Transcript Document

QUANTUM
COMPUTING
Part II
Jean V. Bellissard
Georgia Institute of Technology
&
Institut Universitaire de France
QUANTUM GATES:
a reminder
Quantum gates:
|x>
1-qubit gates
U|x>
U
U is unitary in M2 ( C
1
0
I=
0
1
X=
0
1
0
)
-i
Y=
1
0
1
0
Z=
i
Pauli basis in M2 ( C
0
)
0 -1
Quantum gates:
|x>
1-qubit gates
U|x>
U
U is unitary in M2 ( C
1
1
H =2-1/2
1
0
S=
1
-1
)
1
T=
0
i
Hadamard, phase and p/8 gates
0
0 eip/4
Quantum gates:
controlled gates
|x>
|x>
|y>
U
U is unitary in M2 ( C
Ux|y>
)
Quantum gates:
the CNOT gate
|x>
|x>
|y>
|xy>
Quantum gates:
|x>
|y>
x
x
|y>
|x>
the swap gate
=
FOURIER TRANSFORM:
quantum computers are fast !
Fourier Transform:
• Digital basis given by qubits
|x1x2…xn> = |x1> |x2> … |xn> =|y>
If
y = 2(n-1) x1 + 2(n-2)x2 +…+ xn:= x1x2…xn
Fourier Transform :
• Fourier transform:
F |j> = 1 ∑k=0 e2ip jk/N |k>
N1/2
N=2n,
Fourier Transform :
• Binary decomposition:
jk/2n =
(0.jn)k1+ (0.jn-1jn)k2 + … + (0.j1j2…jn)kn
(modulo 1) where
0.j1j2…jr = j1/2 + j2/22 …+ jr/2r
Fourier Transform :
• Binary decomposition:
F |j> = 1
2n/2
F|j> =
∑k=0 e2ip jk/ |k>
(|0> + e2ip(0.jn) |1>)
...
2n/2
(|0> + e2ip(0.j1…jn) |1>)
Fourier Transform :
• Digital phase gate (1 qubit):
Rk =
1
0
0
e
2ip
2k
Rk|x> = e
2ip x
2k
|x>
Fourier Transform :
|jn>
|jn1>
2ip(0.jn) |1>)
H (|0> + e
H
|j2>
|j1> H
H
R2
Rn
Rn-2 Rn-1
R2
(|0>+e2ip(0.jn-1 jn) |1>)
(|0>+e2ip(0.j2...jn)|1>)
(|0>+e2ip(0.j1...jn) |1>)
Circuit producing the quantum Fourier transform
Fourier Transform :
(|0> + e2ip(0.jn) |1>)
(|0>+e2ip(0.j1...jn) |1>)
x
(|0>+e2ip(0.jn-1 jn) |1>)
x
(|0>+e2ip(0.j2...jn)|1>)
(|0>+e2ip(0.j2...jn)|1>)
x
(|0>+e2ip(0.jn-1 jn) |1>)
(|0>+e2ip(0.j1...jn) |1>)
x
(|0> + e2ip(0.jn) |1>)
Swap gates arrange final qubits in right order
Fourier Transform :
• Fourier transform
~
F ∑j f(j)|j> = ∑kf(k)|k>
f(k)
~ = 2-n/2∑jf(j) e2ipjk/2n
• the Fourier transform of f is given by
the coordinates of the outcome.
• It can then be measured
Fourier Transform :
• The usual FFT requires a time
O(N LnN)
• The number of gates needed is
n2/2 + 2n
• Since the N=2n, the algorithm gives
the result in a time (1 time unit/gate)
O((LnN)2) !!
PHASE ESTIMATION
a key subroutine
Phase estimation
• U is a unitary with an eigenvalue
U|u> = eif |u>
• Goal: compute f .
• Set-up: two registers, one with tqubits, the other one for
representing U.
Phase estimation
• a controlled Un-gate GUn gives
GUn|x> |u> = einxf |x> |u>
• It transfers the phase of |u> on the
component |1> of the first register.
• On the first register one uses a
rotated state H|0> = (|0>+|1>)/√2
instead of |x>.
Phase estimation
(t-1)
|0>
|0> + ei2
H
2
|0>
|0>
H
|0>
H
|u>
f|1>
|0> + ei2 f|1>
H
1
|0> + ei2 f|1>
0
|0> + ei2 f|1>
U
20
U
21
U
22
2(t-1)
U
|u>
Phase estimation
• If f= 2p .j1j2…jt…, the outcome is
(|0> + e2ip(0.jt) |1>)
...
(|0> + e2ip(0.j1…jt) |1>)
2n/2
• Then use a Fourier transform back to
get |j> =| j1j2…jt >, giving the value of
the phase modulo O(2p/2t).
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
t=n+log(2+1/2e)
SHOR’S ALGORITHM:
factorizing integer into primes
Shor’s algorithm
• Input: a composite integer N
• Output: a non trivial factor of N
• Runtime: O((log N)3) operations,
succeeds with probability O(1).
Shor’s algorithm
• First step: order finding.
• If x<N are integers with no common
factors, the order of x modulo N is
the least 0<r such that xr1(mod N).
• Use the unitary U|y> = |xy(mod N)>.
If y  {0,1}L, N<2L, and N≤y<2L, set
U|y> =|y>.
Shor’s algorithm
• Then
|us> = r-1/2k=0r exp(-2ipsk/r)|xr(mod N)>
is an eigenvector of U with phase
f=2p s/r
• A phase-finding computes s/r. A
continuous fraction expansion gives r.
Shor’s algorithm
• It may not be possible to prepare the
initial state of the second register in
the state |us>. But any initial state is
a linear combination of the |us> ‘s.
• The outcome will be s/r for some s. A
continuous fraction expansion will give
r anyway.
Shor’s algorithm
• Factoring procedure
(i) If N is even, return the factor m=2
(ii) Find if N=ab, for a>1, b≥2, integers
(special subroutine)
(iii) Choose randomly x[1,N-1]. If
m=gcd(x,N) >1, then return m.
Shor’s 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/2=1,N), check if one is a
nontrivial factor m. If so return m.
ERROR-CORRECTIONS:
can quantum information be
protected ?
Error-correction codes
• Classical code theory uses redundancy
to transmit bits of information
0
000
Transmission
111
errors
(2nd Law)
coding
1
010
Reconstruction
000
110
at reception
(correction)
111
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 |s> such that
U|y> |s> = |y> |y>
 y H
Error-correction codes
• However it is possible to produce
quantum circuits for which |0>|000>
and |1>|111> for instance:
a|0>+b|1>
|0>
|0>
a|000>+b|111>
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:
H|x> = (|0>+(-1)x|1>)/√2
Error-correction codes
• Phase flip protection
a|0>+b|1>
H
|0>
H
|0>
H
a|000>+b|111>
∑y(a+b(-1)∑yi)|y1y2y3>
2-3/2
Error-correction codes
a|0>+b|1>
H
• Shor’s code
|0>
H
|0>
H
Error-correction codes
•
Shor’s code gives |0>|0L> and
|1>|1L> with:
(|000>+(-)x|111>)(|000>+(-)x|111>)(|000>+(|xL>=________________________
)x|111>)
2√2
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.
PHYSICAL REALIZATIONS
can quantum computers be built ?
Realizations
•
1.
2.
3.
4.
5.
6.
7.
Several devices may produce qubits:
Any quantum harmonic oscillator
Optical photons
Optical cavity quantum electrodynamics: coupling with 2level atoms.
Ion traps
Nuclear magnetic resonance: computation with up to 7qubits have permitted to test Shor’s algorithm 15=3x5 !!
Josephson junctions: quantronium
Double well with quantum dots
Realizations:
•
1-qubit, the quantronium
The quantronium (Esteve & Devoret
Saclay): a Josephson tunneling
junction
Realizations 1-qubit, the quantronium
•
Quantronium :
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.
Realizations 1-qubit, quantum dots
•
Double quantum dots : group of
Kouwenhoven, (U. Delft Holland)
resonant tunneling
µL
5
4
3
b
a
2
1
-|e|V
-|e|f1(N1+1,N2)
-|e|f2(N1, N2+1)
µR
Realizations 7-qubit, NMR
•
Nuclear Magnetic Resonance : IBM
15=3x5 !! (Shor’s algorithm)
CONCLUSIONS
will quantum computers be built ?
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…
To conclude (Part II)
1. Several algorithms are available: Fourier
transform, phase estimation, quantum search,
hidden subgroup, order-finding
2. Shor’s 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
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 !!