Transcript Schrödinger’s Elephants & Quantum Slide Rules Solving NP-complete problems with approximate adiabatic evolution

Schrödinger’s Elephants &
Quantum Slide Rules
Solving NP-complete problems with
approximate adiabatic evolution
A.M. Zagoskin
S. Savel’ev
F. Nori
(FRS RIKEN & UBC)
(FRS RIKEN & Loughborough U.)
(FRS RIKEN & U. of Michigan)
Standard quantum computation



Consecutive application of
unitary transformations
(quantum gates)
Problem encoded in the initial
state of the system
Solution encoded in the final
state of the system


Precise time-domain
manipulations

complex design and
extra sources of
decoherences

Problem and solution encoded in
fragile strongly entangled states of
the system

effective decoherence time
must be large

Quantum error-correction (to
extend the coherence time of the
system)

overhead (threshold
theorems: 104-1010(!))
digital operation
Examples:
Shor’s algorithm
Grover’s algorithm
quantum Fourier transform
Aharonov, Kitaev & Preskill,
quant-ph/05102310
Adiabatic quantum computation
Continuous adiabatic evolution of
the system
 Problem encoded in the
Hamiltonian of the system
 Solution encoded in the final
ground state of the system


“Space-time swap”: the timedomain structure of the algorithm
is translated to the timeindependent structural properties
of the system

Ground state is relatively robust

much easier conditions on
the system and its evolution

Well suited for the realization by
superconducting quantum circuits
Farhi et al., quantph/0001106; Science
292(2001)472

The approach is equivalent to the
standard quantum computing
Aharonov et al., quantph/0405098
Kaminsky, Lloyd & Orlando,
quant-ph/0403090
Grajcar, Izmalkov & Il’ichev,
PRB 71(2005)144501
Travelling salesman’s problem*
N points with distances dij
 Let nia=1 if i is stop #a and 0 otherwise;
there are N2 variables nia (i,a = 1,…,N)


The total length of the tour
1
L   d ij nia (n j ,a 1  n j ,a 1 )
2 ij,a
 nia  1 and  nia  1
a
i
*See e.g. M. Kastner, Proc. IEEE 93 (2005) 1765
Travelling salesman’s problem

The cost function
1
H ts   d ij nia (n j ,a 1  n j ,a 1 ) 
2 ij,a
2
2





 
 1   nia    1   nia  
2  a 
i
i 
a

 
Travelling salesman’s problem

Ising Hamiltonian
1
H ts   d ij ( 1 2   z ia )(1   z j ,a 1   z j ,a 1 ) 
2 ij,a

2
 N
 N
z 
z 
 1     ia    1     ia 
2  a 
2
2 a
i
i 


2



Spin Hamiltonian


1
z
z
H   J jk j k   (t ) h j j
2 jk
k
 H 0  (1   (t ))V

Adiabaticity parameter
  1 / T
Adiabatic optimization
H ( )  H 0  (1   )V
H ( )  H 0  (1   )V


Approximate adiabatic optimization
vs. simulated annealing
H ( )  H 0  (1   )V

Approximate adiabatic optimization
vs. simulated annealing

RMT theory near
centre of spectrum*

Simulated
annealing**
Diffusive behaviour
D T
(   2) / 2
Residual energy
  DT  T
 / 4
β = 1 (GOE); 2 (GUE)
*M. Wilkinson, PRA 41 (1990) 4645
  ln T 

ζ≤6
**G.E. Santoro et al.,
Science 295 (2002) 2427
Running time vs. residual energy

Classical/quantum simulated annealing
(classical computer)
Tanneal  exp 

1/ 

Approximate adiabatic algorithm (quantum
computer)
Tadiab  
4 / 
AQC vs. Approximate AQC

Solution is encoded in the final
ground state

Error produces unusable results
(excited state does not, generally,
encode an approximate solution)

Objective: minimize the probability
of leaving the ground state

Solution is a (smooth enough)
function of the energy of the final
ground state

Error produces an approximate
solution (energy of the excited
state is close to the ground state
energy)

Objective: minimize the average
drift from the ground state

Relevant problems:
finding the ground state
energy of a spin glass
traveling salesman
problem
Generic description of level
evolution: Pechukas gas*
H 0  V  m  Em m
xm  Em ; vm  Vmm ; lmn  ( Em  En )Vmn
d
xm  vm
d
lmn
2
d
vm  2 
3
d
m  n  xm  xn 


d
1
1

lmn   lmk lkn 

2
2 
d
k  m,n
  xm  xk   xk  xn  
*P. Pechukas, PRL 51 (1983) 943
Pechukas gas kinetics
F1 ( x, v)    ( x  x j ) (v  v j )
j
G1 (l )    (l  l jk )
jk
F2 ( x, v; y, u; l )    ( x  x j ) (v  v j ) ( y  xk ) (u  vk ) (l  l jk )
j
***
f1 ( x, v)  F1 ( x, v)
g1 (l )  G1 (l )
f 2 ( x, v; y, u; l )  F2 ( x, v; y, u; l )
***
Pechukas gas kinetics:
taking into account Landau-Zener transitions
 2



min
PLZ  exp 


4  Vm,m1 

Pechukas gas flow simulation
Level collisions and LZ transitions
“Diffusion” from the initial state
Analog vs. digital
4-flux qubit register
*M. Grajcar et al., PRL 96 (2006) 047006
Conclusions





Eigenvalues behaviour is not described by simple
diffusion
Marginal states behaviour qualitatively different:
adiabatic evolution generally robust
Analog operation of quantum adiabatic computer
provides exponential speedup
Advantages of Pechukas mapping: exact, provides
intuitively clear description and controllable
approximations (BBGKY chain)
In future: external noise sources; mean-field theory;
quantitative theory of a specific algorithm realization;
investigation of the class of AA-tractable problems