Introduction to Quantum Computation

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Introduction to Quantum Computation
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What is a qubit

• A quantum bit (qubit) is a quantum system with 2
  discrete states. (0 and 1)
• Possible states of a qubit
• 0
• 1
• Superposition of 0 and 1

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Mathematical framework to describe a quantum
system

• States of a quantum system are represented by
  vectors in a Hilbert space.
• Hilbert space
• An complex vector space with an inner product
  defined on it. (also known as inner product
  space)
• It is also complete with respect to the norm.

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Hilbert space

• Complex Vector space
• A set of vectors
• Complex numbers are used
• Inner product
• A mapping from 2 vectors in a vector space to a
  complex number.
• Given that Vector space is V, then inner product
  is
• (u,v) V x V -gt C

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• Inner product
• A function has to satisfy several axioms in order
  to qualify as an inner product.
• Most commonly used is the standard inner product
  (also known as the dot product)

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Bases

• Orthogonal Basis of a Vector space
• If a set of vectors v1, v2,vk in Vector space
  V, forms a basis of V, then every vector u, in V
  is a linear combination of v1, v2,vk
• I.e.
• All vectors in an orthogonal basis are orthogonal
  (perpendicular) to each other.
• Note If u and v are orthogonal to each other,
• then (u,v) 0

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Bases

• Orthonormal Basis of a vector space
• Same as orthogonal basis except that all the
  vectors in the basis are of length 1 (unit
  length)
• Note Given
• length of a vector (norm),

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Qubit

• Since a qubit can be in a superposition of 2
  values, the basis used should have 2 vectors.
• A qubit is represented by a vector in the Hilbert
  space H2.
• The dimension of H2 is 2, meaning that all bases
  of H2 have 2 vectors each.

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Qubit

• An orthonormal basis for H2 is
• We can represent it using Dirac notation as

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Qubit

• Hence, every qubit can be represented as
• a and ß are called the amplitude of the state.
• If both are not 0, the above equation represents
  a qubit in a superposition of 0 and 1.

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Tensor Product

• Tensor product is used to combine the Hilbert
  space of individual quantum systems
• Used to represent states of systems comprising of
  several quantum systems

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• For example Given 2 qubits
• Note
• Tensor product is not commutative

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Tensor Product

• The product state lies in H4 with basis
• In general with n qubits,
• the dimension of the required Hilbert space is
  2n.
• Each new state can be written as

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Measurement

• For a qubit
• Measurement of a qubit will result in
• Value 0 with probability
• Value 1 with probability
• Hence

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Measurement

• Measurement alters the original state to one of
  the vectors in the basis of measurement.
• After measurement, original superposition state
  cannot be recovered.
• Example Measuring
• will result in with
  probability 0.5
• Then measuring
• will result in with probability 1

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Some Properties of Quantum Systems

• No Exact Measurement
• No Cloning
• Entanglement

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No Exact Measurement

• Given an arbitrary qubit,
• we cannot obtain its amplitudes by measuring it
• Measuring a qubit results in a discrete value of
  0 or 1
• Can only use measurement to distinguish between
  orthogonal states.
• I.e. Given 2 orthogonal states
• and our qubit is in one of these states, we can
  use measurement to determine which one

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No Cloning Theorem

• It is impossible to create copies of an arbitrary
  quantum state.
• I.e. Given an arbitrary state, ?, there doesnt
  exist an operator U such that

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• Proof
• Assume there exist a cloning operation U.
• 2 ways of simplifying
• The 2 ways mean the same thing, yet they yield
  different results. Hence, there does not exist a
  cloning operation U.

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Entanglement

• If 2 particles are entangled, measuring one of
  them will lead to a correlated result when the
  other is measured.
• While measurement of the 1st particle gives a
  random result, the state of the 2nd particle will
  be fixed.
• Example Assume 2 qubits are entangled, If
  measuring the 1st qubit gives a 0 (1) then
  measuring the 2nd qubit will yield a 0 (1)
  too.

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• In general for a system consisting of 2 qubits
• Measuring the 1st qubit will give 0 with
  probability
• The resulting post measurement state is

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Illustration of Entanglement

• Given the following 2 un-entangled qubit
• Measuring the 1st qubit will give 0 with
  probability
• The resultant state is
• The state of 2nd qubit is in a superposition. Its
  state is not fixed by measurement of 1st qubit.

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Illustration of Entanglement

• Given the following entangled 2 qubit
• Measuring the 1st qubit will give 0 with
  probability
• The resultant state is
• The state of 2nd qubit is fixed at 0. Its state
  is determined by measurement of 1st qubit.

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Entanglement

• An quantum system is entangled if it cannot be
  written as a tensor product of its qubits.
• Example
• An un-entangled system
• An entangled system (Cannot be simplified)

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Reversible Operations

• The state of a quantum system may change due to
  time (Time evolution) or by performing an
  operation on it.
• To preserve properties of superposition and
  entanglement, all evolutions of the quantum
  system must be unitary.
• An unitary transformation, U, satisfy

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Reversible Operations

• An operator, U, can be represented as a matrix.
• If U is an unitary operation, then its
  corresponding matrix is unitary.
• Note A matrix is unitary if its inverse is equal
  to its conjugate transpose.
• Applying an operator on a quantum state is
  represented as multiplying a state vector with
  the operator matrix.

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Reversible Operations

• Since operations are unitary, an important
  consequence is that they are reversible.
• Any operation that loses information is not
  reversible and hence not unitary.

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• Example of irreversible operation
• NAND gate
• Output bit cannot give the original input bits.
• Information is lost. By Landauers principle,
  energy is dissipated
• Example of reversible operation
• Toffoli gate (reversible NAND gate)
• No Information lost. No dissipation and power
  expenditure.
• All computation can be made reversible, but
  requires extra bits.

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Difference between Quantum Bit and Classical Bits

• Quantum parallelism
• Qubits cannot be copied
• Reading a qubit changes its state
• Entanglement

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Quantum parallelism

• It is very difficult to simulate a quantum
  computer using a classical computer.
• Supposing a quantum computer operates on N
  qubits. To represent a quantum state of this
  computer, we need 2N complex numbers. Impossible
  for large N.
• When N is large, performing operations (matrix
  multiplication) can be computationally
  infeasible.

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Quantum parallelism

• However looking from an opposite angle, this
  suggest that a quantum computers can perform
  massive amount of classical computation
  efficiently.

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• In general, given
• a function f(x), that takes in N bits
• a quantum transformation
• We can prepare N qubits in the following
  superposition
• This superposition encodes all possible input

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• By applying f once on this superposition
• This superposition encodes all values of f(x).
• All values of f(x) are computed in parallel
• However cannot observe all values of f(x).
• If measurement is made, the superposition will
  collapses and only 1 value of f(x) can be
  obtained.
• Global information about the function can,
  however, be extracted from this state.

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Example of quantum parallelism Deutschs
algorithm

• The problem
• Given a function f 0,1 -gt0,1, compute f(0) Å
  f(1) by calling f once only.
• A classical computer has to call f twice.
• But a quantum computer running Deutschs
  algorithm is able to accomplish this.

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Example of quantum parallelism Deutschs
algorithm

• Operations needed
• Hadamard operator
• Reversible version of f, Uf

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Example of quantum parallelism Deutschs
algorithm

• The algorithm
• Perform a hadamard operator on
• Apply Uf to both qubits
• Apply hadamard to 1st qubit
• Measure 1st qubit

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Deutschs algorithm

• Perform a hadamard operator on
• 2. Apply Uf to both qubits

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• 3. Apply hadamard to 1st qubit

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Quantum parallelism

• A classical computer is limited to computing
  either f(0) or f(1).
• A quantum computer acts on a superposition of 0gt
  and 1gt and extracts global information about the
  function (I.e. information that depends on both
  f(0) and f(1)
• This is quantum parallelism.

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No copying of qubits.

• Classical bits can be easily copied.
• The no cloning theorem states that an arbitrary
  qubit can not be copied.
• Since an arbitrary qubit cannot be copied and its
  internal state cannot be measured, hence it cant
  be broadcasted.
• This limitation can be used to our advantage
  E.g. in Quantum key exchange

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Reading of bits.

• Reading a classical bit leaves it unchanged.
• A qubit can only be read once after which
  measurement will cause the superposition state to
  collapse.
• Also the no exact measurement theorem states that
  a internal state of a qubit in a superposition
  cannot be measured.
• This limitation can be used to our advantage
  E.g. in Quantum key exchange

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Entanglement

• Ability of particles to be entangled gives rise
  to many interesting and useful applications, e.g.
  quantum teleportation.

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Randomness

• True randomness extremely hard to achieve in
  classical computers.
• Randomness is an inherent property of quantum
  computation.
• If we prepare a qubit in this superposition
• Then measuring it will yield 0 (or 1) with
  probability of 0.5. Result is totally
  unpredictable.
• Can be used to build true random number
  generators.