Benchmarking quantum annealing for community detection on synthetic social networks

27 Pages·2014·3.16 MB·English

Photos placed in horizontal position 

with even amount of white space

between photos and header
Benchmarking

 quantum
 annealing
 for
 community

detection
 on
 synthetic
 social
 networks
Ojas
 Parekh

Sandia
 National
 Labs

AQC,
 2014

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin
Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND NO. 2011-XXXXP
A
 question?
Which
 is
 harder
 (pick
 a
 reasonable
 definition
 of
 “hard”)?
Finding
 an
 optimal
 solution
  Finding
 a
 solution
 within

99%
 of
 the
 time 99%
 of
 optimal
 all
 the
 time
Which
 is
 more
 practically
 relevant?
2
A
 question?
Which
 is
 harder
 (pick
 a
 reasonable
 definition
 of
 “hard”)?
Finding
 an
 optimal
 solution
  Finding
 a
 solution
 within

99%
 of
 the
 time 99%
 of
 optimal
 all
 the
 time
“Average”-­‐case
 analysis,
 Approximation
 algorithm,

rather
 than
 worst-­‐case
or
 approximation
 scheme
3
How
 should
 we
 measure
 success?
7
Optimal 99% of the time Within 99% of optimal all of the time
55,000 random {-1,1}-weight instances
on 509-qubit D-Wave Two
40000
30000
20000
10000
0
96 97 98 99 100
% of optimal
Was
 always
 within
 96%
 of
 optimal!
4
FIG. 4. Speedup for ratio of quantiles for the DW2 FIG. 5. Comparing wall-clock times A comparison of the
compared to SA. A) For instances with range r = 1. B) waLlle-cUloc:k
 Rtiomnentoofiwnd
 ethte
 asoll.u,
 tDionewfiinthinprgo
 baabnildit
 ydpe=te0c.9.9ng
 quantum
 speedup,
 arXiv:1401.2910v1
 (2014)
For instances with range r = 7. Shown are curves from the for SA running on a single CPU (dashed lines) compared to
median (50th quantile) to the 99th quantile. 16 gauges were the DW2 (solid lines) using 16 gauges. A) for range r = 1,
used. In these plots we multiplied Eq. (6) by 512 so that B) for range r = 7. Shown are curves from the median (50th
the speedup value at N = 512 directly compares one DW2 quantile) to the 99th quantile. The large constant program-
processor against one classical CPU. ming overhead of the DW2 masks the exponential increase of
time to solution that is obvious in the plots of pure annealing
time. Results for a single gauge are shown in the Supplemen-
tary Material.
the DW2 does not exhibit a speedup over SA for this
particular benchmark.
D. Instance-by-instance comparison
1. Total time to solution
We now focus on the question of whether the DW2
exhibits a limited quantum speedup for some fraction of
3. Wall-clock time the instances of our benchmark set. To this end we per-
form individual comparisons for each instance and show
in Figure 6A-B the ratios of time to solution between
While not as interesting from a complexity theory the DW2 and SA, considering only the pure annealing
point of view, it is instructive to also compare wall-clock time. We find a wide scatter, which is not surprising
timesfortheabovebenchmarks, aswedoinFigure5. We since we previously found that DW1 performs like a sim-
observe that the DW2 performs similarly to SA run on a ulated quantum annealer, but correlates less well with a
single classical CPU, for su�ciently large problem sizes simulated classical annealer [25]. We find that while the
and at high range values. Note that the large constant DW2 is sometimes up to 10 faster in pure annealing
⇥
programming overhead of the DW2 masks the exponen- time, there are many cases where it is 100 slower.
� ⇥
tial increase of time to solution that is obvious in the Considering the wall-clock times, the advantage of the
plots of pure annealing time. DW2 seen in Figure 6A-B for some instances tends to
We
 ask
▪ What
 is
 an
 appropriate
 measure
 of
 success?

▪ What
 classical
 algorithm(s)
 should
 be
 used
 for
 comparison?

▪ How
 should
 one
 select
 appropriate
 benchmark
 instances?
5
Problem
 definitions
▪ Ising:


min J x x + h x
ij i j i i
x 1,1
i
2{� } ij i
X X
!
▪ Quadratic
 binary
 unconstrained
 optimization
 (QUBO):


min A x x + c x
ij i j i i
x 0,1
i
2{ } ij i
X X
6
Comparison*of*Rainier*and*Vesuvius*chips*
Vesuvius*
Rainier*
506/512*
108/128*
spins*
spins*
Images
 from
 D-­‐Wave
 Systems:
 hbp://www.dwavesys.com
 .

7
Complexity
 of
 Ising
 on
 Chimera
▪ (Decision
 version)
 NP-­‐complete
 even
 with
 no
 linear
 term
 and

{-­‐1,
 0,
 1}
 weights
 [Barahona,
 1982]

▪ We
 show
 NP-­‐complete
 with
 no
 linear
 term
 and
 {-­‐1,1}
 weights

▪ Instances
 used
 in
 D-­‐Wave
 benchmarking
 studies

[with
 Benjamin
 Moseley
 at
 Washington
 University]

▪ Tree-­‐width
 (path-­‐width)
 is

 ,
 yielding

 p

 n

 algorithm

⇥(pn) O(2 )
▪ “Subexponential”
 exact
 algorithm
 even
 though
 NP-­‐hard

▪ Approximation
 complexity?

▪ Polynomial-­‐time
 approximation
 scheme
 (PTAS)

[Saket,
 2013,
 arXiv:1306.6943]

▪ PTAS’s
 are
 rarely
 efficient;
 theory
 vs
 practice?

▪ Efficient
 approx
 algorithm
 for
 say,
 getting
 within
 75%?
8
G can be solved in an adiabatic quantum computer that implements the spin-1/2
Ising Hamiltonian, by reduction through minor-embedding of G in the quantum
hardware graph U. By reduction through minor-embedding, we mean that one
can reduce the original Ising Hamiltonian on the input graph G to the embed-
emb
ded Ising Hamiltonian on its minor-embedding G , i.e., the solution
emb
H
to the embedded Ising Hamiltonian gives rise to the solution to the original
Ising Hamiltonian. We proved the correctness of the minor-embedding reduc-
tion. There are two components to the reduction: embedding and parameter
setting. The embedding problem is to find a minor-embedding G of a graph
emb
G in U. The parameter setting problem is to set the corresponding parameters,
qubit bias and coupler strengths, of the embedded Ising Hamiltonian. In [6],
we solved the parameter setting problem. The embedding problem, though, is
dependent on the hardware graph design problem discussed in the following
sections.
Approaches
 to
 problem
 embedding
4 TRIAD: Optimal Hardware Graph for Embedding Complete
Graph K
n
▪ Embedding
 is
 hard:
 O(nn)
 vs
 O(2n)

In this section, we describe a K -minor hardware graph, where K is a complete
n n
▪ Even
 harder
 when
 optimizing
 #
 qubits

graph of n vertices. A triangular layout of a K -minor graph [18], called T ,
RIAD
n
is shown in Figure 3.
▪ Choi:
 worst
 case
 O(n2)
 qubits
 for
 n
 vars

▪ Requires
 (linearly)
 large
 coupler
 weights
1 8
3 4
1 7 2 8
2 5 1 6 2 7 3 8
1 5 2 6 3 7 4 8
1 6
1 4 2 5 3 6 4 7 5 8
1 3 2 4 3 5 4 6 5 7 6 8
8 7
1 2 2 3 3 4 4 5 5 6 6 7 7 8
9
Fig.3: Left, K . Right, a triangular layout of a K -minor. Each vertex of K is mapped to a chain
8 8 8
of 7 “virtual” vertices (with the same color).
4.1 Construction of TRIAD
The idea behind the construction of T is to map each vertex of K to a
RIAD
n
chain of n 1 “virtual” vertices. The inductive construction is illustrated in
�
Figure 4.
Limits
 of
 reducing
 to
 Chimera
▪ Can
 we
 do
 better
 than
 a
 quadratic
 blowup
 in
 qubits?

▪ Probably
 not,
 due
 to
 Exponential
 Time
 Hypothesis

▪ Problems
 like
 Max-­‐Cut
 on
 general
 graphs
 are
 conjectured

n
to
 require

 O

 (

 2

 )

 time

pn
▪ But
 we
 have
 a

 O

 (

 2

 )

 time
 algorithm
 for
 Chimera
 Ising

▪ So
 in
 some
 sense
 quadratic
 factor
 is
 artifact
 of
 Chimera

▪ Weights
 make
 this
 worse:
 Choi
 embedding
 assumes
 (linearly)

large
 weights

2
▪ Reduction
 better
 than

 O

 (
 n

 )

 for
 Max-­‐Cut
 on
 bounded-­‐degree

graphs
 would
 improve
 best-­‐known
 classical
 algorithm

▪ Applies
 to
 any
 reduction,
 not
 just
 minor
 embeddings
10