QUANTUM COMPUTING

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QUANTUM COMPUTING
Part II
Jean V. Bellissard Georgia Institute of
Technology Institut Universitaire
de France
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Hello again everyone !
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QUANTUM GATES
a reminder
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Quantum gates
1-qubit gates
U is unitary in M2 ( C )
Pauli basis in M2 ( C )
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Quantum gates
1-qubit gates
U is unitary in M2 ( C )
1 0 S 0 i
1 0 T 0 eip/4
1 1 H 2-1/2
1 -1
Hadamard, phase and p/8 gates
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Quantum gates
controlled gates
xgt
xgt
U
ygt
Uxygt
U is unitary in M2 ( C )
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Quantum gates
the CNOT gate
xgt
xgt
ygt
xÅygt
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Quantum gates
the swap gate
xgt
ygt
x

ygt
xgt
x
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FOURIER TRANSFORM

• quantum computers are fast !

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Fourier Transform

• Digital basis given by qubits
• x1x2xngt x1gt x2gt xngt ygt
• If
• y 2(n-1) x1 2(n-2)x2 xn x1x2xn

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Fourier Transform

• Fourier transform
• F jgt 1 ?k0 e2ip jk/N kgt
• N1/2
• N2n,

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Fourier Transform

• Binary decomposition
• jk/2n
• (0.jn)k1 (0.jn-1jn)k2 (0.j1j2jn)kn
• (modulo 1) where
• 0.j1j2jr j1/2 j2/22 jr/2r

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Fourier Transform

• Binary decomposition
• F jgt 1 ?k0 e2ip jk/ kgt
• 2n/2

(0gt e2ip(0.jn) 1gt)
(0gt e2ip(0.j1jn) 1gt)
. . .
Fjgt
2n/2
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Fourier Transform

• Digital phase gate (1 qubit)

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Fourier Transform
jngt
(0gt e2ip(0.jn) 1gt)
jn-1gt
(0gte2ip(0.jn-1 jn) 1gt)
j2gt
(0gte2ip(0.j2...jn)1gt)
j1gt
(0gte2ip(0.j1...jn) 1gt)
Circuit producing the quantum Fourier transform
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Fourier Transform
(0gte2ip(0.j1...jn) 1gt)
x
(0gt e2ip(0.jn) 1gt)
(0gte2ip(0.jn-1 jn) 1gt)
(0gte2ip(0.j2...jn)1gt)
x
(0gte2ip(0.jn-1 jn) 1gt)
(0gte2ip(0.j2...jn)1gt)
x
(0gte2ip(0.j1...jn) 1gt)
(0gt e2ip(0.jn) 1gt)
x
Swap gates arrange final qubits in right order
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Fourier Transform

• Fourier transform
• F ?j f(j)jgt ?kf(k)kgt
• f(k) 2-n/2?jf(j) e2ipjk/
• the Fourier transform of f is given by the
  coordinates of the outcome.
• It can then be measured

2n
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Fourier Transform

• The usual FFT requires a time
• O(N LnN)
• The number of gates needed is
• n2/2 2n
• Since the N2n, the algorithm gives the result in
  a time (1 time unit/gate)
• O((LnN)2) !!

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PHASE ESTIMATION

• a key subroutine

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Phase estimation

• U is a unitary with an eigenvalue
• Uugt eif ugt
• Goal compute f .
• Set-up two registers, one with t-qubits, the
  other one for representing U.

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Phase estimation

• a controlled Un-gate GUn gives
• GUnxgt ugt einxf xgt ugt
• It transfers the phase of ugt on the component
  1gt of the first register.
• On the first register one uses a rotated state
  H0gt (0gt1gt)/v2 instead of xgt.

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Phase estimation
0gt
0gt
0gt
0gt
ugt
ugt
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Phase estimation

• If f 2p .j1j2jt, the outcome is

• Then use a Fourier transform back to get jgt
  j1j2jt gt, giving the value of the phase modulo
  O(2p/2t).

(0gt e2ip(0.jt) 1gt)
(0gt e2ip(0.j1jt) 1gt)
. . .
2n/2
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Phase estimation

• To get n digit of f accurate, with probability of
  success (1-e), it can be shown that t must be
  chosen as
• tnlog(21/2e)

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SHORS ALGORITHM

• factorizing integer into primes

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Shors algorithm

• Input a composite integer N
• Output a non trivial factor of N
• Runtime O((log N)3) operations, succeeds with
  probability O(1).

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Shors algorithm

• First step order finding.
• If xltN are integers with no common factors, the
  order of x modulo N is the least 0ltr such that
  xrº1(mod N).
• Use the unitary Uygt xy(mod N)gt. If y Î
  0,1L, Nlt2L, and Nylt2L, set Uygt ygt.

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Shors algorithm

• Then
• usgt r-1/2åk0r exp(-2ipsk/r)xr(mod N)gt
• is an eigenvector of U with phase
• f2p s/r
• A phase-finding computes s/r. A continuous
  fraction expansion gives r.

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Shors algorithm

• It may not be possible to prepare the initial
  state of the second register in the state usgt.
  But any initial state is a linear combination of
  the usgt s.
• The outcome will be s/r for some s. A continuous
  fraction expansion will give r anyway.

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Shors algorithm

• Factoring procedure
• (i) If N is even, return the factor m2
• (ii) Find if Nab, for agt1, b2, integers
  (special subroutine)
• (iii) Choose randomly xÎ1,N-1. If mgcd(x,N)
  gt1, then return m.

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Shors algorithm

• Factoring procedure (continued)
• (iv) Find the order r of x mod N.
• (v) If r is even xr/2-1?-1 (mod N), compute
  gcd(xr/2-1,N) gcd(xr/21,N), check if one is a
  nontrivial factor m. If so return m.

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ERROR-CORRECTIONS

• can quantum information be protected ?

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Error-correction codes

• Classical code theory uses redundancy to transmit
  bits of information

0 1
000 111
010 110
000 111
Transmission
Reconstruction at reception (correction)
coding
errors (2nd Law)
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Error-correction codes

• Quantum computer are submitted to the no-cloning
  theorem!
• there is no Hilbert space H neither any unitary
  operator U on H H for which there is a state
  sgt such that
• Uygt sgt ygt ygt yÎ H

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Error-correction codes

• However it is possible to produce quantum
  circuits for which 0gt000gt and 1gt111gt for
  instance

a000gtb111gt
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Error-correction codes

• The previous circuit protects against index
  flips. How can one protects the signal against
  phase flips ?
• Hadamard gates transform index into a phase
• Hxgt (0gt(-1)x1gt)/v2

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Error-correction codes

• Phase flip protection

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Error-correction codes

• Shors code

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Error-correction codes

• Shors code gives 0gt0Lgt and 1gt1Lgt with
• xLgt________________________
• 2v2

(000gt(-)x111gt)(000gt(-)x111gt)(000gt(-)x111gt
)
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Error-correction codes

• Kitaev proposed in 1997 to replace digital
  degrees of freedom by topological ones.
• Tunneling effect between topological sectors is
  unlikely, leading to a better code protection.

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PHYSICAL REALIZATIONS

• can quantum computers be built ?

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Realizations

• Several devices may produce qubits
• Any quantum harmonic oscillator
• Optical photons
• Optical cavity quantum electrodynamics coupling
  with 2-level atoms.
• Ion traps
• Nuclear magnetic resonance computation with up
  to 7-qubits have permitted to test Shors
  algorithm 153x5 !!
• Josephson junctions quantronium
• Double well with quantum dots

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Realizations 1-qubit, the quantronium

• The quantronium (Esteve Devoret Saclay) a
  Josephson tunneling junction

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Realizations 1-qubit, the quantronium

• Quantronium

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Realizations

• Quantronium

RABI OSCILLATIONS Coherent manipulation of the
Quantronium state a microwave resonant pulse
with duration t and amplitude URF is applied to
the gate. The Quantronium undergoes Rabi
oscillations. The probability of measuring the
Quantronium in its excited state, i.e. the
switching probability of the measuring junction,
oscillates accordingly as a function of t and
URF. Each dot is an average over 50000
measurements. The decoherence time is about 5µs.
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Realizations 1-qubit, quantum dots

• Double quantum dots group of Kouwenhoven, (U.
  Delft Holland)

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Realizations 7-qubit, NMR

• Nuclear Magnetic Resonance IBM
• 153x5 !! (Shors algorithm)

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CONCLUSIONS

• will quantum computers be built ?

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To conclude (from Part I)

1. The elementary unit of quantum information is the
   qubit, with states represented by the Bloch ball.
2. Several qubits are given by tensor products
   leading to entanglement.
3. Quantum gates are given by unitary operators and
   lead to quantum circuits
4. Law of physics must be considered for a quantum
   computer to work measurement, dissipation

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To conclude (Part II)

1. Several algorithms are available Fourier
   transform, phase estimation, quantum search,
   hidden subgroup, order-finding
2. Shors algorithm for factoring shows enormous
   efficient and threaten present cryptography
3. Error-correcting codes are now available
4. Few qubits computer have been realized with NRM
   experiments

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To conclude (other topics)

1. A theory of quantum information and code theory
   is also available even though incomplete
2. Quantum cryprography exists (Gisin, Geneva)
3. Need for developments in quantum complexity
   theory are notions of P- NP- completeness
   obsolete ?
4. Main problem putting qubits together in concrete
   machines. Can one control entanglement and /or
   decoherence on a large scale ? Not clear !!

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Will quantum computers be built ?
YES of course !!