Quantum Computing - University of North Carolina … · What is Quantum Computing? A quantum...
Transcript of Quantum Computing - University of North Carolina … · What is Quantum Computing? A quantum...
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Quantum Computing
Robert Sizemore
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Outline
Introduction: What is quantum computing?What use is quantum computing?
Overview of Quantum SystemsDirac notation & wave functionsTwo level systems
Classical ComputationTuring machine model
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Outline (cont.)
Circuit modelQuantum Computation
QubitsCircuit model of quantum computationDeutsch's algorithm, Shor's algorithmPhysical implementation
SummaryWorks Cited
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Introduction
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What is Quantum Computing?
A quantum computer is any device that uses quantum mechanical phenomena to perform
calculations and manipulate data.
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Why Quantum Computers?
More efficient algorithmsQuantum ParallelismCryptography (breaking codes & securing data)Simulating quantum systems
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Cryptography
RSA public key encryption relies on the difficulty of factoring large integers It is conjectured that factorization is impossible to do efficiently with a classical computer.Shor's factoring algorithm for quantum computers would render this encryption method useless
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An Inevitability?
Moore's Law: The number of transistor's
on a silicon chip doubles every two
yearsAtomic size is a fundamentally limit to the size of possible transistors.
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An Overview of Quantum Systems
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Schrödinger's Equation
Quantum systems are described by a wave function (ψ)
For a given potential (V(x)), we find all solutions to Schrödinger's equation.
These solutions form a basis of a vector space called a Hilbert space.
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Dirac (Bra-ket) Notation
Notation introduced by Dirac to represent objects in Hilbert space.
Bra – row vectorKet – column vectorBra-kets – inner
products
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Wave Functions
Hilbert spaces have an orthogonal basis.
A wave function is a linear combination of the basis vectors
Wave functions should be normalizable.
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Useful Wave Function Properties
InterferenceLike classical waves, the wave function can exhibit
constructive/destructive interference Exploited in quantum algorithms (Deutsch's)
EntanglementMultiple particle's wave functions may become
“entangled”Measuring or manipulating one particle will
necessarily affect the other.
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Born (statistical) Interpretation
A quantum system can in a superposition of the allowed states until a measurement is done.
When a measurement is performed, the wave function accepts one of the allowed states.
You cannot know which state the system will be in before the measurement, only the probability of it being in a given state.
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Statistical Interpretation
The complex modulus squared of each weight corresponds to probability of finding the system in that state.
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Useful Wave Function Properties
InterferenceLike classical waves, the wave function can exhibit
constructive/destructive interference Exploited in some quantum algorithms
EntanglementMultiple particle's wave functions may become
“entangled”Measuring or manipulating one particle will
necessarily affect the other.
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Operators
Operators act on wave functions
Linear operators can be represented as square matrices.
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Special Operators
Inverse operatorsHermitian operators – are
self adjoint. (Observables)
Unitary operators – adjoint is its inverse
Example: Pauli matrices
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2 Level Systems
A system with two basis states.
Examples:Nuclear spin (up, down)Polarization of a photon
(left, right)
Qubits (|1>, |0>)
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Bits
A bit is an elementary unit of information.
A bit has two possible states: 1 or 0.
A bit is unambiguous, never in a superposition of states.
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Qubits
A qubit is the elementary unit of quantum information.
A qubit has two allowed states: |1>, |0>.
A qubit can be in a superposition of the states of a bit.
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Overview of Classical Computation
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Classical Computation
Computation is done by means of an algorithm, which can be thought of as a set of instructions for solving a specific problem.
The mathematician Alan Turing (1912-1954) devised a hypothetical machine that could execute any algorithm called a Turing Machine.
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Turing Machine
Tape – infinite number of cells, every cell is blank or contains a letter in a finite alphabet
Control Head – has a finite number of states and a halting state
Read/Write – reads and writes or erases a letter in a given cell then moves left or right.
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Operation of a Turing Machine
Programs are defined by a set of instructions (a function)An instruction can be written as T:(s,a) → (s',a',d)
s is the current state and s' is the final state of the head. a is the letter in the cell, a' is the letter to be written.d is the direction the head will move
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Input of a Turing Machine
The input is the initial state of the Turing machine.a finite number of non-blank cellsthe read/write head in an initial positionthe control head in an initial state.a set of instructions for every preceding state
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Output of a Turing Machine
The output of a Turing machine is the final state of the Turing machine, if any.Control head reaches the halting state (H)Finite length of non-blank cells
For a Turing machine it is impossible to know for all inputs whether the control head will reach a halting state or not. This is the known as the halting problem.
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Example: Unary Addition
Top: A table listing the instructions for mapping each state of the Turing machine (s,a) to a sucessive state (s',a',d).
Bottom: Initial and Final state of the Turing Machine.
s a s' a' ds1 b s2 b ls2 b s3 b ls2 1 s2 1 ls3 b H b 0s3 1 s4 b rs4 b s2 1 l
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Circuit Model of Computation
Circuits are made of wires and gates. Each wire carries one bit of information Gates perform logical operations on one or more
wires.In any computation, a circuit can be represented
as mapping a n-bit input to a m-bit output.
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Logic Gates
AND – outputs 1 iff both inputs
are 1OR - outputs 0 iff both inputs
are 0NOT -outputs 1 iff input is 0
outputs 0 iff input is 1
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Logic Gates
FANOUT - outputs two bits
identical to input.AND,OR,NOT, and
FANOUT constitute a universal set of gates. That is, any n-bit to m-bit function can be represented by them.
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Gates as Linear Operators
Consider the NOT gate, which returns the opposite of the input.
We may represent this gate as a 2x2 matrix.
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Example: Binary Addition
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Binary Addition
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Quantum Computation
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Qubits
2 level system in a superposition of bit states.
We will manipulate qubits in a manner analogous to the circuit model of classical computation.
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Representing a Two-Qubit Input
Given two wires going into a gate, you can represent both bits as the tensor product of the two bit.
Two 2-dimensional vectors becomes one 4-dimensional vector.
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Representing Two Qubits
We can write a basis for a two qubit system as the tensor products of the individual qubit's basis vectors
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Entanglement of Two Qubits
A two-qubit state is separable if it can be written as the tensor product of two one-qubit states.
A two-qubit state is entangled otherwise.
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n-Qubit Quantum Register
A quantum computer may be thought of as a collection of n-qubits called a quantum register.
A n-qubit quantum register has a basis of 2n allowed states, and any state of the quantum computer is in a superposition of these states.
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Prerequisites for Quantum Computation
Be able to prepare system in a well defined initial state.
Be able to manipulate the wave function via unitary transformations.
Be able to measure the final states of each qubit.
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Hadamard Gate (Single Qubit)
Changes the basis to a superposition of the computation basis.
Hadamard gate is a Hermitian operator.
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Controlled Gates (CNOT)
The first qubit acts as the control qubit.
The second qubit acts as the target qubit.
Flips the state of the target qubit if the first qubit is found in a certain state. ( |1>)
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Quantum CNOT Gate
Can be used to introduce entangled state into a two-qubit system.
CNOT along with some of its variants form a universal set of quantum logic gates.
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Deutsch's Algorithm (1992)
Acts on a boolean functionIs the function balanced?Is the function constant?Can determine whether the
function is balanced or constant in one step!
Classical computers require two runs!
x f1 f2 f3 f40 0 1 0 11 0 0 1 1
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Deutsch's Algorithm (1992)
Start with |01> stateApply Hadamard gates
to both qubits.Apply U, then another
Hadamard gate on the first bit.
Measure the first bit.|0> => f is constant.
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Shor's Algorithm (1994)
A hybrid algorithm that is able to a factor a large integer n in polynomial time.
We want to factor a large composite integer n.Find the periodicity of F(a) = xa (mod n) for x relatively
prime to n.Test all the exponents 'a' up to a power of two between
n2 and 2n2 using quantum parallelism.Given this period we can more easily determine possible
factors.
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Other Curiosities
Grover's sorting algorithm (1996)Searching through an unsorted databaseBest classical algorithm is brute force
Universal quantum simulatorProposed by Richard Feynman (1982)Quantum simulation requires exponential resources
classicallyCan be done efficiently with a quantum computer.
(Seth Lloyd - 1996)
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Implementation
Quantum logic gate implemented (1995 - NIST)C-NOT gate using trapped ions
Quantum algorithm – 7 qubits (2001 - IBM)Used Shor's algorithm to factor 15
Largest qubit register – 12 qubits (2006 ICQ)Benchmarked a 12 qubit register
Decoherence – the bane of quantum computing
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Summary
Quantum computing is promising fieldEfficient solutions to classically difficult problemsInevitable conclusion of Moore's lawDecryption & EncryptionComputer modeling of quantum mechanicsPhysical implementation requires solving or
bypassing the problem of decoherence.
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Works Cited
“Principle of Quantum Computation and Information” - Benenti, et al.
“An Introduction to Quantum Computing” - Kaye, et al.
“Quantum Computing and Shor's Algorithm” - Hayward (http://alumni.imsa.edu/~matth/quant/299/paper/)
Images from Wikipedia (http://en.wikipedia.org/*)
Slide 1 – Bloch Sphere; Slide 6 – BQP; Slide 8 – Moore's Law; Slide 24 – Turing Machine; Slides 31/32 – Logic Gates;
Actual Turing machine from (http://aturingmachine.com)
LaTeX by the Online LaTeX Generator: (http://codecogs.com/)