Quantum Computation and Quantum Information Lecture 2

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Quantum Computation and Quantum Information
Lecture 2

• Part 1 of CS406 Research Directions in Computing

Dr. Rajagopal Nagarajan Assistant Nick
Papanikolaou
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Lecture 2 Topics

• Physical systems on the atomic scale
• State vectors and basis states Qubits
• Systems of many qubits
• Quantum Measurement
• Entanglement
• Quantum gates
• Quantum coin-flipping and teleportation

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Quantum physics and Nature

• There exists a vast array of minute objects on
  the atomic scale electrons, protons, neutrons,
  photons, quarks, neutrinos,
• Quantum mechanics is a system of laws that
  describes the behaviour of such objects
• With computer chips getting smaller and smaller,
  by 2020 we will store 1 bit of data on objects of
  that size!

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Quantum physics and Nature (2)

• Atom-sized objects behave in unusual ways their
  state is generally unknown at any given time,
  and changes if you try to observe it!
• Several properties of these systems can be
  manipulated and measured.

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Qubits

• A qubit is any quantum system with exactly two
  degrees of freedom we use them to represent
  binary 0 and 1
• Hydrogen atom
• Spin-1/2 electron

Ground state
Excited state
Spin-down (-h/2) state
Spin-up (h/2) state
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Qubits (2)

• In general, the state of a qubit is a
  combination, or superposition, of two basis
  states
• The rest state and the excited state are the
  basis states of the hydrogen atom
• The spin-up and spin-down states are basis states
  for the spin-1/2 particle

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The State Vector

• The state of a quantum system is described by a
  state vector, written yñ
• If the basis states for a qubit are written 0ñ
  and 1ñ, then the state vector for the qubit is
• yñ a 0ñ b 1ñ
• where a and b are complex numbers with
• a2 b2 1

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Basis States

• Instead of 0ñ and 1ñ we can use any other basis
  states, as long as we can distinguish clearly
  between the two.
• Mathematically, basis states must be given by
  orthogonal vectors.

The inner product of the two vectors must be
0 á0 1ñ 0
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Basis states (2)

• For example, we could use the basis ñ, -ñ
  to describe the state of a qubit

1ñ
Now yñ g ñ d -ñ orthogonality á
-ñ 0
-ñ
ñ
0ñ
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Systems of many qubits

• If we know the individual states of the electrons
  in the system below

y1ñ 0 0ñ 1 1ñ 1ñ y2ñ 1 0ñ 0 1ñ
0ñ y3ñ 0 0ñ 1 1ñ 1ñ

• ... then what is the overall state of the
  three-particle system?

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Systems of many qubits (2)

• The state of a composite quantum system, when all
  the component states are known, is their tensor
  product
• yñ y1ñ Ä y2ñ Ä y3ñ
• This is the outer product of vectors
• Note that this is different from the inner
  product áf½cñ

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Systems of many qubits (3)

• We have
• yñ y1ñ Ä y2ñ Ä y3ñ
• (0 0ñ 1 1ñ) Ä (1 0ñ 0 1ñ) Ä (0 0ñ
  1 1ñ)
• 1ñ Ä 0ñ Ä 1ñ
• By convention, we write 1ñ Ä 0ñ Ä 1ñ as
  101ñ

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Quantum Measurement

• To extract any information out of a quantum
  system, you have to perform a physical
  measurement
• By measuring a quantum system
• you automatically change its state, the very
  state youre trying to measure
• you obtain, in general, a random result, which
  may be different from the original state

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Quantum Measurement (2)

• When you try to measure a qubit
• yñ a 0ñ b 1ñ
• ... you will never be able to obtain the values
  of a and b.
• A measurement has to be made with respect to a
  particular basis.

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Quantum Measurement (3)

• If you measure with respect to the 0ñ, 1ñ
  basis
• if yñ 0ñ the answer will be 0ñ with
  probability 100
• if yñ 1ñ the answer will be 1ñ with
  probability 100
• in all other cases (e.g. a2b20.5), the result
  will be probabilistic.
• After measurement, the value of yñ will change
  permanently to the result obtained.

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Quantum Measurement (4)

• If you measure with respect to a different basis,
  things are worse!
• Measuring yñ a 0ñ b 1ñ with respect to
  ñ, -ñ will give one of the results ñ and
  -ñ with particular probabilities.
• Also, the value of yñ will change permanently to
  the result obtained.

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Quantum Measurement, Formally

• Formally, when you measure
• yñ a 0ñ b 1ñ
• with respect to 0ñ, 1ñ you will get
• result 0ñ with probability a2
• result 1ñ with probability b2
• If you use a different measurement basis, the
  result will be one of the basis states, with
  different probabilities

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Measuring many qubits

• We want to know the possible outcomes of
  measuring the two qubit state
• yñ (a 0ñ b 1ñ) Ä (g 0ñ d 1ñ)
• ag 00ñ ad 01ñ bg 10ñ bd 11ñ

prob. ag2 ad2
prob. bg2 bd2
the first measurement will reduce yñ to one of
these smaller states
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Measuring many qubits (2)

• The second measurement will reduce yñ to one of
  the four states 00ñ, 01ñ, 10ñ, 11ñ.

ag 00ñ ad 01ñ
bg 10ñ bd 11ñ
00ñ
01ñ
10ñ
11ñ
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Measuring many qubits (3)

• By multiplying the branches in the overall tree,
  we can obtain the probability of each result. So
  for the state
• yñ ag 00ñ ad 01ñ bg 10ñ bd 11ñ
• two consecutive measurements will give
• result 00ñ with probability ag2
• result 01ñ with probability ad2
• result 10ñ with probability bg2
• result 11ñ with probability bd2

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Entanglement

• There exist states of many-qubit systems that
  cannot be broken down into a tensor product
• E.g. there do not exist a, b, g, d for which
• m 00ñ n 11ñ (a 0ñ b 1ñ) Ä (g 0ñ d
  1ñ)
• These are termed entangled states.

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The Bell states

• For a two-qubit system, the four possible
  entangled states are named Bell states

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Measuring Entangled States

• After measuring an entangled pair for the first
  time, the outcome of the second measurement is
  known 100

1
0ñ
0ñ
0.5
1
1ñ
1ñ
0.5
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Review

• Thus far we have seen
• how qubits are represented
• how many qubits can be combined together
• what happens when you measure one or more qubits
• where entangled pairs come from, and what happens
  when you measure them
• Now we will take a look at quantum gates

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Quantum gates

• As in classical computing, a gate is an operation
  on a unit of data, here a qubit
• A quantum gate is represented by a matrix that
  may be applied to a state vector
• We will talk about this in more detail next time
  for now we will look at some examples of commonly
  used quantum gates
• the Hadamard gate (H)
• the Pauli gates (I, sx, sy, sz)
• the Controlled Not (CNot)

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The Hadamard gate

• The Hadamard gate acts on one qubit, and places
  it in a superposition of 0ñ and 1ñ

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The Pauli gates

• The Pauli gates act on one qubit, as follows
• phase shift, sz
• sz(a 0ñ b 1ñ) a 0ñ - b 1ñ
• bit flip, sx
• sx(a 0ñ b 1ñ) a 1ñ b 0ñ
• phase shift and bit flip, sy
• sy(a 0ñ b 1ñ) a 1ñ - b 0ñ
• identity, I, does not change the input

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The Controlled Not Gate

• The CNot gate acts on two qubits
• CNot( 00ñ ) 00ñ
• CNot( 01ñ ) 01ñ
• CNot( 10ñ ) 11ñ
• CNot( 11ñ ) 10ñ

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Quantum Coin Flipping

• Quantum coin flipping is based on the following
  game
• Alice places a coin, head upwards in a box.
• Alice and Bob then take turns to optionally turn
  the coin over (without looking at it).
• At the end of the game, the box is opened and and
  Bob wins if the coin is head upwards.
• In the quantum version of the game, the coin is a
  quantum state

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Quantum Coin Flipping (2)

• Assume that Alice can only perform a flipping
  operation, i.e. gate sx
• Remember sx(a 0ñ b 1ñ) a 1ñ b 0ñ
• There is a strategy that allows Bob to win
  always he must perform Hadamard operations.
• Thus Bob places the state of the coin in a
  superposition of heads and tails!

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Quantum Coin Flipping (3)
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The No-cloning principle

• It has been proved by Wootters and Zurek that it
  is impossible to clone, or duplicate, an unknown
  quantum state.
• However, it is possible to recreate a quantum
  state in a different physical location through
  the process of quantum teleportation.

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Quantum Teleportation The Basics

• If Alice and Bob each have a single particle from
  an entangled pair, then
• It is possible for Alice to teleport a qubit to
  Bob, using only a classical channel
• The state of the original qubit will be destroyed
• How?
• Using the properties of entangled particles

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Quantum Teleportation

• Alice wants to teleport particle 1 to Bob
• Two particles, 2 and 3, are prepared in an
  entangled state
• Particle 2 is given to Alice, particle 3 is given
  to Bob

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Quantum Teleportation (2)

• In order to teleport particle 1, Alice now
  entangles it with her particle using the CNot and
  Hadamard gates
• Thus, particle 1 is disassembled and combined
  with the entangled pair
• Alice measures particles 1 and 2, producing a
  classical outcome 00, 01, 10 or 11.

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Quantum Teleportation

• Depending on the outcome of Alices measurement,
  Bob applies a Pauli operator to particle 3,
  reincarnating the original qubit
• If outcome00, Bob uses operator I
• If outcome01, Bob uses operator sx
• If outcome11, Bob uses operator sy
• If outcome10, Bob uses operator sz
• Bobs measurement produces the original state of
  particle 1.

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Quantum Teleportation (Summary)

• The basic idea is that Alice and Bob can perform
  a sequence of operations on their qubits to
  move the quantum state of a particle from one
  location to another
• The actual operations are more involved than we
  have presented here see the standard texts on
  quantum computing for details
• Recommended S. Lomonaco, A Rosetta Stone for
  Quantum Computation see www

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Review

• Quantum gates allow us to manipulate quantum
  states without measuring them
• Quantum states cannot be cloned
• Teleportation allows a quantum state to be
  recreated by exchanging only 2 bits of classical
  information
• Quantum coin flipping is more fun than classical
  coin flipping!