use of genetic algorithm for quantum information processing by...
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Use of Genetic Algorithm for Quantum Information Processing
by NMR
V.S. Manu and Anil Kumar
Centre for quantum Information and Quantum ComputingDepartment of Physics and NMR Research Centre
Indian Institute of Science, Bangalore-560012
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The Genetic Algorithm
John Holland
Charles Darwin 1866 1809-1882
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“Genetic Algorithms are good at taking large, potentially huge, search spaces and navigating them, looking for optimal combinations of things, solutions you might not otherwise find in a lifetime”
Here we apply Genetic Algorithm to Quantum Computing and Quantum Information Quantum Computing and Quantum Information
ProcessingProcessing
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Quantum Algorithms1. PRIME FACTORIZATION
1010years (Age of the Universe)
Classically :exp [2(ln c)1/3(ln ln c)2/3]
400 digit
Shor’s algorithm : (1994)(ln c)3
3 years
2. SEARCHING ‘UNSORTED’ DATA-BASE
Classically : N/2 operationsGrover’s Search Algorithm : (1997) N operations
3. DISTINGUISH CONSTANT AND BALANCED FUNCTIONS:
Classically : ( 2N-1 + 1) stepsDeutsch-Jozsa(DJ) Algorithm : (1992) . 1 step
4. Quantum Algorithm for Linear System of Equation:Harrow, Hassidim and Seth Lloyd; Phys. Rev. Letters, 103, 150502 (2009). Exponential speed-up
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5. Simulating a Molecule: Using Aspuru-Guzik Algorithm
(i) J.Du, et. al, Phys. Rev. letters 104, 030502 (2010).
Used a 2-qubit NMR System ( 13CHCl3 ) to calculated the ground state energy of Hydrogen Molecule up to 45 bit accuracy.
(ii) Lanyon et. al, Nature Chemistry 2, 106 (2010).
Used Photonic system to calculate the energies of the ground and a few excited states up to 20 bit precision.
Recent Developments
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Experimental Techniques for Quantum Computation:1. Trapped Ions
4. Quantum Dots
3. Cavity Quantum Electrodynamics (QED)
6. NMR
Quantum Dots:Quantum Dots:
µmElectrodesDot
Circle
http://theorie5.physik.unibas.ch/qcomp/node3.html
http://news.uns.purdue.edu/html4ever/010917.Chang.quantum.html
7. Josephson junction qubits
8. Fullerence based ESR quantum computer
5. Cold Atoms
2. Polarized Photons Lasers
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Photograph of a chip constructed by D-Wave Systems Inc., designed to operate as a 128-qubit superconducting adiabatic quantum optimization processor, mounted in a sample holder. 2011
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0
1. Nuclear spins have small magnetic moments (I) and behave as tiny quantum magnets.
2. When placed in a large magnetic field B0 , they oriented either along the field (|0⟩ state) or opposite to the field (|1⟩ state) .
4. Spins are coupled to other spins by indirect spin-spin (J) coupling, and controlled (C-NOT) operations can be performed using J-coupling.
Multi-qubit gates
Nuclear Magnetic Resonance (NMR)
3. A transverse radiofrequency field (B1) tuned at the Larmor frequency of spins can cause transition from |0⟩ to |1⟩ (NOT Gate by a 1800 pulse). Or put them in coherent superposition (Hadamard Gate by a 900 pulse). Single qubit gates.
SPINS ARE QUBITS
B1
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DSX300
AMX 400
AV 500
AV 700
DRX 500
NMR Research Centre, IISc1 PPB
Field/ Frequency stability = 1:10 9
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Why NMR?Why NMR?
> A major requirement of a quantum computer is that the coherence should last long.
> Nuclear spins in liquids retain coherence ~ 100’s
millisec and their longitudinal state for several seconds.
> A system of N coupled spins (each spin 1/2) form an N qubit Quantum Computer.
> Unitary Transform can be applied using R.F. Pulses and J-evolution and various logical operations and quantum algorithms can be implemented.
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1. Preparation ofPseudo-Pure States
2. Quantum Logic Gates
3. Deutsch-Jozsa Algorithm
4. Grover’s Algorithm
5. Hogg’s algorithm
6. Berstein-Vazirani parity algorithm
7. Quantum Games
8. Creation of EPR and GHZ states
9. Entanglement transfer
Achievements of NMR - QIP
√
√
√
10. Quantum State Tomography
11. Geometric Phase in QC
12. Adiabatic Algorithms
13. Bell-State discrimination
14. Error correction
15. Teleportation
16. Quantum Simulation
17. Quantum Cloning
18. Shor’s Algorithm
19. No-Hiding Theorem
√
√√
√
√√
√
√
√√
Maximum number of qubits achieved in our lab: 8
√ Also performed in our Lab.
√
In other labs.: 12 qubits; Negrevergne, Mahesh, Cory, Laflamme et al., Phys. Rev. Letters, 96, 170501 (2006).
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NMR sample has ~ 1018 spins.
Do we have 1018 qubits?
No - because, all the spins can’t beindividually addressed.
Spins having different Larmor frequencies can beindividually addressed as many “qubits”
Progress so far
One needs resolved couplings between the spins in order to encode information as qubits.
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NMR Hamiltonian
H = HZeeman + HJ-coupling
= ∑ ω i Ι zi
+ ∑ Jij Ii • Ij
i i < j
Weak coupling Approximation
ω i − ω j>> Jij
Two Spin System (AM)
A2 A1 M2 M1
ω A ω M
M1= | 0 A⟩
M2= | 1 A⟩
A1= | 0 M⟩
A2= | 1 M⟩
| α α ⟩ = | 00⟩
| β β ⟩ = | 11⟩
| α β ⟩ = | 01⟩
| β α ⟩ = | 10⟩
H = ∑ ω i Ι zi + ∑ Jij Izi Izj
Under this approximation all spins having same Larmor Frequency
can be treated as one Qubit
i i < j
Spin States are eigenstates
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13CHFBr2
An example of a three qubit system.
A molecule having three different nuclear spins having different Larmor frequencies all coupled to each other
forming a 3-qubit system
Homo-nuclear spins having different Chemical shifts (Larmor frequencies) also form multi-qubit systems
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1 Qubit1 Qubit
00
0110
11
0
1
CHCl3
000
001010
011
100
101110
111
2 Qubits2 Qubits 3 Qubits3 Qubits
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Unitary Transforms in NMR
1. Rational Pulse design. (using RF Pulses and coupling (J)-evolution)
2. Optimization Techniques
σinitial
U1
σ1
Goodness criterion
C1 = l σ1 – σf lIterate to minimize C1
σfinal
Various optimization Techniques used in NMR
(a) Strongly Modulated Pulses (SMP) (Cory, Mahesh et. al)(b) Control Theory (Navin Kheneja et. al (Harvard))(c) Algorithmic Technique (Ashok Ajoy et. al)(d) Genetic Algorithm (Manu)
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1. Rational Pulse design.
using RF Pulses and coupling (J)-evolution
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The two methods
Coupling (J) Evolution Transition-selectivePulses
Examples
I1
I2
y x
1/4J 1/4J
00
0110
11πXOR/C-NOT
I1
I2
NOT1
00
0110
11
π
π
I1z+I2z
I1z+I2x
I1z+2I1zI2y
I1z+2I1zI2z
y
(1/2J)
x
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I1
I2
y x -y
y x -y
00
0110
11
π 1
SWAP
I1
I2
y y -x -x
I3
Toffoli
000
001010
011
100
101110
111
I1
I2
y y -x x
I3
OR/NOR
000
001010
011
100
101110
111non-selective π pulse + a π on 000 ⇔ 001
π 2π 3
π
π
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CNOT GATE
1
2y
2
π
x
2
ππ
π
z
x
y
22
11 zzeq II γγρ +∝
22
112 xz II γγρ +=
[ ]212
113 2 yzz III −+= γγρ
[ ]212
114 2 zzz III γγρ +=
IN OUT
|00> |00>
|01> |01>
|10> |11>
|11> |10>
( )212
1 γγ +
( )212
1 γγ −
( )212
1 γγ −−
( )212
1 γγ +−
( )212
1 γγ +
( )212
1 γγ −−
( )212
1 γγ −
( )212
1 γγ +−
eqρ4ρ
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Logic GatesUsing 1D NMR
NOT(I1)
C-NOT-2XOR2
C-NOT-1XOR1
Kavita Dorai, PhD Thesis, IISc, 2000.
�
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1
2y
2
π
x
2
ππ
πx
2
π
π
π
y−
2
π
y−
2
πy
2
π
1 2 3 4 5
11
01
00
10
( )212
1 γγ +
( )212
1 γγ − ( )212
1 γγ −−
( )212
1 γγ +−
22
11 zzeq II γγρ +∝
1
SWAP GATE
2
22
111 xx II γγρ +=
[ ] [ ]212
2112 22 zyzy IIII −+−= γγρ
[ ] [ ]212
2113 22 yzyz IIII −+= γγρ
[ ] [ ]12
214 xx II γγρ +=
12
215 zz II γγρ +=
3
4
2,1
2 x
π
J
2,1
2 x−
π
|00>
|01>
|10>
|11>
( )212
1 γγ +
( )212
1 γγ −
( )212
1 γγ −−
( )212
1 γγ +−
( )212
1 γγ +
( )212
1 γγ −−
( )212
1 γγ −
( )212
1 γγ +−
5
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Logical SWAP
Kavita, Arvind, and Anil KumarPhys. Rev. A 61, 042306 (2000).
0 0⟩
0 1⟩
1 0⟩
1 1⟩
0 0⟩
1 0⟩
0 1⟩
1 1⟩
INPUT OUTPUT
π
XOR+SWAP
0
| 11⟩
1
| 10⟩
2
| 00⟩
1
| 01⟩ π2
π1 π
3
π
ε 1 , ε 2 ⟩ ε 2 , ε 1⟩
ε 1
ε 2
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Kavita Dorai, PhD Thesis, IISc, 2000.
Toffoli Gate = CToffoli Gate = C22-NOT-NOTε 1 , ε 2 , ε 3 ⟩
ε 1 , ε 2 , ε 3 ⊕ (ε 1
ε 2)⟩Input Output
000 000001 001010 010011 111100 100
101 101 110 110111 011
AND
NANDπ
^
ε 1
ε 3
ε 2
Eqlbm.
Toffoli
ε 2
ε 3
ε 1
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Strongly Modulated Pulses (SMP)
(Cory, Mahesh et. al)
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Adiabatic Satisfibility problem using Strongly Modulated Pulses
Avik Mitra
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In a Homonuclear spin systems
spins are close(~ kHz) in frequency
space
Pulses are oflonger duration
Decoherence effects cannot
be ignored
Strongly Modulated Pulses circumvents the above problems
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NMR Implementation, using a 3-qubit system.
The Sample.
C CIF
F F
a
b c
Hz8128J
Hz948J
Hz168J
bc
ac
ab
.
.
.
−===
Equilibrium Specrum.
Iodotrifluoroethylene(C2F3I)
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Implementation of Adiabatic Evolution
xI=++= 3x
2x
1xB IIIH zI=++= 3
z2z
1zF IIIH
πππ 2
180
M
m1i
180
M
mi
2
180
M
m1i
m
xzx
eeeU
−−
−
−−
⋅⋅≈III
( ) FB HM
mH
M
m1mH +
−=
pulse 3,2,1
x
m
2
α [ ] pulse 3,2,1
zmβpulse
3,2,1
x
m
2
α
mth step of the interpolating Hamiltonian .
mth step of evolution operator
•Pulse sequence for adiabatic evolution• Total number of iteration is 31• time needed = 62 ms (400µ s x 5 pulses x 31 repetitions)
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Strongly Modulated Pulses.
( ) ( ) ( ) lllrf
leffiH
l1
zl
llSMP eUU τφωωτδ −−⋅∆= ∏ω
rf
∆τ
φ
slope: ω
[ ] 2
SMPT
N
UUTrF
⋅=
Nedler-Mead Simplex Algorithm (fminsearch)
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Using Concatenated SMPs
Avik Mitra et al, JCP, 128, 124110 (2008)
Duration: Max 5.8 ms, Min. 4.7 ms
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Avik Mitra et al, JCP, 128, 124110 (2008)
Results for all Boolean Formulae
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Algorithmic Technique (Ashok Ajoy et. al PRL under review)
Applied for provingQuantum No-Hiding Theorem
by NMR
Jharana Rani Samal, Arun K. Pati and Anil Kumar,Phys. Rev. Letters, 106, 080401 (25 Feb., 2011)
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Quantum Circuit for Test of No-Hiding Theorem using State Randomization (operator U).
H represents Hadamard Gate and dot and circle represent CNOT gates.
After randomization the state |ψ> is transferred to the second Ancilla qubit proving the No-Hiding Theorem.
(S.L. Braunstein, A.K. Pati, PRL 98, 080502 (2007).
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U =
The Randomization Operator is obtained as
Blanks = 0
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Conversion of the U-matrix into an NMR Pulse sequence has been achieved here by a Novel Algorithmic Technique, developed in our laboratory by Ajoy et. al (to be published). This method uses Graphs of a complete set of Basis operators and develops an algorithmic technique for efficient decomposition of a given Unitary into Basis Operators and their equivalent Pulse sequences.
The equivalent pulse sequence for the U-Matrix is obtained as
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Experimental Result for the No-Hiding Theorem. The state ψ is completely transferred from first qubit to the third qubit
325 experiments have been performed by varying θ and φ in steps of 15o
All Experiments were carried out by Jharana (Dedicated to her memory)
Input State
Output State
s
s
S = Integral of real part of the signal for each spin
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Genetic Algorithm
We present here our latest attempt to use Genetic Algorithm (GA) for direct numerical optimization of rf pulse sequences and devise a probabilistic method for doing universal quantum computing using non-selective (hard) RF Pulses.
We have used GA for
Quantum Logic Gates ( Operator optimization)
and
Quantum State preparation (state-to-state optimization)
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Representation Scheme
Representation scheme is the method used for encoding the solution of the problem to individual genetic evolution. Designing a
good genetic representation is a hard problem in evolutionary computation. Defining proper representation scheme is the first
step in GA Optimization.
In our representation scheme we have selected the gene as a combination of
(i) an array of pulses, which are applied to each channel with amplitude (θ) and phase (φ),
(ii) An arbitrary delay (d).
It can be shown that the repeated application of above gene forms the most general pulse sequence in NMR
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The Individual, which represents a valid solution can be represented as a matrix of size (n+1)x2m. Here ‘m’ is the number of genes in each individual and ‘n’ is the number of channels (or spins/qubits).
So the problem is to find an optimized matrix, in which the optimality condition is imposed by a “Fitness Function”
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Fitness function
In operator optimization
GA tries to reach a preferred target Unitary Operator (Utar) from an initial random guess pulse sequence operator (Upul).
Maximizing the Fitness function
Fpul = Trace (Upul Χ Utar )
In State-to-State optimization
Fpul = Trace { U pul (ρin) Upul (-1) ρtar † }
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Two-qubit Homonuclear case
H = 2π δ (I1z – 12z) + 2π J12 (I1zI2z)
Single qubit rotation
δ = 500 Hz, J= 3.56 Hz φ1 = 2π, φ2 = π, Θ = π/2, φ = π/2
φ1 = π, φ2 = 0Θ = π/2, φ = π/2
π/2 π/2
Simulated using J = 0
Hamiltonian used
Non-Selective (Hard) Pulses applied in the centre
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Fidelity for finite J/δ
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Controlled- NOT:
Equilibrium
00
01 10
11
1
-1
00
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Pseudo Pure State (PPS) creation
All unfilled rectangles represent 900 pulseThe filled rectangle is 1800 pulse. Phases are given on the top of each pulse.
Fidelity w.r.t. to J/δ
00 01 10 11
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Controlled- Hadamard:
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Bell state creation: From Equilibrium (No need of PPS)
Bell states are maximally entangled two qubit states.
The Singlet Bell State
Experimental Fidelity > 99.5 %
Shortest Pulse Sequence for creation of Bell States directly from Equilibrium
All blank pulses are 900 pulses. Filled pulse is a 1800 pulse.Phases and delays Optimized for best fidelity.
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CC-NOT
Controlled SWAP:
Creating GHZ state using nearest neighbor coupling
Three qubit system :
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We plan to use these GA methods for implementation of various Algorithms
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Thank Thank YouYou
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