Introduction to Quantum Computing

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Quantum Computation
Dr. Richard B. Gomez rgomez_at_gmu.edu

Introduction to Quantum Computing
Lecture 9
George Mason University School of Computational
Sciences
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Quantum Mechanics Courses and Superposition Or
why Students in QIS should study QM Maria
DworzeckaProfessor of Physics and Chairman of
Department of the Physics and Astronomy George
Mason UniversitySeminar on Quantum Information
Systems October 24th, 2005
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From Cbits to Qbits Teaching computer scientists
quantum mechanicsN. David Mermin

• Quantum information scientists need quantum
  mechanics
• But how much quantum mechanics? Here are two
  possible views from

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• I remarked to the Director that I spent the
  first four or five lectures of my course2 on
  quantum computation teaching the necessary
  quantum mechanics to the computer scientists in
  the class. His response was that any application
  of quantum mechanics that could be taught after
  only a four hour introduction to the subject
  could not have serious intellectual content.
  After all, he remarked, it takes any physicist
  years to develop a feeling for quantum
  mechanics.

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• I do not know it is for you to decide but here
  are few options of courses we are offering
• Physics 402/502 Introduction to Quantum Mechanics
  taught every fall combined course for
  undergraduate physics majors and graduate
  students with little previous preparation

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• Physics 632/732 ( to be approved)
• Advanced Quantum Mechanics
• New graduate 2 semesters sequence
• Offered every year or every second year
  (depends on students needs)

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• Both courses cover similar topics but of
  different mathematical and physical depths
• Typical topics ( incomplete)
• Postulates of QM
• Operators
• Superposition
• 1D solutions to Schrödinger equation
• Spin states

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Textbooks used

• Griffiths
• Liboff
• Sakurai
• Shankar
• Depends on instructor

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Postulate 1

• How to describe quantum states of a closed system
  ?
• At each instant the state of a physical system
  is represented by a (ket) state vector ?gt which
  lies in the Hilbert space. All information about
  the system is contained in the state vector
• This postulate implies that the superposition of
  two states is again a state of the system. If ?1
  gt and ?2 gt are possible states of a system, then
  so is
• ? gt a1?1gt a2?2gt
• where a1 and a2 are complex numbers.

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Postulate 2

• How to describe measurements of a quantum system
  ?
• Every observable A of a physical system is
  associated with an operator A such that the
  measurement of A yields values a which are
  eigenvalues A.
• Aagtaagt

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Operators

• The form of operators is a part of the postulate
  2.
• There are 3 basic operators
• All others can be constructed from these three
• X - position operator
• P - momentum operator
• S spin operator
• ( vectors in 3D)

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How to define general operators

• Take classical function, say
• f(x,p)? f(x,p)
• keep the form and substitute for x and p the
  operator x and p
• Example Hamiltonian ( total energy of the
  physical system in functional form)
• H p2 /2m v(x)

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Operators in x representation

• Operator x become multiplication by x
• x f(x) x f(x)
• Operator p becomes derivative with respect to x
• p f(x) -ih (?f/?x,...)
• hence H becomes
• Hf(x) -h2 /2m ? 2 f/?x2 v(x)f(x)

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Operators without classical equivalence

• Spin operators are represented by matrix
• Size of matrix depends on postulated ( measured)
  value of spin for given system
• Spin is quantized that why size of the matrix
  is defined by the spin value
• Electron spin 1/2h
• Photon spin 1h

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How do we know that spin is quantized

• Stern Gerlach experiment electron spin is a
  vector, with two and only two projections on the
  specific axis's which has values h/2 and - h/2
• Theory angular momentum is quantized is
  derived from commutation relations among
  components of angular momentum values are the
  same as experimental

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Stern-Gerlach experiment spin of electron
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Postulate 3

• What is the effect of measurement on the state of
  the system
• The measurement of the observable A which yields
  the value a leaves the system in the state agt
  which is the eigenstate of an operator A

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Magnetic field is along z-axis
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Explanation of the Stern-Gerlach experiment

• Use classical analogy to construct operator
  associated with Stern-Gerlach experiment
• H -µB? H- µ0 SB
• where µ0 is a constant
• Solve eigenvalue problem and use the results of
  experiment to find S matrix

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Eigenvalue problem

• Aagtaagt
• Operator (eigenfunction) eigenvalue (
  eigenfuncion)
• Assume that field B is along z-axis
• µ0 SzB gt - µ0 B h/2gt
• - µ0 SzB - gt - µ0 B(-h/2)-gt

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or in matrix notation

• a b 1 h/2 1
• c d 0 0
• then
• a h/2 , c 0
• and from other eigenvalue
• b0 , d -h/2

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• These two vectors span 2D vector space

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• These four matrices span the2x2 complex matrix
  space

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• So Sz is one of base matrices
• and similarly all other are also proportional to
  spin matrices or Pauli matrices

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Postulate 4

• How to describe quantum dynamics?
• The state of the system evolves in time by the
  time dependent Schrödinger equation
• H ?gt (ih/2p ) ?/?t?gt
• where H is the Hamiltonian of the system

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Postulate 1 example in the language of qubits
photons electron spin nuclear spin etcetera
0gt gt 1gt -gt
Normalization
All we do is draw little arrows on a piece of
paper - that's all. - Richard Feynman
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QUBIT states

• All states of a qubit are superpositions of two
  basis states
• The superposition is a purely quantum property
  unlike a classical bit, a qubit can have an
  infinite of states

Thaller
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Spinor Wavefunction

• It makes no sense to speak about the position of
  the qubit when the cs are just numbers
• But if we let the cs depend on position, then
  can get a spinor wavefunction

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Postulate 2 and 3 measuring a qubit
Einstein God doesnt play dice
Measuring in the computational basis
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Postulate 2 and 3 -Measuring a qubit
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Postulate 4 dynamics quantum logic gates
Quantum not gate
Input qubit
Output qubit
Matrix representation
Dynamics of a closed quantum system (including
logic gates) can be represented as a unitary
matrix.
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Pauli gates
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qcomputers use multiple-qubit systems
Measurement in the computational basis
General state of n qubits
classical
Hilbert

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The measurement problem
Quantum system
Measuring apparatus
Rest of the Universe
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Postulate 4
The state space of a composite physical system is
the tensor product of the state spaces of the
component systems.
Example
Properties
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Compound Systems tensor products
A tensor product is a larger vector space formed
from two smaller ones simply by combining
elements from each in all possible ways that
preserve both linearity and scalar multiplication
If V is a vector space of dimension n W is a
vector space of dimension m then V W is a vector
space of dimension mn

• E.g. let CA?B be a system composed of two
  subsystems A,B each with vector spaces A, B with
  bases ai?, bj?

Alice
Bob
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Compound Systems

• E.g., if Alice has state ?aa0? ß1?,while
  Bob has state ?b?0? d1?, then Cs state is

• Product states are separable because each qubit
  has a well-defined state

Thaller
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• E.g. if the state of compound system C can be
  expressed as a tensor product of states of two
  independent subsystems A and B, i.e.,
  ?c ?a ? ?b
• Then, we say that A and B are not entangled, and
  they have individual states, e.g.,
• 00?01?10?11? OR 00?0??0?
• (0?1?)?(0?1?)

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Some conventions implicit in Postulate 4
Alice
Bob
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Examples
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In general superpositions of product states are
Quantum entangled
Alice
Bob
Impossible to write this as product of states
Schroedinger (1935) I would not
call entanglement one but rather the
characteristic trait of quantum mechanics, the
one that enforces its entire departure from
classical lines of thought.
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Bell states non-separable kinematic nonlocality
In this example, no arrows are specified because
each individual QUBIT does not have a separate
state. The 2 QUBITS, together, have a single
state, but not individually cannot understand
as being a composite of a state of Alice and
state of Bob
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History of Entanglement and nonlocality I

• Einstein-Podolsky-Rosen (1935) God does not
  play dice suppose we have a wavefunction

• Same preparation (same wavefunction) different
  results
• Connection between causes and effects (basis of
  science) violated by quantum mechanics

• Bohm (1950), Bell (1964) and Aspect (1982)
  entanglement indicates that no locally realistic
  theory of the world (i.e. classical) is possible

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History of Entanglement and nonlocality II

• EPR QM is incomplete - Hidden variables exist
  which can explain why different results are
  obtained given identical preparations

Alice
Bob

• QM says that the particles state is not definite
  until hits measuring device so send Bob far
  away
• No time for Alices particle to inform Bobs
• Always green-green or red-redHow possible?

• Bohm (1950), Bell (1964) and Aspect (1982)
  entanglement indicates that no locally realistic
  theory of the world (i.e. classical) is possible

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History of Entanglement and nonlocality III

• EPR solution particles agree at the source (when
  together) that ½ the time they are green-green

Alice
Bob

• The other ½ the time they are red-red

• But QM never talks about these hidden variables

• Bohm (1950), Bell (1964) and Aspect (1982)
  entanglement indicates that no locally realistic
  theory of the world (i.e. classical) is possible

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History of Entanglement and nonlocality IV

• Bohm (1950), Bell (1964) and Aspect (1982)
  entanglement indicates that no locally realistic
  theory of the world (i.e. classical) is possible

Alice
Bob

• Any run might be XY or YX, so if 1 particle
  allows Y detector to flash G, its companion must
  require a X det to flash R (otherwise XG, YG or
  YG, XG)
• Thus in any YY run in which both flash G, then
  each particle must require an X detector to flash
  R. Thus, if XX were measured then both detectors
  would have flashed R
• But XR, XR is never observed