A Physically-motivated Algorithm for the Graph Isomorphism Problem

1. 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

2. Outline • Graphs and Graph Isomorphism • A little computer science, classical and quantum • Quantum physics algorithm • Implications for quantum computing

3. 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 1 2 4 6 7 3 7 8 2 3 5 4 1 5 6 8

4. 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)

5. GRAPH ISOMORPHISM G′ G 1 2 4 6 3 7 7 2 8 3 5 4 1 6 5 8 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.

6. 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.

7. Two non-isomorphic strongly-regular graphs (16,9,4,6) – the smallest known such pair ! 10 12 10 5 2 12 2 6 5 6 4 1 3 4 1 3 7 11 9 14 14 8 11 9 8 7 15 13 15 16 16 13

8. 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

9. 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 NP-complete. 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?

10. 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 18

11. ‘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.

12. Quantum Random Walks on a Graph • The Hamiltonian is given by 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.

13. Numerical test of the quantum-mechanical algorithms Compute One-particle GF (doesn’t work!) 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

14. 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.

15. 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 ! 5

16. Soft-core bosons work, too R and I for the two non-isomorphic SRGs with N = 16. 4

17. 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?

18. 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 two-particle operators, and on the sudden failure of the fermion algorithm 3

19. 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 N-dimensional space. We regard these as mass points and let them move according to: 12

20. Testing for Isomorphism 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. 11

21. 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 14

22. Pattern recognition in Shor’s algorithm to factor 91 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. 13

23. 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 GI P? QP? HARDER P 15

24. G’ PG 4 5 3 6 7 8 1 2 G’ PG 4 4 5 6 3 3 7 6 2 7 5 8 1 1 8 2 19

25. Pattern Recognition “To understand is to perceive patterns” - Isaiah Berlin Objects (highly spatial patterns) Physical Laws (sometimes spatial patterns) Personalities (not very spatial patterns) 24

26. “I have no use for computers – they only give you answers.” • Pablo Picasso • (A better artist than critic!) 1