Loading...

PPT – Quantum Computer Overview PowerPoint presentation | free to download - id: 78b0f3-MDc0M

The Adobe Flash plugin is needed to view this content

Quantum Computer Overview

What is a quantum computer? A quantum computer

is any device for computation that makes use

of quantum mechanical phenomena to perform

operations on data, implemented with quantum

binary digits (qubits) in particle spin

states. What are the advantages? Search of

unstructured data base with N items would take N

steps on Classical Computer and only vn steps on

a quantum computer. (Grover algorithm) Quantum

computers can perform parallel computations on

superposed data. -Shors algorithm

(Factorization with Fourier transform)

Quantum computers are highly efficient for

- Factoring large numbers!

- Solving discrete logarithms!

- Simulation of quantum systems!

Quantum Registers, Ket Notation

Typical state vector of single-particle system

Where a2 ß2 1 and a2 or ß2 give the

probability of observing each state. Since all

possible 2n binary states of an n-qubit quantum

register can be superposed, the register can be

described by a state vector ?gt a0000gt

a1001gt a2010gt a3011gt an?gt or

in a column with 2n entries correlating binary

states to amplitudes.

?1 a10gt ß11gt

Example (With arbitrary amplitudes abi is

v(a2b2))

State Amplitude Probability

- abi a2b2

000 .37.04i .14

001 .11.18i .04

010 .09.31i .10

011 .30.30i .18

100 .35.43i .31

101 .40.01i .16

110 .09.12i .02

111 .15.16i .05

Quantum Array Notation vs. Classical

All quantum circuits are reversible (have a

one-to-one correspondence between input

and output vectors), so quantum array gates do

not have an implicate direction like standard

Boolean gates.

Example (Toffoli EXOR-middle)

Quantum Permutative Gates Overview

All quantum gates are represented by unitary

matrices that define transformations on

qubits. Permutative gates are gates whose

unitary matrices are simply permutative

matrices. Five major types and their

tautological equivalents -Inverter (NOT also

a quantum primitive) -Feynman (Controlled-NOT) -To

ffoli (Controlled-Controlled-NOT) -Fredkin

(Controlled-Swap) -Swap These gate-matrices can

also be described by how many inputs are fed to

the output with no transformations, in which case

they are said to be k-through. A reversible

gate with n inputs/outputs is described by a

2n2n matrix.

Inverter (NOT)

Schematic (zero-through)

Unitary Matrix

Logic

Feynmann (CNOT)

Unitary Matrices

Logic

Schematic (one-through)

EXOR-Down

EXOR-Up

Toffoli (CCNOT)

Schematic of EXOR-down (two-through)

Unitary Matrix

Logic

Fredkin (CSWAP)

Unitary Matrix

Schematic of Fredkin-Up (one-through)

Logic

Swap Gate

Schematic (zero-through)

Feynmann Gate Realizations

Unitary Matrix

Logic

Pb, Qa

Universal Controlled Gate

Pa if a0 then Qb if a1 then QU(b)

Matrices Overview

Testing for Unitary Matrix U (Example Phase

gate matrix)

-Identity Matrix (Square matrix with 1 in

entries along main diagonal, 0s elsewhere)

1) Take the hermitian transpose U of U

-Permutation Matrix Square matrix with entry of

1 in each row/column

2) Verify that UU I (the corresponding nn

identity matrix)

-Inverse Matrix Square matrix A-1 of A such that

AA-1A-1AIn

-Orthogonal Matrix Square matrix Q whose

transpose is its inverse QQTQTQIn

Where 1) a2c2 1 2) b2d2 1 3) abcd 0

Standard Product

The product AB of two quantum gate-matrices A

and B in a serial connection is used to derive

their collective unitary matrix. Example

Kronecker Product

The Kronecker product A B of two quantum

gate-matrices A and B in a parallel connection is

used to derive their collective unitary

matrix. Example

Y

X (NOT)

Matrix Vector

A vector is a line segment with a direction and

magnitude spanning some set of axes. In quantum

mechanics, a complex vector space of n dimensions

(axes) is denoted Cn (Hilbert space).

Column Notation for Vectors (With entries

representing coordinates of the terminal point in

complex vector space)

Vector Addition

Dual/Adjoint Vector ltu of Ket Vector

ugt (Transpose column and conjugate entries)

ltu ugt u1, , un

Truly Quantum Primitives

Most truly quantum primitive gates cannot be

decomposed into simpler matrix operators. They

include complex entries to reflect state

vector amplitudes rather than only standard bits,

thus introducing quantum phenomena into

reversible circuits (and can also transform

qubit inputs into superposed states).

Hadamard

Pauli-X (NOT)

Pauli-Y

Pauli-Z

Phase

V (vNOT)

Hadamard, Phase, V (vNOT)

Unitary Matrices

Pauli- X (NOT), Y, Z

Unitary Matrices

Simple Circuit Equivalences

U U I

Karnaugh Maps and Testing

In testing for constant or balanced Boolean

functions, 2 tries is the best-case scenario for

a regular computer while n/21 tries is the worst

(for a function with n possible output bits). A

quantum computer requires only one step using the

Deutsch-Jozsa algorithm.

Truth Table (Displays input/output function)

EXOR Operation

Balanced Function f(n)

Karnaugh Map (Displays input/output coordinates)

EXOR Operation

Bloch Sphere!!

On the Bloch sphere, a qubits state vector is

described as ?gt cos? 0gt eif sin? 1gt

for p/2 ? lt p/2 and 0 f lt 2p

The corresponding Bloch vectors terminal point

on the spheres surface is given by the

coordinates

Ternary Logic

- Ternary logic uses three basic bits 0, 1, and

2. Ternary quantum - registers can have 3n superposed states for

n-qubit inputs.

- Thus, the state vector would be ? a0gt ß1gt

?2gt where

-Ternary NOT gates include 1, 2, (01), (02),

and (12) with parentheses denoting permutations

on single-bit inputs (in addition to equivalent

vNOT gates)

Design Realizations

Proposed Quantum Computer Architectures

Criteria (DiVincenzo checklist)

- Superconductor arrays
- Ion traps
- Nuclear magnetic resonance (NMR)
- on solutions of molecules
- Solid-state NMR
- Quantum dot surfaces
- Cavity quantum electrodynamics
- (CQED) structures
- Molecular magnets

- Physically scalable (qubits can be increased)

- Qubits can be initialized to some values

- Operates much faster than decoherence time

- Computer uses quantum gates and logic

H?(x,t) ihd?(x,t)/dt

- Has a means of reading qubits

Phenomena

-Superposition Blend of eigenstates in QM

system (QM database searching) -Entanglement

Instantaneous particle correlation (QM

teleportation, communication) -Interference

Disruption of QM systems (QM parallelism, joint

computations) -Non-clonability Impossible to

copy unknown QM state (QM cryptography)

Conclusion