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Adiabatic Quantum Computation
Dorit Aharonov
Hebrew Univ. & UC Berkeley
1
Ground State Solutions







Which  spin distribution
minimizes the number of
red edges with similar spins
and green edges with
opposite spins?
(1 violation.)
1) A combinatorial minimization problem.
2) A lowest energy question for magnetic materials.
The ground state of the magnet is the solution to
our optimization problem.
2
Properties of Adiabatic Computation
• Language of Hamiltonians.
• New approach to designing quantum
algorithms
• Equivalent in power to quantum ckts.
• Natural fault-tolerance properties
• Laid back approach!
3
The Conventional Model of
Quantum Computers
|  ( L)  U LU L1   U1 |  (0)
U4
U3
|  (0) | 011...10
….
U1
Input
Output:
measure
U5
U2
Quantum Computing of “Classical” functions
4
“Quantum states”
Ground States
Schrodinger’s Equation:
i
d | ( t )
dt
 H (t ) |  (t )
The Hamiltonian (A Hermitian Matrix)
H (t )   H k ,l (t )
k ,l
Eigenvectors (eigenstates)
Eigenvalues (Energies)
| j
Ej
5
Ground state: Eigenvector with lowest eigenvalue
Classical Optimization
in terms of
Quantum states
Given: f: {0,1}n N, f(x) for x=x1,…..xn,
Objective: find xmin which minimizes f
| x
 f ( x000 )

.

H 



.
.






f ( x111)
are the eigenvectors
f(x) are the eigenvalues
6
The answer = state with minimal eigenvalue
Special Quantum States
1. Graph Isomorphism
[AharonovTa-Shma’02]
2. Closest Lattice Vector
v
v1
v2
0
1
n!
|  (G )


S n
As well as Factoring, Discrete Log…
7
[A’TaShma’02]
Apply a Hamiltonian with the desired
ground state
AND….
?
Adiabatic Computation
A method to help the system reach
a desired groundstate
8
Adiabatic Evolution
i
d | ( t )
dt
 H (t ) |  (t )
Adiabatic theorem: [BornFock ’28, Kato ’51]
H(0)
|  (0) Ground state of H(0)
T 
1
2
min s { ( t )}
H(T)
|  (T ) ground state of H(T)
 (t )  E1 (t )  E0 (t )
9
Adiabatic Systems as Computation Devices
H (s)  (1  s) H 0  sH T
|  (0)
HT
H0
| (T )
Input
Output
Algorithm:
• HT Hamiltonian with ground state |Y(T)i
• H0 Hamiltonian with known ground state |Y(0)I
• Slowly transform H0 into HT
Efficient: T<
nc
i.e.  ( s) 
1 10
nc
Remark 1: Non Negligible Spectral Gaps
Physics: Periodic Hamiltonians, n∞
γ > const or γ0
Adiabatic computation:
Tailored Hamiltonians , n∞
The interesting line is  ( s ) 
1
poly( n )
Allow it to go to zero if sufficiently slowly.
11
Remark 2: Connection to Simulated Annealing
i
d | ( t )
dt
H0
|  (0)
Adiabatic
Computation
Hamiltonian
Groundstate
Spectral gap
 H (t ) |  (t )
HT
| (T )
Rapidly mixing
Markov Chains
 Transition rate matrix
 Limiting Distribution
 Spectral gap for rapid mixing
Quantum Simulated Annealing
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Remark 3:
Adiabatic Optimization [FGGS’00]
Adiabatic Computation [ADKLLR’03]
HT   H i, j
H T   f ( x) | x x |
x{0 ,1}n
i, j
Diagonal HT
General Local HT
Final state is a basis state
Final state is
the groundstate
of a local Hamiltonian
Without increasing the physical resources:
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A Natural Model of Computation
Adiabatic Computation
The set of computations that can be performed by
Quantum systems, evolving adiabatically under the
action local Hamiltonians with non negligible
spectral gaps.
What is the
computational power of
Adiabatic Computers
?
What are the
possible dynamics of
Adiabatic systems
?
14
Overview
1 Adiabatic Computation
2 Previous Results Adiabatic Optimization
3 Main Result:
Adiabatic Computers Can perform any
Quantum Computation
4 Adding Geometry: True even if the adiabatic
computation is on 2 dim grid, nearest neighbor interactions
Implications and Open Questions
15
2.
Examples:
Adiabatic Optimization
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Adiabatic Algorithms for Optimization
[FarhiGoldstoneGutmanSipser’00].
Given: f: {0,1}n N, f(x) for x=x1,…..xn,
Objective: find xmin which minimizes f
HT 
| (T ) | xmin
 f ( x) | x
x|
x{0 ,1}n
F ( x1...xn )  ( x1  x2  x3 )  ( x2  x4  x7 )  ...
f(x) is number of unsatisfied clauses
H (T ) 
H
Clauses c
c
 |0001, 2 , 3  |101 2, 4, 7  ....
17 x
Energy Penalty: Project on Unsatisfying values of
Adiabatic Algorithms for Optimization (Cont’d)
[FarhiGoldstoneGutmanSipser’00].
| (T ) | xmin
| (0) 
H0
|0 |1
2

HT 
 f ( x) | x
x|
x{0 ,1}n
|0 |1
2
 ..... 
HT
|0 |1
2
n
H 0   ( |02|1 )(  0|2 1| ) j
j 1
H (s)  (1  s) H 0  sH T
 ( s)  poly1( n ) ?
• 20 bits: promising simulation [Farhi et al.’00,’01…]
• Mounting evidence that γ(s) is exponentially small in worst case
[vanDamVazirani’01, Reichhardt’03].
• Quadratic speed up: Adiabatic algorithm to solve NP in √2n. Classical NP
18
algorithm: 2n [RolandCerf’01,vanDamMoscaVazirani’01]
Tunneling:
Simulated Annealing vs Adiabatic Optimization
[FGGRV’03]
E(x)
E(x)
w(x)
w(x)
0
0
n
n
E ( x)  w( x)  Number of 1' s | (T ) | x | 0  | 0  .... | 0
min
n
xmin  00
....
0

H T   | 11 | j
n
j 1
Adiabatic optimization is
|0 |1
|0 |1
|0 |1
|

(
0
)



.....

Exponentially faster than
2
2
2
n
simulated annealing!
H  ( |0|1 )(  0|1| ) 19
But finding 0 is easy….
0

j 1
2
2
j
3.
How to Implement any Quantum
Algorithm
Adiabatically
20
Result
[A’TaShma’02,A’02,A’vanDamKempeLandauLloydRegev’03]
All of Quantum Computation can be done
adiabatically!
Condensed matter &
Mathematical Physics
Implication for Quantum computation:
Equivalence: New Language, new tools !
New vantage point to tackle the challenges of quantum computation:
1. Designing new algorithms: change of langauge, new tools.
2. Adiabatic Computation is resilient to certain types of errors
[ChildsFarhiPreskill’01]  Possible applications for
fault tolerance. (2-dim architecture)
Implications for Physics:
Understanding ground states, Adiabatic Dynamics from
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an information perspective.
What’s the Problem?
….
U1
H(T)
U4
U5
H(0)
U3 U
2
U1 ,  ,U L Local unitary gates
|  ( L)  U L   U1 | 0110...1
First try: Make
|  ( L)
Want to construct
adiabatic computation
with γ(t)>1/Lc from
which we can deduce
the answer.
the ground state of H(T).
Problem: To specify such a Hamiltonian
we need to know |  ( L) !
22
Key Idea
Kitaev’99, based on Feynman:
Time
steps
Classical computation:
|  (k )
:
|  (1)
|  (0)
Correct History can be
checked locally.
Instead of |  ( L) , use a local Hamiltonian H(T)
whose ground state is the History.
23
Key Idea
Time
steps
Kitaev’99, based on Feynman:
| history  
Classical computation:
Correct History can be
checked locally.
L
1
L 1
 |  (k )
| k
k 0
| k | 11
..100..0

k
L k
Instead of |  ( L) , use a local Hamiltonian H(T)
whose ground state is the History.
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The Hamiltonian H(s)
HT:
|  (T ) 
● Test correct
propagation:
Energy penalty
1
L 1
 |  (k )  | k 
k 0
L
H
HT 
1
2
Hk
 I  | k k |  I  | k  1k  1 |
k 0
k
 U k  | k k  1 |  U k  | k  1k |
Local interaction:
H0:
L
| k k  1 |  | 110100 |k 1,k ,k 1
| (0) |  (0) | 0..0 | 01..0 | 0..0
● Test that input is 0
n
L
H 0   | 11 | j | 00 |1  | 1251 |k
4.
Adding Geometry:
Adiabatic Computation
on a
Two-D Lattice
26
Particles on a 2-d Lattice
Wanted:
Evolution of the form
Problem:
|  (k ) | k , k  0,..., L
Not enough interaction between clock and computer
to have terms like: H k  I  | k k |  I  | k  1k  1 |
Solution:
 U k  | k k  1 |  U k  | k  1k |
Relax notion of computation/clock particles.
Each particle will have both types of degrees of freedom.
States will no longer be tensor products but will encode
time in their geometric shape.
To do this we use a
like evolution.
27
The 2-Dim Lattice Construction
Six states particles:
Unborn
0
0
1
1
First Phase
*
*
Second Phase
*
*
*
*
*
*
*
**
**
**
*
*
*
*
*
*
*
*
*
0*
1*
0*
*1
0*
*0
n
Dead
R
28
The Hamiltonian
As before: Check correct propagation by checking each
move; Each move involves only two particles.
Except: Moves may seem correct locally but are not.
Space of legal states is no longer invariant.
0
0
0
0
0
0
0
0
0
0
0
0
Solution: Add penalty for all “forbidden” shapes:
Fortunately, can be checked by checking nearest neighbors:
Hclock=∑
0
0
0
0
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To Summarize
Saw how to implement any Q algorithm adiabatically.
Algorithm Design:
New language: Ground states, spectral gaps.
What states can we reach?
What states are ground states of local Hamiltonians?
Methods from Mathematical Physics?
Fault Tolerance: Adiabatic comp. is naturally robust.
Adiabatic Fault Tolerance?
Ground states:
All states are ground states of local Hamiltonians,
30
Adiabatic dynamics are general.
Slow
down,
you
move
too
fast……
31