows and Macintosh versions. Windows version Descriptions of setting items Initialization Qubits: Gates: Specify the initial number of qubits used in a circuit to be opened. Specify the initial number of gates used in a circuit to be opened. Quantum Computer Simulator User Guide 67

72 Color Specify a color for gates and matrices.gates and matrices will be displayed in the color specified here. Gates movement Specify a gate movement mode. Check the box, and a matrix will be moved together with control points attached to it when you drag it, as shown in the figure below. When the box is not checked, a matrix will be moved without control points when you drag it. Unitary check Specify the effective number of digits for unitary check. A more precise unitary check is performed when a larger number is specified. This unitary check is performed automatically when a custom gate is placed or a matrix is set for a basic gate. 68 Quantum Computer Simulator User Guide

73 Grids Show grids Check this box to display grids as shown in the figure below. When the box is not checked, grids are not displayed as shown below. Size: Specify the size of a grid in a range between 16 and 36 points. Probability Window Separate qubits: Check this box to set separators for qubits displayed in the Numeric window as shown: Quantum Computer Simulator User Guide 69

74 When the box is not checked, separators are not set for qubits as shown: Decimal notation: Check this box to set decimal notation for qubits displayed in the Numeric window. When the box is not checked, binary notation is set for qubits as shown: Update probability: Check this box to update probability distribution in the Graphic or Numeric window each time a gate is simulated. When the box is not checked, only the final result is displayed. 70 Quantum Computer Simulator User Guide

75 Notification Start simulation: Check this box to display the dialog as shown below before starting simulation. When the box is not checked, the dialog is not displayed. End simulation: Check this box to display the dialog as shown below at the end of simulation. When the box is not checked, the dialog is not displayed. NoteBook file Click the [...] button to specify a notebook file of Mathematica used when a custom gate is imported from or exported to Mathematica. Quantum Computer Simulator User Guide 71

76 Macintosh version The Circuit category specifies settings to edit quantum circuits. When Gate Moves... Also Moves Controls check box Check this box to move a gate together with a qubit to be controlled. When checked, a gate moves ttogether with a qubit to be controlled. Namely, the relation between a unitary gate and control will not change. When not checked, the position of control will not change even if a gate is moved. Only the unitary portion, therefore, can be moved. A gate can be freely moved to any position except to a qubit for which a control is placed. The Distribution category specifies the display properties for numeric values. Precision Specify the number of digits as a precision of numeric values to be displayed. A larger number will give higher precision. The precision specified here does not only determine the number of digits for numeric values in the probability distribution window but also influences the precision of numeric values in all windows including Unitary Property and Custom Property. Note: The precision of numeric values specified here influences only numeric values to be displayed and does not do so those values retained in the Simulator. When numeric values cannot be displayed in the specified number of digits, they are rounded. If this happens, all values displayed in the probability distribution window may not be summed to exactly Quantum Computer Simulator User Guide

77 The Mathematica Connection category specifies settings to link with Mathematica. Specify... Specify a Mathematica s notebook necessary to link with Mathematica. The notebook specified is displayed at Notebook:. In most cases, specify the Mathematica Connection.nb. Quantum Computer Simulator User Guide 73

78 3.22 Other features Print Windows version A circuit created can be printed using the Print command in the File menu. When the Print command is selected, the following dialog box will appear. Click OK to start printing. For print setup, use the Print Setup command in the File menu as shown in the example figure below. 74 Quantum Computer Simulator User Guide

79 For print preview, use the Print Preview command in the File menu as shown in the example figure below. Quantum Computer Simulator User Guide 75

80 Macintosh version The forefront (active) window can be printed by selecting the Print... command in the File menu. A dialog displayed may depend on the version of your MacOS or the type of a printer driver used by you. Copying a circuit in BITMAP or PICT The data of a circuit can be converted to BITMAP or PICT data by copying with the Copy command. Select the part of a circuit you want to convert it to BITMAP and execute the Copy command in the File menu to copy that part. The copied part now converted to BITMAP can be pasted to various files of other applications. For example, such part will be pasted to a new Paint file as shown in the figure below. 76 Quantum Computer Simulator User Guide

81 Note: If all gates are selected and copied, their qubits are copied together. When, for example, all the gates contained in a circuit are selected and copied as shown in figure 1 below, their qubits that are not inclued in the selected area are also copied together with the gates (see figure 2 below). Figure 1 Copying after selecting all gates σ x 0 σ x σ x Figure 2 Copied image of a circuit containing qubits Quantum Computer Simulator User Guide 77

82 DESCRIPTION OF FUNCTIONS 4. DESCRIPTION OF FUNCTIONS This chapter describes all the commands and functions of the Quantum Computer Simulator. 4.1 Windows Version MAIN SCREEN The figure below shows the main screen of the Quantum Computer Simulator. The above screen is composed mainly of the seven parts listed below: 1. Menu bar 2. Toolbar 3. Qubit display area 4. Circuit area 5. Graphic/Numeric window 6. Result display area 7. Status bar 78 Quantum Computer Simulator User Guide

83 DESCRIPTION OF FUNCTIONS MENU BAR File menu New Opens a new quantum circuit. Open Opens an existing quantum circuit in a file. Close Closes a quantum circuit. Save Saves a quantum circuit. Save As... Saves a quantum circuit with a new name. Save Result... Saves the simulation result of a quantum circuit with a new name. Preferences... Specifies the settings. Print Prints the active quantum circuit. Quantum Computer Simulator User Guide 79

84 DESCRIPTION OF FUNCTIONS Print Preview Previews the print image of the active quantum circuit. Print Setup Sets up the print conditions. Recent files area Shows upto four (4) recently opened files. Exit Quits the application. Edit menu Cut Cuts the selected gates, which can be then pasted by the Paste command. The gates cut by the Cut command are copied on a clipboard as BITMAP data. Copy Copies the selected gates, which can be then pasted by the Paste command. The gates copied by the Copy command are transferred to a clipboard as BITMAP data. Paste Pastes the cut or copied selected gates at the designated position. Select All Selects all gates in the active quantum circuit. 80 Quantum Computer Simulator User Guide

85 DESCRIPTION OF FUNCTIONS Insert Inserts a new gate or qubit at the designated position. Insert Matrix Inserts a new matrix or control point in the selected blank gate. Remove Removes the selected gate, matrix or qubit. Composite Gate - Create Integrates the selected gates into a single gate. Composite Gate - Expand Expands the selected composite gate. Swap Gate Inserts a swap gate at the designated position. Custom Gate Inserts a custom gate at the designated position. Measurement Gate Inserts a measurement gate at the designated position. Exponential Gate Inserts an exponential gate at the designated position. DFT Gate Inserts a DFT (Discrete Fourier Transform) gate at the designated position. Quantum Computer Simulator User Guide 81

86 DESCRIPTION OF FUNCTIONS Simulator menu Run Starts simulation. Abort Aborts simulation. Break Suspends simulation. Step Gate Starts gate by gate simulation. Step Cursor Starts simulation upto the cursor position. Sort - By State Sorts qubits in the Numeric window by state. Sort - By Probability Sorts qubits in the Numeric window by probability. Notation - Decimal Shows qubits in the Numeric window in decimal notation. Notation -Binary Shows qubits in the Numeric window in binary notation. 82 Quantum Computer Simulator User Guide

87 DESCRIPTION OF FUNCTIONS Window menu Probability Window - Graphic Displays the Graphic window of probability distribution. Probability Window - Numeric Displays the Numeric window of probability distribution. Toolbar Displays or hides the toolbar. Status Bar Displays or hides the status bar. Cascade Arranges windows in cascade. Tile Horizon Arranges windows horizontally. Tile Vertical Arranges windows vertically. Arrange Icons Arranges icons in order. Windows Shows a list of currently opened windows. Quantum Computer Simulator User Guide 83

88 DESCRIPTION OF FUNCTIONS Help menu About QCS Shows information of the QCS version. TOOLBAR New Opens a new quantum circuit. Open Opens an existing quantum circuit in a file. Save Saves a quantum circuit. Save Result... Saves the simulation result of a quantum circuit with a new name. Preferences... Specifies the settings. Cut Cuts the selected gates, which can be then pasted by the Paste command. The gates cut by the Cut command are copied on a clipboard as BITMAP data. Copy Copies the selected gates, which can be then pasted by the Paste command. The gates copied by the Copy command are transferred to a clipboard as BITMAP data. Paste Pastes the cut or copied selected gates at the designated position. Insert Inserts a new gate or qubit at the designated position. 84 Quantum Computer Simulator User Guide

89 DESCRIPTION OF FUNCTIONS Insert Matrix Inserts a new matrix or control point in the selected blank gate. Remove Removes the selected gate, matrix or qubit. Swap Gate Inserts a swap gate at the designated position. Custom Gate Inserts a custom gate at the designated position. Composite Gate - Create Integrates the selected gates into a single gate. Composite Gate - Expand Expands the selected composite gate. Measurement Gate Inserts a measurement gate at the designated position. Exponential Gate Inserts an exponential gate at the designated position. DFT Gate Inserts a DFT (Discrete Fourier Transform) gate at the designated position. Run Starts simulation. Abort Aborts simulation. Step Gate Starts gate by gate simulation. Step - Cursor Starts simulation upto the cursor position. Break Suspends simulation. Quantum Computer Simulator User Guide 85

90 DESCRIPTION OF FUNCTIONS STATUS BAR Status/Simplified help Shows the status of simulation or simplified help. Current Gate Shows the selected gate. Qubits Shows the total number of qubits. Gates Shows the total number of gates. GRAPHIC WINDOW 1. Range of probability 2. Range of qubit 3. Magnifier 4. Graph 5. Number of states 86 Quantum Computer Simulator User Guide

91 DESCRIPTION OF FUNCTIONS NUMERIC WINDOW 1. Qubit 2. Probability 3. Complex number (Real number) 4. Complex number (Imaginary number) 5. Number of states POPUP MENU Gate menu Cut Cuts the selected gates, which can be then pasted by the Paste command. The gates cut by the Cut command are copied on a clipboard as BITMAP data. Quantum Computer Simulator User Guide 87

92 DESCRIPTION OF FUNCTIONS Copy Copies the selected gates, which can be then pasted by the Paste command. The gates copied by the Copy command are transferred to a clipboard as BITMAP data. Paste Pastes the cut or copied selected gates at the designated position. Insert Gate Inserts a new gate at the designated position. Insert Matrix Inserts a new matrix or control point in the selected blank gate. Remove Removes the selected gate, matrix or qubit. Composite - Create Integrates the selected gates into a single gate. Composite - Expand Expands the selected composite gate. Swap Gate Inserts a swap gate at the designated position. Custom Gate Inserts a custom gate at the designated position. Measurement Gate Inserts a measurement gate at the designated position. Exponential Gate Inserts an exponential gate at the designated position. DFT Gate Inserts a DFT (Discrete Fourier Transform) gate at the designated position. 88 Quantum Computer Simulator User Guide

93 DESCRIPTION OF FUNCTIONS Matrix menu Edit Edits a matrix. Remove Removes a matrix. Qubit menu Insert Inserts a qubit at the designated position. Remove Removes the selected qubit. Set 0 Set 0 in the selected qubit. Set 1 Set 1 in the selected qubit. Quantum Computer Simulator User Guide 89

94 DESCRIPTION OF FUNCTIONS 4.2 Macintosh Version MAIN SCREEN The figure below shows the main screen of the Quantum Computer Simulator, Macintosh version. 1. Menu bar 2. Simulation status display area 3. Quantum circuit edit area 4. Distribution status display area 5. Distribution status switching tab 6. Distribution status (Shows a graph in the Graphical mode or numeric values in Numerical mode.) 90 Quantum Computer Simulator User Guide

95 DESCRIPTION OF FUNCTIONS Note: The Macintosh version has a quantum circuit edit window and distribution display window separately. The latter corresponding to the former is always attached to the latter. Which distribution display window corresponds to the quantum circuit edit window is shown in the title bar. The distribution display window which has no close box is automatically closed when the quantum circuit edit window is closed. MENU BAR File menu New Opens a new quantum circuit. Open... Opens an existing quantum circuit in a file. Close Closes the active quantum circuit. (The distribution display window is closed together.) Save Saves a quantum circuit. Save As... Saves a quantum circuit with a new name. Revert Cancels any changes made in a quantum circuit and reverts to the last saved state. Quantum Computer Simulator User Guide 91

96 DESCRIPTION OF FUNCTIONS Import Algorithm... Imports a quantum algorithm in the form of tensor product from Mathematica. (Not yet incorporated in this version) Export Algorithm... Exports a quantum algorithm in the form of tensor product to Mathematica. (Not yet incorporated in this version) Page Setup... Sets up the print conditions. Print... Prints the active quantum circuit. Quit Quits the application. Edit menu Undo Undoes the last edit work. (Not yet incorporated in this version) Cut Cuts the selected gates amd others. (The bit map image is created.) Copy Copies the selected gates amd others. (The bit map image is created.) 92 Quantum Computer Simulator User Guide

97 DESCRIPTION OF FUNCTIONS Paste Pastes at the designated position the gates and other stored in the clipboard. Select All Selects all gates and others in an active quantum circuit. Property... Edits the selected gates. Put Unitary Gate Creates a unitary gate at the selected position. Put Swap Gate Creates a swap gate at the selected position. Put Custom Gate Creates a custom gate at the selected position. Put Measurement Gate Creates a measurement gate at the selected position. Put Exponential Gate Creates an exponential gate at the selected position. Put DFT Gate Creates a DFT (Discrete Fourier Transform) gate at the selected position. Group as Composite Groups the selected gates as a composite gate. De-Group as Composite De-groups the selected composite gate. Expand Adds a qubit if a qubit is selected or a gate if a gate is selected. Reduce Removes a qubit if a qubit is selected or a gate if a gate is selected. Preferences... Specifies the settings. Quantum Computer Simulator User Guide 93

98 DESCRIPTION OF FUNCTIONS Simulator menu Run Starts simulation. Stop Stops simulation. Continue Restarts simulation from the stop point. Sort Specifies a display type in the Numerical window. Display Mode Specifies a display type of state vectors. Sort submenu by State Sorts state vectors by state. by Probability Sorts state vectors by probability. 94 Quantum Computer Simulator User Guide

99 DESCRIPTION OF FUNCTIONS Display Mode submenu Binary Shows qubits in binary notation with delimiters. (Standard expression in quantum computation) Binary w/o Separator Shows qubits in binary notation without delimiters. Decimal Shows qubits in decimal notation. Graphical Window 1. Range of qubits displayed in the vertical axis of a graph 2. Range of state vectors displayed in the horizontal axis of a graph 3. Magnifier (1 to 32 times) of a graph 4. Distribution of state vectors by probability 5. Number of state vectors of which probabilities are not zero Quantum Computer Simulator User Guide 95

100 DESCRIPTION OF FUNCTIONS Numerical Window 1. State vectors 2. Probabilities (Real and imaginary numbers in parentheses respectively) 3. Number of state vectors of which probabilities are not zero 96 Quantum Computer Simulator User Guide