Quantum Computer Simulation Alex Bush Matt Cole James Hancox Richard Inskip Jan Zaucha

Aims Develop an understanding of quantum computing Implement a simulation of a quantum computer Simulate Grover’s algorithm using our implementation of a quantum computer Learn group work skills and techniques

Background Feynman – 1982 Deutsch – 1994 Shor – 1994 Grover – 1996

Qubits Classically 0 or 1 Quantum bits: qubits Information obtained by measurement

Register: System of Qubits For n qubits 2 n states Basis vectors by convention in binary notation Example : basis states for a 3 qubit register

Gates Unitary linear operators which act on the register to alter its state Universal set of gates: Single qubit Hadamard Phase Two qubit CNOT

Quantum Algorithms Ingredients: registers, universal set of gates Advantage over classical computations: quantum interference

Grover’s Algorithm 1996, Lov Grover Search problem: find particular element in an unordered list of N entries Need to be able to assess if search found correct element, all elements in list have to be known beforehand

Grover’s Algorithm: technical description A register of n qubits is initialized. 2 n = L basis states corresponding to elements of the unordered list Walsh-Hadamard transform equal superposition of basis states Grover Iteration: 1) Oracle: marks searched state 2) Inversion about mean: Increases amplitude of searched state and decreases amplitudes of the other states

Inversion about the mean 1) Walsh-Hadamard transform 3) Walsh-Hadamard transform 2) NAND: The NAND gate returns false only if all qubits are in state We define this gate to apply a phase shift of -1 to all states that return true

Geometrical Representation: Rotations in plane of complex Hilbert Space

Geometrical Representation: action of the oracle

Geometrical Representation: inversion about the mean

Implementation of the Quantum Computer Register – the register is implemented as a matrix of complex numbers Gate – gates are implemented in both matrix and functional form

Basic Structure GroverCircuit Register UserInterface prepareRegister() nextIteration() measureRegister() Matrix GateFactory

Gates {interface} Gate CompositeGate{abstract} FunctionGate {abstract} MatrixGate HadamardMatrixHadamardFunction Matrix apply() GroverComposite

UML Diagram

Grover Iteration The GroverComposite gate is a special gate that when applied to the register performs a Grover iteration. groverIteration[0] = oracle; groverIteration[1] = hadamardTransform; groverIteration[2] = nand; groverIteration[3] = hadamardTransform;

Implementation: The Quantum Register, Quantum Gates; - Matrix, - Function, - Composite, Factory Method for creating Gates, Quantum Algorithm and the GUI.

Implementation: Register The Quantum Register: An extension of the Matrix class, 2 N entries, A storage medium for the Probability Amplitudes of each state. An example of a 2-qubit register: |ac| a |0> + b|1> ac|00> + ad|01> + bc|10> + bd|11> |ad| c |0> + d|1>|bc| |bd|

Implementation: Gates A Quantum Gate: Gates were implemented in a variety of ways: Matrix Functional Composite Gates were packaged with a private constructor to make use of a factory method for creating the gates.

Implementation: Gates 1-qubit Matrix Gate; A specific 2x2 matrix was constructed to perform a single bit operation depending on the role of the gate. This could then be extended by applying(tensor product) Identity matrices to the 2x2 gate. An example of such would be to apply a “ GATE ” to a 4- qubit register, with the qubit to be operated on being the second:

Implementation: Gates Higher order Matrix Gates and Functional Gates: Both for higher order matrix gates and functional gates a defined method was not so obvious for the operation of the gates. The use of patterns were used to analysis the expected change on the register and a matrix or function created from this. Functional gates using Java operations such as bitwise shift operators.

Implementation: Factory Factory method is an abstract class which can be used to create gates – matrix or functional – all from one location. All Gate constructors are private, therefore can only be created from inside the Factory. The factory allows control over all gates being matrix or functional or overloading certain gates to a different representation if desired..

Implementation: Quantum Algorithm and the GUI; The algorithm and GUI sections of the code are unique to Grover’s algorithm. When the GUI is run it prompts the user for specific values such as register size and the element to search for in the register. The register is then prepared by applying a Hadamart gate to every qubit. Iterations of Grover’s are then applied to the register. A measurement can then be taken.

Demonstration

Need for managment tools Share code between members Ensure members have up-to-date copies of files Backups of previous versions Version control systems

Subversion (SVN) Widely used – high compatability Free and open-source Integrates well with Eclipse Central repository for files Full version history of files

SVN (cont.) Repository Me Check out Check in Edit files Repository Me Check out Update Edit files Jan Check out Check in Edit files Me Check in Merge Changes

Google Code Provide SVN service View file diffs online (changes in files between versions) Mailing list Issues/bug tracker File hosting

Eclipse GUI to develop under Code completion Speed up routine operations Automatically compiles code Graphical interface for SVN (Subclipse)

Conclusion All of our aims achieved Scope for further work using existing code Improved team work skills Improved programming skills