QUANTUM COMPUTATION

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QUANTUM COMPUTATION

• The History and the State-of-Art
• Part 1

2
Plan

• 1. Conventional and Quantum Logic
• 2. Quantum algorithms
• 3. Quantum Error Correction
• 4. Physical implementation of quantum computation
• 5. Adiabatic quantum computer
• 6. Quantum Simulator

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1. Conventional and Quantum Logic
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Turing, 1936 Computer Science

• All computers are equivalent, and could all be
  simulated by each other.
• A problem is either computable or not, regardless
  of what computer we use
• Assumption
• Computer operation is based on conventional
  logic

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• Conventional (classical) computer
• Operates with the digital units bits
• Number in binary notation
• N a5a4a3a2a1a0
• ak 0 or 1
• N a020 a121

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• Physical implementation of a bit
• Voltage of a capacitor
• Magnetic moment of a domain

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Classical Computation

• Change of the values of bits
• Logic gates
• Logic gates are based on the conventional logic
• Negation (NOT)
• NOT (True) False
• NOT (False) True

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Classical logic gates
Operation of N-gate
Final state of bit
Initial state of bit
0
1
0
1
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2-Bit Logic Gates

• (True) AND (True) True
• Any other combination with AND False
• AND gate
• 00?0
• 01?0
• 10?0
• 11?1
• AND and NOT are universal for classical
  computation

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Landauer, 1961 Logic and Thermodynamics

• We use irreversible two-bit gates
• ? Conventional logic is
• thermodynamic logic
• Shannon entropy
• S -kB?plnp

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Classical Computer is a Heat Generator

• In a single computational step
• Entropy S of the available information decreases
  by kBln2
• Entropy of the environment ?S gt kBln2
• ? Classical computer generates heat Q gt kBT
  ln2
• ? Classical computer consumes energy E gt
  kBT ln2

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Classical Computer vs Heat Engine

• If a classical computer were built before the
  heat engine
• the 2nd law of thermodynamics would be formulated
  as
• A computer based on a classical logic cannot
  operate without generation of heat

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Bennett, 1973 Reversible Logic

• A computer can operate using a novel
• reversible logic!
• Toffoli, 1980
• Universal Control-Control-Not 3-bit gate
• 111?110
• 110? 111
• All other combinations do not change

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Benioff, 1980 Quantum Computer (QC)

• Computer based on the non-dissipative classical
  physics is possible
• Advantage compare to the conventional computer
  no basic computational heat generation
• (Real heat generation is much greater)
• Computer based on the quantum physics is possible

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• Bit stationary state of an atom or ion
• Advantage
• Huge possible information density
• 1022 bits/cm3

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Deutsch, 1985 Superpositional States

• The most important power of QC
• Using superpositional states
• C00gt C11gt
• Quantum measurement
• C02 probability to collapse
  to the state 0gt
• C12 probability to collapse
  to the state 1gt

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Quantum Logic

• Quantum bit (qubit)
• Superposition of true and false
• A Quantum logic gate must change all the states
  of superposition simultaneously

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Operations with Qubits

• Examples
• One qubit
• Initially the ground state 0gt
• One-qubit quantum gate can create the state
• (0gt 1gt)/v2
• Physical implementation
• electromagnetic (EM) pulse

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• Two qubits
• Two one-qubit gates can create the state
• (00gt 01gt 10gt 11gt)/2
• Three qubits
• Three one-qubit gates can create the state
• (000gt 001gt 111gt)/v8

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Loading Numbers

• We can create superposition of all possible
  numbers!
• L qubits we can load 2L numbers at a time using
  L qubits and L one-qubit gates!

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Quantum Logic Gates

• Quantum logic gate - any unitary transformation
  (transformation, which does not change ?Ck2
  1)
• One qubit gate (one qubit rotation) changes the
  state of a single atom

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• Example
• One-qubit Hadamard gate H
• H0gt (0gt 1gt)/v2
• H1gt (0gt - 1gt)/v2

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Hadamard (H) Gate

• 1) H creates the superposition of all possible
  numbers from the ground state.
• 2) H also implements the simplest quantum
  interference
• H(0gt 1gt)/v2
• 0gt 1gt)/v2 (0gt - 1gt)/v2/v2
• 0gt
• We got one number from two numbers!

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Universal Quantum Computations Using Quantum
Logic Gates

• Barenco et al., 1995
• Theorem
• For universal QC we need one-qubit rotations and
  a two-qubit gate.
• The most popular two-qubit gate is the
  Control-Not (CN) gate

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Control Not (CN-gate)
Operation of CN-gate
At least two qubits are required
0
0
0
0
Control qubit c
1
0
0
1
Target qubit t
1
0
1
1
0
1
1
1
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An Implementation of the CN Gate

• Interacting atoms.
• The resonant frequency of one atom depends on the
  state of the other atom
• Frequency is ? if the neighbors state is 0gt
• Frequency is ? if the neighbors state is 1gt
• Using electromagnetic pulse of frequency ? we
  drive the atom only if the neighbors state is
  1gt!

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Two Registers

• Using two registers we can load 2L values of a
  function
• Example
• 2 qubits are in the x-register, 1 qubit is in the
  y-register

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• x,ygt 00,0gt
• H1H200,0gt
• (00,0gt 01,0gt 10,0gt 11,0gt)/2
• Now CN23 gate changes the value of y
• (2-control qubit, 3-target qubit)
• (00,0gt 01,1gt 10,0gt 11,1gt)/2
• Periodic function y(x)!
• It is an example of the entangled state

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Computing a Function

• In general
• We can load 2L values of a function with a
  polynomial number of gates!
• How to use this computational power?

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Quantum Interference

• Use quantum interference.
• Eliminate all the numbers except for the desired
  ones
• Make quantum measurement
• Get the desired parameter of a function!

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2. Quantum Algorithms
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Problems for a Classical Computer

• Input L bits
• Classical computers can solve
• Polynomial time (P) problems
• number of computational steps increases as a
  polynomial function Ln
• (typically n 3).

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Polynomial time (P) problems

• Addition, multiplication, solution of an ordinary
  differential equation are the
• P problems
• Tractable problems
• Efficient algorithm

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Problems for QC

• Non-deterministic polynomial time problems (NP)
• Input L bits (qubits)
• The number of computational steps needed to
  verify a solution increases as a polynomial
  function Ln.
• Number of computational steps needed to solve the
  problem increases typically as 2L (exponentially)

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Prime Factorization

• Example 29083 127 x 229
• Gauss
• The dignity of science demands that every aid to
  the solution of such a celebrated problem be
  zealously cultivated.

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• Statement There is no efficient classical
  algorithm for prime factorization
• (It is not proved!)
• Since 1970th the main cryptosystem (RSA) is based
  on this statement.
• Bank cards, diplomatic secret codes are based on
  this statement.

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Financial Security

• No one can fake bank cards
• A computer chooses at random two prime factors N1
  and N2,
• multiply them,
• assign the product N N1N2 to your account
  (visible for banks employees),
• encode your bank card with one of the factors,
• and erase both factors.

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Prime Factorization on a Classical Computer

• N - 200-digit number
• The best supercomputer
• IBM Blue Gene 1015 operations/second
• If we try about 10100 numbers
• It would take about 1085 s, more than 1077 years
• The best classical algorithm takes
• about 2x109 years
• It is still inefficient

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Shor, 1994 The First Practical Quantum Algorithm

• 1. Create superposition of all possible numbers
  in the x-register
• 2. Compute f(x) ax mod(N) in the y-register
• (a-is any co-prime to N)

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• 3. Use Shors gate (discrete Fourier transform
  DFT) to eliminate all the numbers in the
  x-register except for multiples of (2L /T).
• (L - number of qubits in the x-register)
• 4. Make quantum measurement on the x-register.
• 5. Repeat procedure to get multiples of (2L/T)
  and derive the value of T.
• Find a factor GCD (N, aT/2 1)

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• The Shors algorithm is efficient!
• Polynomial number of computational steps (less
  than L3)
• A few hours instead of 2x109 years!

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Complete NP Problems

• Is there is a hope to find efficient algorithms
  for other NP problems?
• Cook, Karp, and Levin (1970th)
• There exist complete NP problems
• with the following properties
• an efficient algorithm for a complete NP problem
  can be used to construct an efficient algorithm
  for any other NP problem!

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• Examples of a complete NP problem
• Given the dimensions of boxes how to pack them
  into the trunk?
• Given a finite set of integers. Whether any
  subset sums to zero?

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• There is no efficient classical algorithm.
• No one proved that it does not exist
• Clay Math Institute in Massachusetts
• 1,000,000.00 reward for the algorithm
• What about an efficient quantum algorithm?

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Grover, 1996

• Quantum speed-up for a very large class of the
  NP problem
• vN steps instead of N/2 steps
• Example
• We need 1026 operations (1011/s)
• More than 1000 years

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• Using Grovers algorithm
• 1013 operations
• Less than 1 second!
• Impressive but still inefficient!

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How Grovers Algorithm Works

• 1. Create superposition of all possible numbers
  in the x-register
• 2. In the y-register enter y 1 for x x,
  where x is the problems solution (it is easy
  to check if x is a solution) and y -1 for all
  other x.

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• 3. Use Grovers gate to increase the amplitude
  C for the state ltxx,y-1.
• 4) Make a quantum measurement to find x.

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• Still exponential number of steps!
• Is it possible to find an efficient quantum
  algorithm for a complete NP problem?
• Nobody could find it.
• No one proved that it is impossible

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New Physical Principle

• Aaronson, 2008
• A physical computer, which can efficiently solve
  a complete NP problem is impossible.
• Informational principle

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Physical applications of the Informational
Principle

• Example non-linear correction to the Schrodinger
  equation non-linear quantum mechanics
• Abrams and Lloyd,1998
• If a non-linear term is added to the Schrodinger
  equation then a quantum computer can solve
  efficiently a complete NP problem

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3. Quantum Error Correction
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Error Correction in a Classical Computer

• Error correction in a classical compute -
  redundancy
• Take two additional (ancillary) bits,
• All three bits are supposed to have the same
  value
• Check the values of all the bits
• If one of the bits has a value different from two
  other bits change its value

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• How to correct the amplitudes Ck for qubits?
• Classical redundancy cannot be used!

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Steane and Shor, 1995 Quantum Error Correction

• The most complicated quantum algorithm!
• Example
• (C00gt C11gt)00gt C0000gt C1100gt
• Assumption Error occurs independently on
  different qubits

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Procedure for Quantum Error Correction

• 1. Use a special sequence of quantum gates to the
  three qubits
• 2. Measure the state of the ancillary qubits.
• 3. Depending on the measurement result apply a
  sequence of quantum gates to the main qubit.
• As a result the main qubit returns to the initial
  state (C00gt C11gt)
• with accuracy to a non-significant phase
  constant exp(ikt)

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Collective Decoherence

• What happens if the errors on different qubits
  are correlated?
• Collective decoherence
• Example
• Thermal electromagnetic wave with the wavelength
  longer than the distance between the qubits

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Super-decoherence and sub-decoherence

• Collective decoherence depends on the type of
  states
• For example, in some cases for L atoms
• Difference ? between the integrated values in a
  superposition is about L super-decoherence
  quantum superposition quickly destroys
• ? ltlt L sub-decoherence

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• Example with two atoms
• (00gt 11gt)/v2 super-decoherence
• ? 2 0 2
• (01gt 10gt)/v2 sub-decoherence
• ? 1 1 0

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Palma, Suominen, and Ekert, 1996 Logical Qubits

• Two physical qubits one logical qubit
• 0gt ? 01gt
• 1gt ? 10gt
• (00gt 11gt)/v2 ? (0101gt 1010gt)/v2
• Sub-decoherence instead of super-decoherence!
• Decoherence-free subspace