Quantum Computation and Quantum Information Lecture 3

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Quantum Computation and Quantum Information
Lecture 3

• Part 1 of CS406 Research Directions in Computing

Nick Papanikolaou
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Motivation

• Quantum computers are built from wires and logic
  gates, just as classical computers are
• The potential of such devices stems from the
  ability to manipulate superpositions of states
• Quantum algorithms solve problems which are not
  known to be solvable classically!

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Lecture 3 Topics

• Quantum logic gates
• Simple quantum circuits
• Quantum teleportation as a circuit
• Deutschs quantum algorithm

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Quantum vs. classical gates

• The simplest boolean gate is NOT, with truth
  table
• Quantum gates have to be defined not only on the
  equivalents of 0 and 1, but on their
  superpositions too!

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Quantum NOT gate Linearity

• Suppose we define a quantum NOT gate as follows
• The action of the quantum NOT gate on a
  superposition must then be
• All quantum operations are linear

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The NOT Gate as a Matrix

• Because all quantum operations have to be linear,
  we can represent the action of a quantum gate by
  a matrix
• The quantum NOT, or Pauli-X gate, is written

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Quantum State Vectors

• Remember that a quantum state is represented by a
  vector
• Notation

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Quantum NOT

• We can express the NOT operation on a general
  qubit as matrix multiplication

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Other Single Qubit Gates

• The Pauli-X gate works on only one qubit
• Other common single qubit gates are
• Pauli-Z gate
• Pauli-Y gate
• Hadamard gate

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Summary of Simple Gates
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Reversibility Requirement

• All quantum operations have to be reversible
• Boolean operations are not necessarily so
• A reversible operation is always given by a
  unitary matrix, i.e. one for which

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The Controlled NOT Gate

• The CNOT gate is the standard two-qubit quantum
  gate
• It is defined like this

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The Controlled NOT Gate (2)

• CNOT is a generalisation of the classical XOR
• The CNOT gate is drawn like this

control qubit
target qubit
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The Controlled NOT Gate (3)

• The matrix corresponding to the CNOT gate is
• The CNOT together with the single qubit gates are
  universal for quantum computing

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Quantum Circuits

• Using the conventions for control and target
  qubits, we can build interesting circuits
• Example A Qubit Swap Circuit

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Qubit Swap Circuit
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Features of Quantum Circuits

• No loops are allowed quantum circuits are
  acyclic
• Fan-in is not allowed
• Fan-out is not allowed

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Generalised Control Gate

• Any quantum gate U can be converted into a
  controlled gate

One control qubit
n target qubits
U
If the control qubit is high, U is applied to
the targets. CNOT is the Controlled-X gate!
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Quantum Measurement

• Measurement in a quantum circuit is drawn as

M
(classical bit representing outcome of
measurement)
M 0 with prob. or M 1 with prob.
If
then
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A Qubit Cloning Circuit?

• Using the XOR gate, it is possible to copy a
  classical bit

Can we build a quantum circuit that performs does
this with qubits?
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A Qubit Cloning Circuit? (2)
OK here
entangled!!
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A Qubit Cloning Circuit? (3)
It is impossible to clone a qubit! Also note that
unwanted terms!
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The Bell State Circuit
x
y
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The Bell State Circuit By Example
?
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Quantum Teleportation Circuit
M1
H
M2
XM2
ZM1
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Quantum Teleportation Circuit (2)
M1
H
M2
XM2
ZM1
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Quantum Teleportation Circuit (3)
M1
H
M2
XM2
ZM1
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Quantum Teleportation Circuit (4)
M1
H
M2
XM2
ZM1
00, 01, 10 or 11
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Quantum Teleportation Circuit (5)
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What have we achieved?

• The teleportation process makes it possible to
  reproduce a qubit in a different location
• But the original qubit is destroyed!
• Next topic Quantum Parallelism and Deutschs
  quantum algorithm

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Quantum Parallelism

• Quantum parallelism is that feature of quantum
  computers which makes it possible to evaluate a
  function f(x) on many different values of x
  simultaneously
• We will look at an example of quantum parallelism
  now how to compute f(0) and f(1) for some
  function f all in one go!

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Quantum Circuits for Boolean Functions

• It is known that, for any boolean function
• it is possible to construct a quantum circuit Uf
  that computes it
• Specifically, to each binary function f
  corresponds a quantum circuit

binary addition
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Quantum Circuits for Boolean Functions (2)

• What can this circuit Uf do? Example

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Quantum Circuits for Boolean Functions (3)

• But what if the input is a superposition?

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Quantum Parallelism Summary

• So, a superposition of inputs will give a
  superposition of outputs!
• We can perform many computations simultaneously
• This is what makes famous quantum algorithms,
  such as Shors algorithm for factoring, or
  Grovers algorithm for searching
• Simple q. algorithm Deutschs algorithm

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Deutschs Algorithm

• David Deutsch famous British physicist
• Deutschs algorithm allows us to compute, in only
  one step, the value of
• To do this classically, you would have to
• compute f(0)
• compute f(1)
• add the two results
• Remember

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Circuit for Deutschs Algorithm
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Circuit for Deutschs Algorithm (2)
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Circuit for Deutschs Algorithm (3)
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End of Lecture 3

• Congratulations! If you are still awake, you have
  learned something about
• quantum gates (X, Y, Z, H, CNOT)
• quantum circuits (swapping, no-cloning problem)
• teleportation
• quantum parallelism
• and Deutschs algorithm