Download Introduction to Adiabatic Quantum Computation

Introduction to
Adiabatic Quantum Computation
Mohammad Amin
D-Wave System Inc.
Topics:
1. Introduction to AQC
2. Effect of noise on AQC
3. Quantum annealing and quantum phase transition
4. Quantum annealing with superconducting qubits
•2•
Quantum Evolution
Schrödinger equation:
ih
dψ
dt
=Hψ
Planck constant
Time evolution operator:
For time-independent H:
•3•
Hamiltonian
ψ (t ) = U (t ) ψ (0)
U (t ) = e
− iHt / h
Quantum Gates
ψ
Uψ
H
0
δt
t
Hamiltonian is applied only at the time of gate operation
•4•
Eigenstates and Eigenvalues:
H ψ n = En ψ n
Eigenstate
Eigenvalue
Question:
What happens if the system starts in an eigenstate?
ψ (0) = ψ n
ψ n (t ) = e −iHt / h ψ n = e − iE t / h ψ n
n
Answer: Nothing!
The system will stay in the same eigenstate
•5•
Adiabatic Quantum Computation (AQC)
E. Farhi et al., Science 292, 472 (2001)
E
.
.
.
System Hamiltonian:
H = (1− λ) HB + λ HP
gm
.
.
.
E1
E0
0
• Prepare the system in the ground state of HB
• Slowly vary λ(t) from 0 to 1
• Read out the final state which is ground state of HP
•6•
1
λ
Computation Time
Energy-time
uncertainty relation:
δt ⋅ δE > h
Planck constant
Uncertainty of energy
(superposition in energy basis)
Time spent on the energy level
•7•
Computation Time
E
Energy-time
uncertainty relation:
Ef
δt ⋅ δE > h
δE < g m
h
⇒ δt >
gm
Uniform time sweep:
Non-uniform time sweep:
•8•
δt
tf
t
Minimum gap
δt
gm
=
tf Ef
⇒ tf >
hE f
g m2
h
t f ~ δt ⇒ t f >
Optimal
gm
Landau-Zener Problem
P1
E
2-State Hamiltonian:
1
H = − ( g mσ x +νtσ z )
2
P0
0
P1 (t = −∞) = 0 ⇒ P1 (t = +∞) = e
ν=
•9•
Ef
tf
⇒ P1 (t f ) ≈ e
−
π 2 g m2
hE f
tf
π 2 g m2
−
hν
t
Exact solution
<< 1 ⇒ t f >>
hE f
g m2
Adiabatic Theorem
t f >>
max λ∈[ 0,1] 0 dH / dλ 1
g m2
Controversy about adiabatic theorem:
K.-P. Marzlin and B. C. Sanders, PRL 93,160408 (2004)
M.S. Sarandy, L.-A. Wu, D.A. Lidar, Quant. Info. Proc. 3, 331 (2004)
D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, PRL 95, 110407 (2005)
Z. Wu and H. Yang, PRA 72, 012114 (2005)
S. Duki, H. Mathur, and O. Narayan, PRL 97,128901 (2006)
M.H.S. Amin, PRL 102, 220401 (2009)
• 10 •
.
.
.
Adiabatic Theorem
t f >>
max λ∈[ 0,1] 0 dH / dλ 1
g m2
Rigorous versions of adiabatic theorem:
A. Ambainis, O. Regev, arXiv:quant-ph/0411152
S. Jansen, M.-B. Ruskai, and R. Seiler, J. Math. Phys. 48, 102111 (2007)
D.A. Lidar, A.T. Rezakhani, and A. Hamma, J. Math. Phys 50, 102106 (2009)
.
.
.
• 11 •
Two Types of Computation
1. Universal Adiabatic Quantum Computation
D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev,
SIAM J. Comput. 37, 166 (2007)
A. Mizel, D.A. Lidar, M. Mitchell, Phys. Rev. Lett. 99, 070502 (2007)
2. Adiabatic Quantum Optimization
E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda,
Science 292, 472 (2001)
Or Quantum Annealing
A. B. Finnila, M. A. Gomez, C. Sebenik, C. Stenson, and J. D. Doll,
Chemical Physics Letters 219, 343 (1994)
• 12 •
Universal AQC
Seth Lloyd, arXiv:0805.2757
Consider a quantum algorithm with n gates
represented by unitary operators: U1,U2 , ... ,Un
After l gates:
Therefore
ψ l = U l ... U 2U1 ψ 0
ψ l +1 = U l +1 ψ l
−1
ψ
=
U
and
l −1
l ψl
Solution:
ψ n −1 = U n −1 ... U 2U1 ψ 0
Reset:
ψ n = U n ψ n −1 = ψ 0
Write down a Hamiltonian
• 13 •
with ψ n −1 being in its ground state
Pointer or Clock Qubits
Define orthogonal states:
l , l = 0, ... , n
They can be represented by n additional qubits:
0 = 0...0001
1 = 0...0010
2 = 0...0100
...
n − 1 = 1...0000
n = 0
• 14 •
Hamiltonian
n −1
[
H P = −∑ U l +1 ⊗ l + 1 l + U l−+11 ⊗ l l + 1
l =0
1 n −1 i 2πkl / n
Ψk =
e
ψl ⊗ l
∑
n l =0
Define states
1
H P Ψk = −
n
1
=−
n
(
=−e
n −1
i 2πkl / n
e
ψ l +1
∑
l =0
∑ (e
n −1
i 2πk ( l −1) / n
l =0
i 2πk / n
+e
−i 2πk / n
1
⊗ l +1 −
n
n
i 2πkl / n
e
ψ l −1 ⊗ l − 1
∑
l =1
)
+ ei 2πk ( l +1) / n ψ l ⊗ l
) ∑e
= −2 cos(2πk / n ) Ψk
• 15 •
]
1
n
n −1
l =0
i 2πkl / n
ψl ⊗ l
Eigenstates of H
Ground State
H P Ψk = Ek Ψk
Eigenstates:
Eigenvalues:
Ground state:
First excited state:
Energy gap:
• 16 •
1 n −1 i 2πkl / n
Ψk =
e
ψl ⊗ l
∑
n l =0
Ek = −2 cos(2π k / n)
k = 0 ⇒ E0 = −2
2π
k = 1 ⇒ E1 = −2 cos
n
2π  2π 2

E1 − E0 = 21 − cos
≈ 2
n  n

Adiabatic Evolution
H = (1 − λ ) H B + λH P
Initial Hamiltonian:
H B = − l = 0 l = 0 + η (1 − ψ 0 ψ 0 ) ⊗ l = 0 l = 0
Final Hamiltonian:
n −1
[
H P = −∑ U l +1 ⊗ l + 1 l + U l−+11 ⊗ l l + 1
l =0
]
2π 2
Minimum gap happens at λ = 1, therefore g m ≈ 2
n
• 17 •
Computation time ~ n4
n = # of gates
Universality
AQC is computationally equivalent
to gate model quantum computation
They can be mapped to each other with
polynomial overhead
• 18 •
Practicality
n −1
[
H P = −∑ U l +1 ⊗ l + 1 l + U l−+11 ⊗ l l + 1
l =0
2-qubit interaction
=
+
]
2-qubit interaction
4-qubit interaction
Not practical!
Using perturbation it can be reduce to 2-qubit interactions
J. Kempe, A. Kitaev, and O. Regev, SIAM J. Computing 35(5), 1070 (2006)
Only XZ coupling or XX + ZZ couplings is enough
J.D. Biamonte and P.J. Love, Phys. Rev. A 78, 012352 (2008)
• 19 •
Hp is not diagonal in computation basis
Adiabatic Quantum Optimization
Hamiltonian:
H = (1− λ) HB + λ HP
Off-diagonal
(in computation basis)
Diagonal
(in computation basis)
The final state is a classical state
that minimizes the energy of Hp
• 20 •
Adiabatic Grover Search
Unstructured search problem:
Find state m in a data base containing
N = 2n entries with no structure
• 21 •
Classical search algorithm:
t f = O(N )
Grover search algorithm
t f = O( N )
Adiabatic Grover Search
(N-2)-fold degenerate
J. Roland and N. Cerf, PRA 65, 042308 (2002)
H = (1− λ) HB + λ HP
H B = 1− + +
1
+ =
N
H P = 1− m m
N −1
∑
En
N = 2n
z,
1
gm =
N
z =0
λ
Linear time sweep:
1
tf ~ 2 ~ N
gm
Classical speed
Optimal time sweep:
1
tf ~
~ N
gm
Grover speed
• 22 •
Not practical!
Ising Problem
Find a set of si = ±1 that minimizes
n
n
r
F ( s ) = ∑ hi si + ∑ J ij si s j ,
i =1
i , j =1
r
s = [ s1 , s2 ,..., sn ]
Ising problem is NP-Hard
• 23 •
Ising Hamiltonian
Find a set of si = ±1 that minimizes
n
n
r
F ( s ) = ∑ hi si + ∑ J ij si s j ,
i =1
i , j =1
r
s = [ s1 , s2 ,..., sn ]
H = (1− λ) HB + λ HP
Hamiltonian:
n
n
i =1
i , j =1
H P = ∑ hiσ iz + ∑ J ijσ izσ zj
Diagonal
(in σz basis)
n
H B = −∆ ∑ σ ix
i =1
• 24 •
Off-diagonal (transverse)
Open Question
How does computation time scale with n?
G. Rose et al., arXiv:1006.4147
• 25 •