CSI 789002 Quantum Computation

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Quantum Computation
Dr. Richard B. Gomez rgomez_at_gmu.edu

Introduction to Quantum Computing
Lecture 3
George Mason University School of Computational
Sciences
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Quantum Computing Quantum Mechanics
OverviewWhats Quantum Mechanics All About?

3
Quantum ComputingReview of Fundamental Quantum
Concepts
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Classical vs. Quantum Computing

• For any digital computer, its set of
  computational states is some set of mutually
  distinguishable abstract states
• The specific computational state that is in use
  at a given time represents the specific digital
  data currently being processed within the machine
• Classical computing is computing in which
• All of the computational states (at all times)
  are pointer states of the computer hardware
• Quantum computing is computing in which
• The computational state is not always a pointer
  state

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What is Quantum Computing?

• Non-pointer-state computing
• Harnesses these quantum effects on a large,
  complex scale
• Computational states that are not just pointer
  states, but also, coherent superposition of
  pointer (observable) states
• States having non-zero amplitude in many pointer
  states at the same time! Quantum parallelism
• Entanglement (quantum correlations)
• Between the states of different subsystems
• Unitary (thus reversible) evolution through time
• Interference (reinforcement and cancellation)
• Between convergent trajectories in pointer-state
  space

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Why Quantum Computing?

• It is, apparently, exponentially more
  time-efficient than any possible classical
  computing scheme at solving some problems
• Factoring, discrete logarithms
• Simulating quantum physical systems accurately
• This application was the original motivation for
  quantum computing research first suggested by
  famous physicist Richard Feynman in the early
  80s
• However, this has never been proven yet!
• If you want to win a sure-fire Nobel prize find
  a polynomial-time algorithm for accurately
  simulating quantum computers on classical ones

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Status of Quantum Computing

• Theoretical experimental progress is being
  made, but slowly
• There are many areas where much progress is still
  needed
• Physical implementations of very small (e.g.,
  7-bit) quantum computers have been tested and
  work as predicted
• Scaling them up is difficult
• There are no known fundamental theoretical
  barriers to large-scale quantum computing
• Guess It will be a real technology in 20 yrs.
  or so

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Early History

• Quantum computing was largely inspired by
  reversible computation work from the 1970s
• Bennett, Fredkin, Toffoli, Margolus
• Early quantum computation pioneers (1980s)
• Early models not using quantum parallelism to
  gain performance
• Benioff 80, 82 - Quantum serial TM models
• Feynman 86 - Q. models of serial reversible
  circuits
• Margolus 86,90 - Q. models of parallel rev.
  circuits
• Performance gains w. quantum parallelism
• Feynman 82 - Suggested faster quantum sims with
  QC
• Deutsch 85 - Quantum-parallel Turing machine
• Deutsch 89 - Quantum logic circuits

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More Recent History

• There was a rapid ramp-up of quantum computing
  research throughout the 1990s
• Some developments, 1989-present
• Refining quantum logic circuit models
• What is a minimal set of universal gates for QC?
• Algorithms Shor factoring, Grover search, etc.
• Developing quantum complexity theory
• What is the ultimate power of quantum
  computation?
• Quantum information theory
• Communications, Cryptography, etc.
• Error correcting codes, fault tolerance, robust
  QC
• Physical implementations
• Numerous few-bit implementations demonstrated

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Quantum Logic Networks

• Invented by Deutsch (1989)
• Analogous to classical Boolean logic networks
• Generalization of Fredkin - Toffoli reversible
  logic circuits
• System is divided into individual bits, or qubits
• 2 orthogonal states of each bit are designated as
  the computational basis states, 0 and 1
• Quantum logic gates
• Local unitary transforms that operate on only a
  few state bits at a time
• Computation via predetermined seq. of gate
  applications to selected bits

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Gates without Superposition

• All classical input-consuming reversible gates
  can be represented as unitary transformations
• E.g., input-consuming NOT gate (inverter)

in out0 11 0
in
out
in
out
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Controlled-NOT

• Remember the CNOT gate?

A
A
A
A
B
B A?B
B
B A?B
Example
A B
A B
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Toffoli Gate (CCNOT)
A B C A B C0 0 0 0 0 00 0 1
0 0 10 1 0 0 1 00 1 1 0
1 11 0 0 1 0 01 0 1 1 0
11 1 0 1 1 01 1 1 1 1 1
A
AA
B
BB
A
A
B
B
C
C C?AB
C
C
(XOR)
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The Square Root of NOT

• If you put in either basis state (0 or 1) you get
  a state that appears random when measured
• But if you feed the output back into another N1/2
  without measuring it, you get the inverse of the
  original value!
• How is thatpossible?

0 (50)
0 (50)
0
1
N1/2
N1/2
1 (50)
1 (50)
0 (50)
0
1
N1/2
N1/2
1 (50)
0 (50)
0
0
N1/2
N1/2
1 (50)
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Key Points to Remember

• An abstractly-specified system may have many
  possible states only some are distinguishable
• A quantum state/vector/wavefunction ? assigns a
  complex-valued amplitude ?(si) to each
  distinguishable state si (out of some basis set)
• The probability of state si is ?(si)2, the
  square of ?(si)s length in the complex plane
• States evolve over time via unitary (invertible,
  length-preserving) transformations
• Statistical mixtures of states are represented by
  weighted sums of density matrices ?????

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System Descriptions

• Classical physics
• A system could be completely described by giving
  a single state S out of the set ? of all possible
  states
• Statistical mechanics
• Give a probability distribution function
  p??0,1 stating that the system is in state S
  with probability p(S)
• Quantum mechanics
• Give a complex-valued wavefunction ?? ? C,
  ?(S)?1, implying the system is in state S with
  probability ?(S)2

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State Vectors Hilbert Space

• Let S be any maximal set of distinguishable
  possible states s, t, of an abstract system A
• I.e., no possible state that is not in S is
  perfectly distinguishable from all members of S
• Identify the elements of S with unit-length,
  mutually-orthogonal (basis) vectors in an
  abstract complex vector space H, i.e., the
  Hilbert space
• Postulate 1 The possible states ? of Acan be
  identified with the unitvectors of H

t
s
?
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Hilbert Space

• A Hilbert space is a vector space in which the
  scalars are complex numbers, with an inner
  product (dot product) operation ? HH ? C
• See Hirvensalo p. 107 for definition of inner
  product
• x?y (y?x) ( complex conjugate)
• x?x ? 0
• x?x 0 if and only if x 0
• x?y is linear, under scalar multiplication
  and vector addition within both x and y

Componentpicture
y
Another notation often used
x
x?y/x
bracket
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Review The Complex Number System

• It is the extension of the real number system via
  closure under exponentiation.
• (Complex) conjugate
• c (a bi) ? (a ? bi)
• Magnitude or absolute value
• c2 cc a2b2

i
The imaginaryunit
c
b

?
a
Real axis
Imaginaryaxis
?i
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Review Complex Exponentiation

• Powers of i are complex units
• Note
• e?i/2 i
• e?i ?1
• e3? i /2 ? i
• e2? i e0 1

e?i
i
?
?1
1
?i
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Vector Representation of States

• Let Ss0, s1, be a maximal set of
  distinguishable states, indexed by i.
• The basis vector vi identified with the ith such
  state can be represented as a list of numbers
• s0 s1 s2 si-1 si si1
• vi (0, 0, 0, , 0, 1, 0, )
• Arbitrary vectors v in the Hilbert space can then
  be defined by linear combinations of the vi
• And the inner product is given by

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Diracs Ket Notation

• Note The inner productdefinition is the same as
  thematrix product of x, as aconjugated row
  vector, timesy, as a normal column vector.
• This leads to the definition, for state s, of
• The bra ?s means the row matrix c1 c2
• The ket s? means the column matrix ?
• The adjoint operator takes any matrix Mto its
  conjugate transpose M ? MT, so?s can be
  defined as s?, and x?y xy.

Bracket
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Distinguishability of States

• State vectors s and t are (perfectly)
  distinguishable or orthogonal (write s?t) iff
  st 0. (Their inner product is zero.)
• State vectors s and t are perfectly
  indistinguishable or identical (write st) iff
  st 1. (Their inner product is one.)
• Otherwise, s and t are both non-orthogonal, and
  non-identical. Not perfectly distinguishable.
• We say, the amplitude of state s, given state t,
  is st. Note amplitudes are complex numbers.

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Probability and Measurement

• A yes/no measurement is an interaction designed
  to determine whether a given system is in a
  certain state s.
• The amplitude of state s, given the actual state
  t of the system determines the probability of
  getting a yes from the measurement.
• Postulate 2 For a system prepared in state t,
  any measurement that asks is it in state s?
  will return yes with probability Prst
  st2
• After the measurement, the state is changed, in a
  way we will define later.

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A Simple Example

• Suppose abstract system S has a set of only 4
  distinguishable possible states, which well call
  s0, s1, s2, and s3, with corresponding ket
  vectors s0?, s1?, s2?, and s0?.
• Another possible state is then the vector
• Which is equal to the column matrix
• If measured to see if it is in state s0,we have
  a 50 chance of getting a yes.

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Wavefunctions

• Given any set S of system states (whether all
  mutually distinguishable, or not),
• A quantum state vector can also be translated to
  a wavefunction ? S ? C, giving, for each state
  s?S, the amplitude ?(s) of that state.
• When s is another state vector, and the real
  state is t, then ?(s) is just st.
• ? is called a wavefunction because its time
  evolution obeys an equation (Schrödingers
  equation) which has the form of a wave equation
  when S ranges over a space of positional states.

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Linear Operators

• V,W Vector spaces.
• A linear operator A from V to W is a linear
  function AV?W. An operator on V is an operator
  from V to itself.
• Given bases for V and W, we can represent linear
  operators as matrices.
• An operator A on V is Hermitian iff it is
  self-adjoint (AA), its diagonal elements are
  real.

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Eigenvalues Eigenvectors

• v is called an eigenvector of linear operator A
  iff A just multiplies v by a scalar x, i.e. Avxv
  
• eigen (German) characteristic
• x, the eigenvalue corresponding to eigenvector v,
  is just the scalar that A multiplies v by
• x is degenerate if it is shared by 2 eigenvectors
  that are not scalar multiples of each other
• Any Hermitian operator has all real-valued
  eigenvectors, which are orthogonal (for distinct
  eigenvalues)

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Observables

• Hermitian operator A on V is called an observable
  if there is an orthonormal (all unit-length, and
  mutually orthogonal) subset of its eigenvectors
  that forms a basis of V
• Postulate 3 Every measurable physical property
  of a system is described by a corresponding
  operator A. Measurement outcomes correspond to
  eigenvalues.
• Postulate 4 The probability of an outcome is
  given by the squared absolute amplitude of the
  corresponding eigenvector(s), given the state.

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Compound Systems

• Let CAB be a system composed of two separate
  subsystems A,B each with vector spaces A, B with
  bases ai?, bj?
• The state space of C is a vector space CA?B
  given by the tensor product of spaces A and B,
  with basis states labeled as aibj?
• E.g., if A has state ?aca0a0 ? ca1
  a1?,while B has state ?bcb0b0 ? cb1 b1?,
  thenC has state ?c ?a??b ca0cb0a0b0?
  ca0cb1a0b1? ca1cb0a1b0? ca1cb1a1b1?

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Entanglement

• 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, i.e.,
• 00?01?10?11?(0?1?)?(0?1?)
• Otherwise, A and B are entangled (basically,
  correlated) their states are not independent,
  i.e.,
• 00?11?

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Unitary Transformations

• A matrix (or linear operator) U is unitary iff
  its inverse equals its adjoint U?1 U
• Some properties of unitary transformations
• Invertible, bijective (both injective and
  surjective), one-to-one correspondence
• The set of row vectors is orthonormal
• The set of column vectors is orthonormal
• Preserves vector length U? ?
• Therefore, also preserves total probability over
  all states
• Corresponds to a change of basis, from one
  orthonormal basis to another
• Or, to a generalized rotation of? in Hilbert
  space

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Time Evolution

• Postulate 5 (Closed) systems evolve (change
  state) over time via unitary transformations.
• ?t2 Ut1?t2 ?t1
• Note that since U is linear, a small-factor
  change in amplitude of a particular state at t1
  leads to a correspondingly small change in the
  amplitude of the corresponding state at t2
• Chaos (sensitivity to initial conditions)
  requires an ensemble of initial states that are
  different enough to be distinguishable (in the
  sense we defined)
• Indistinguishable initial states never beget
  distinguishable outcomes - ?analog computing is
  infeasible?

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After a Measurement?

• After a system or subsystem is measured from
  outside, its state appears to collapse to exactly
  match the measured outcome
• the amplitudes of all states perfectly
  distinguishable from states consistent w. that
  outcome drop to zero
• states consistent with measured outcome can be
  considered renormalized so their probs. sum to
  1
• This collapse seems nonunitary ( nonlocal)
• However, this behavior is now explicable as the
  expected consensus phenomenon that would be
  experienced even by entities within a closed,
  perfectly unitarily-evolving world (Everett,
  Zurek).

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Density Operators

• For a given state ??, the probabilities of all
  the basis states si are determined by an
  Hermitian operator or matrix ? (the density
  matrix)
• The diagonal elements ?i,i are the probabilities
  of the basis states.
• The off-diagonal elements are coherences.
• The density matrix describes the state exactly.

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Mixed States

• Suppose one only knows of a system that it is in
  one of a statistical ensemble of state vectors vi
  (pure states), each with density matrix ?i and
  probability Pi. This is called a mixed state.
• This ensemble is completely described, for all
  physical purposes, by the expectationvalue
  (weighted average) of density matrices
• Note even if there were uncountable many vi,
  the state remains fully described by ltn2 complex
  numbers, where n is the number of basis states!

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Entropy

• Trace Tr means sum of diagonal elements
• ln of a matrix M denotes the inverse function to
  exp(M).
• Exponential of a matrix M is defined via the
  Taylor-series expansion of the exp function.

(Shannon)
(Boltzmann)
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Pointer States

• For a given system interacting with a given
  environment,
• The system-environment interactions can be
  considered measurements of a certain observable
  of the system by the environment, and vice-versa
• For each observable there are certain basis
  states that are characteristic of that observable
• The eigenstates of the observable
• A pointer state of a system is an eigenstate of
  the system-environment interaction observable
• The pointer states are the inherently stable
  states