Download JOYNT

A Physically-motivated Algorithm
for the Graph Isomorphism
Problem
Robert Joynt
University of Wisconsin-Madison
Work in collaboration with
Shiue-yuan Shiau, Sue Coppersmith
Thanks to Eric Bach and Dieter van Melkebeek
Quantum Information and Computation 4, 492 (2005)
NSF presentation
Washington, DC, September 10, 2007
Outline
• Graphs and Graph Isomorphism
• A little computer science, classical and
quantum
• Quantum physics algorithm
• Implications for quantum computing
AN ABSTRACT PATTERN:
THE GRAPH
A graph is defined geometrically by :
A set of N points in space vi, some
pairs of which are connected by lines:
a line from v1 to v4, from v8 to v6, etc.
G′
G
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THE GRAPH AS AN ALGEBRAIC
PATTERN
A graph is defined algebraically by
an N X N adjacency matrix Aij in which:
Aij = 1, if i and j are connected by an edge
Aij = 0, otherwise
Note that A is
(a) Binary (i.e., consists of zeros and ones)
(b) Symmetric (and therefore Hermitian)
GRAPH ISOMORPHISM
G′
G
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G goes into G′ if we move:
1→ 4, 2 → 5, 3 → 7, 4 → 8, 5 → 2, 6 → 1, 7 → 3, and 8 → 6.
If such a transformation exists, then we say that G and G’ are isomorphic.
The problem of determining whether two graphs are isomorphic is called
the graph isomorphism (GI) problem and it is a classic problem of computer
science, a pattern recognition problem in a decisional form.
GI has applications to optimization, communications,
enumeration of compounds and atomic clusters, fingerprint
matching, etc.
Strongly Regular Graphs (SRGs)
• A SRG with parameters (N, k, λ, µ) is a
graph with N vertices in which each vertex
has k neighbors, each pair of adjacent
vertices has λ neighbors in common, and
each pair of non-adjacent vertices has µ
neighbors in common.
• The one at right has N = 9, k = 4, λ = 1,
µ= 2.
• Non-isomorphic pairs of SRGs with the
same parameter sets are known to be
very difficult to distinguish: many simple
algorithms fail – this is where the really
serious algorithms are tested.
Two non-isomorphic strongly-regular graphs
(16,9,4,6) – the smallest known such pair !
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Complexity theory
• P is the set of problems that are soluble in polynomial
time
• NP is the set of problems whose solutions are
checkable in polynomial time – it has never been
shown that P ≠ NP
• NP-complete problems are the hardest ones in NP:
those whose solution would guarantee, via a
polynomial mapping, the solution of all NP problems
in polynomial time – most well-known problems in
NP have been shown to be NP-complete, but GI is an
exception, as is factoring
WHERE DOES THE GI PROBLEM SIT
IN THIS SCHEME?
• Naively, GI is difficult – to search the set of all
permutations would take N! operations!
• It is not presently known whether GI can be solved in
polynomial time: the best existing algorithm takes a
time of order exp [(cN log N)1/2], with c = constant.
• GI is certainly in NP but is thought to be not NPcomplete. It therefore occupies a somewhat unusual
intermediate position (NP-intermediate?) among the
unsolved problems in classical complexity theory, as
does factoring.
• Can we put physics at the service of computer science
here, specifically the ability of QCs to efficiently
simulate quantum systems?
Why might there be physics here?
• In matrix terms, the GI problem is: given two N X N
adjacency matrices A and A′, does there exist a
permutation matrix P such that
A′ = PAP-1 ?
• Symmetry problem with a QM flavor
• Similarity to tight-binding and other models used in
condensed-matter physics
• We will use physical processes to compute graph
invariants: quantities that are the same when
computed for isomorphic graphs of course we hope
they are also different for non-isomorphic graphs
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‘Quantum’ Algorithms
[also see T. Rudolph, quant-ph/0206068].
• One-particle quantum random walk on the graph
• Two-particle quantum random walk on the graph
with the particles being
– Two non-interacting fermions
– Two non-interacting bosons
– Two hard-core bosons
We have calculated the energy eigenvalues
(following Rudolph) and the full sorted set of
walk amplitudes (defined below) in position
space, to determine whether these invariant sets
will distinguish non-isomorphic graphs.
Quantum Random Walks on a Graph
• The Hamiltonian is
given by
 
H   Aij ci c j  U  ci ci

i

2
i
The ci can be:
fermion operators: cicj+ + cjci+ = δij
or boson operators: cicj+ - cjci+ = δij
U=0 for the noninteracting particles, but U → ∞
for the hard-core bosons.
Numerical test of the quantum-mechanical algorithms
Compute
Oi , j  i exp( iHt ) j
One-particle GF
(doesn’t work!)
Oij,kl  ij exp( iHt ) kl
Two-particle GF
(Similarly for the two-particle case)
R and I are the distances between the sorted
amplitudes for two-non-isomorphic SRG’s
SINGLE-PARTICLE
AMPLITUDES DON’T WORK !
The adjacency matrix of a SRG has the following properties:
• For a general graph, the (a, b) entry of A2 is the number of
vertices adjacent to both a and b. For SRGs,this number is (A2)ab =
k if a = b, (A2)ab = λ if a is adjacent to b, and (A2)ab = µ if a is not
adjacent to b.
• Hence A2 = kI + λA + µ(J -I - A), where I is the identity matrix
and J is the matrix consisting entirely of 1’s.
• J2 = NJ
• A and J also have the properties that AJ = JA = kJ.
The matrices A, I, and J form a closed algebra whose properties
depend only on the set (N, k, λ, µ), and the dynamical process can
be mapped into an orbit in this algebra. Non-isomorphic SRGs
with the same parameters follow the same orbit and this implies
that the sorted walk amplitudes are the same. We have verified
this theorem numerically.
Results for quantum case
R = Σ |Re Oij – Re Oij′| and I = Σ |Im Oij – Im Oij′|
R = I= 0 means that the algorithm has failed !
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Soft-core bosons work, too
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R and I for the two non-isomorphic SRGs with N = 16.
Implications for Quantum Computing
• The two-particle interacting boson algorithms are
polynomial-time even on a classical computer and
certainly would be on a quantum computer. It seems
likely that they will not distinguish all graphs, but
proving this is a pressing issue.
• N/2-particle algorithms (which have an exponentially
large Hilbert space dimension) might very well
distinguish all graphs. A single N/2-particle quantum
walk can be easily implemented in polynomial time
on a quantum computer but would take exponential
time on a classical computer.
• However, we need a smaller output than the
quantities Oij above, since the number of these grows
exponentially with N if the particle number is N/2. Is
there a quantum algorithm that would work by
interfering the two graphs?
Current Direction: Distinguishing Operators
• The adjacency matrix A for an SRG has only three distinct
eigenvalues, implying that A satisfies a cubic equation:
(A-λ1I) (A-λ2I) (A-λ3I)=0,
so that exp(iHt) = aA2+bA+c for some a,b,c.
Generalizing this, we find that noninteracting bosons
(fermions) have 6 (5) independent operators, while interacting
bosons have 16, acting in the two-particle space.
• Only a small subset of the operators actually distinguish
between graphs, in the sense that their matrix representations
can be distinguished in polynomial time by our procedures. In
the fermion case, only two operators are distinguishing, in this
sense.
• We now focus on the construction and diagnosis of twoparticle operators, and on the sudden failure of the fermion
algorithm
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Classical Dynamical Algorithm
[V. Gudkov and S. Nussinov, cond-mat/0209112].
Place the vertex v1 initially at the point r1 (t=0) = (1,0,…,0),
v2 at r2 (t=0) = (0,1,0,…,0),…and vN at rN (t=0) = (0,0,…,1) in Ndimensional space. We regard these as mass points and let them move
according to:
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Testing for Isomorphism
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Aij is the adjacency matrix of the graph G. The
evolution is computed numerically for some
interval T. After this time, we compute the set of
distances dij = |ri-rj| and sort them in increasing
order.
We do the same for the graph G’, obtaining another
sorted set of distances dij’. If the set sets are not the
same, then the graphs are clearly not isomorphic.
But, can two non-isomorphic graphs produce the
same sorted set dij ? If so, then this fails as a test
for GI.
Quantum computing
• The state of a classical computer is given
by, e.g., 011100101010….
• The state of a quantum computer is given
by a linear combination of all such strings
• The most powerful quantum algorithms,
e.g., Shor’s algorithm, depend on the
quantum Fourier transform to find the
period of a discrete function, i.e., to
recognize a pattern
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Pattern recognition in Shor’s algorithm to
factor 91
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This is the function y(x) = 4x (mod 91).
Shor’s quantum algorithm determines that y(x+7) = y(x)
So 4(x+7) = 4x, or 46 = 1 (mod 91)
Factoring this: 0 = 46 – 1 = (43+1) ∙ (43-1) = 65 ∙ 63 = 0 (mod 91), and taking the
greatest common denominator of 65 and 63 gives 7 ∙ 13 = 91.
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IF P≠NP
(“It’s a harder to find a good idea than to be able to recognize
good idea.” OR “Good artists are rarer than good critics.”)
NP
NP-complete
NPI
HARDER
GI
P?
QP?
P
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Pattern Recognition
“To understand is to perceive patterns”
- Isaiah Berlin
Objects (highly spatial patterns)
Physical Laws (sometimes spatial patterns)
Personalities (not very spatial patterns)
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“I have no use for computers –
they only give you answers.”
-Pablo Picasso
(A better artist than critic!)
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