Introduction to Quantum Computing

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Introduction to Quantum Computing
Quantum Computation
Dr. Richard B. Gomez rgomez_at_gmu.edu

• Lecture 2

George Mason University School of Computational
Sciences
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Previous Lecture Topics

• Why Quantum Computing?
• What is Quantum Computing?
• History
• Quantum Weirdness
• Quantum Properties
• Quantum Devices

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The Need for Quantum Computers

• The size of components will drop down to the one
  atom per device level by 2020

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Superposition

• The Principal of Superposition states if a
  quantum system can be measured to be in one of a
  number of states then it can also exist in a
  blend of all its states simultaneously
• RESULT An n-bit qubit register can be in all 2n
  states at once
• Massively parallel operations

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Outline

• Quantum Logic Gates
• Quantum Dots
• Quantum Error Correction

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Quantum Logic Gates
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Controlled NOT

• One of the first quantum logic gates proposed was
  the Controlled-NOT gate which implements an XOR
• It has two inputs and two outputs (required for
  reversibility)

The target, t, is inverted when the control, c,
is 1
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Toffoli Gate

• Example of a reversible AND sometimes called
  controlled-controlled-NOT gate
• It has three inputs and three outputs
• The target input is XORed with the AND of the two
  control inputs

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Quantum Gate Operation

• Suppose the control input is in a superposition
  state, what happens to the target, does it get
  flipped or not?
• The answer is that it does both
• In fact, c and t become entangled

Entangled states that isa superposition of
states in which c and t are either both spin up
or spin down
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Quantum Dots
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Quantum Dots

• Quantum dots are small metal or semi-conductor
  boxes that hold well defined number of electrons
• The number of electrons in a box may be adjusted
  by changing the dots electrostatic environment
• Dots have been made which vary from 30 nm to 1
  micron
• They hold from 0 to 100 electrons

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Quantum Dot Wireless Logic

• Lent and Porod of Notre Dame proposed a wireless
  two-sate quantum dot device called a cell
• Each cell consists of 5 quantum dots and two
  electrons

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Quantum Dot Wire

• By placing two cells adjacent to each other and
  forcing the first cell into a certain state, the
  second cell will assume the same state in order
  to lower its energy

The net effect is that a 1 has moved on to the
next cell
By stringing cells together in this way, a
pseudo-wire can be made to transport a signal
In contrast to a real wire, however, no current
flows
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Quantum Dot Majority Gate

• Logic gates can be constructed with quantum dot
  cells
• The basic logic gate for a quantum dot cell is
  the majority gate

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Quantum Dot Inverter

• Two cells that are off center will invert a
  signal

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Quantum Dot Logic Gates

• AND, OR, NAND, etc can be formed from the NOT and
  the MAJ gates

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Quantum Error Correction
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Quantum Errors

• PROBLEM When computing with a quantum computer,
  you cant look at what it is doing
• You are only allowed to look at the end
• RESULT What happens if an error is introduced
  during calculation?
• SOLUTION We need some sort of quantum error
  detection/correction procedure

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Classical Error Codes

• In standard digital systems bits are added to a
  data word in order to detect/correct errors
• A code is e-error detecting if any fault which
  causes at most e bits to be erroneous can be
  detected
• A code is e-error correcting if for any fault
  which causes at most e erroneous bits, the set of
  all correct bits can be automatically determined
• The Hamming Distance, d, of a code is the minimum
  number of bits in which any two code words differ
• the error detecting/correcting capability of a
  code depends on the value of d

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Parity Checking

• PROCESS Add an extra bit to a word before
  transmitting to make the total number of bits
  even or odd (even or odd parity)
• at the receiving end, check the number of bits
  for even or odd parity
• It will detect a single bit error
• Cost extra bit
• Example Transmit the 8-bit data word 1 0 1 1 0
  0 0 1
• Even parity version 1 0 1 1 0 0 0 1 0
• Odd parity version 1 0 1 1 0 0 0 1 1

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Quantum Schemes

• In 1994 the first paper on Quantum error
  correction was presented at a conference in
  England
• It required the quantum computer to run
  simultaneous copies of a calculation
• If no errors occurred all the separate copies
  would produce the same answer
• Using an inefficient procedure a wrong answer
  could be restored

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Improvements

• In 1995, Peter Shor developed a better procedure
  using 9 qubits to encode a single qubit of
  information
• His algorithm was a majority vote type of system
  that allowed all single qubit errors to be
  detected and corrected

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Example

• A 3-bit quantum error correction scheme uses an
  encoder and a decoder circuit as shown below

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Encoder

• The encoder will entangle the two redundant
  qubits with the input qubit

If the input state is 0gt then the encoder
does nothing so the output state is 000gt
If the input state is 1gt then the encoder
flips the lower states so the output state
is 111gt
If the input is an superposition state, then the
output is the entangled state a000gt b111gt
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Decoder

• Problem Any correction must be done without
  looking at the output
• The decoder looks just like the encoder

Corrected output
If the input to the decoder is 000gt or 111gt
there was no error so the output of the decoder
is
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Example

• Consider the possible error conditions

No Errors
a000gt b111gt decoded to a000gt b100gt
(a0gt b1gt)00gt
Top qubit flipped
a100gt b011gt decoded to a111gt b011gt
(a1gt b0gt)11gt
So, flip the top qubit (a0gt b1gt)11gt
Middle qubit flipped
a010gt b101gt decoded to a010gt b110gt
(a0gt b1gt)10gt
Bottom qubit flipped
a001gt b110gt decoded to a001gt b101gt
(a0gt b1gt)01gt
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Decoder without Measurement

• The prior decoder circuit requires the
  measurement of the two extra bits and a possible
  flip of the top bit
• Both these operations can be implemented
  automatically using a Toffoli gate

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Summary

• Quantum Logic Gates
• Quantum Dots
• Quantum Error Correction