A Quantum Observable for the Graph Isomorphism Problem

Home , Hidden subgroup problem

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2 The Observable

Let G = Sn S2. Since the wreath product is a semidirect product (Sn ≀ × Sn) ⋊ S2 we write an element as a triple (σ, τ, b). We refer to any element of G of the form k = (g, g−1, 1) as an involutive swap. Let = C[Gm]. Note m m 2m H that dim( ) = G = 2 (n!) . For each k G, we deﬁne a k-vector to H | | ∈ be a vector of the form: 1 ( c1 + c1k ) ( cm + cmk ) √2m | i | i ⊗···⊗ | i | i for some c1,...,cm G. Deﬁne (k) to be the subspace spanned by all ∈ H k-vectors. Notice that if v1 and v2 are unequal k-vectors then they are |G| m orthogonal. Therefore dim( (k)) = ( 2 ) . Let 1 = k (k) be the H H P H

2 |G| m sum over all n! involutive swaps. Notice that dim( 1) n!( ) . Let H ≤ 2 0 be the orthogonal complement to 1 in . Our observable is deﬁned H H H as L = λ0P0 + λ1P1 where P0 and P1 are projections onto 0 and 1 H H respectively, and λ0, λ1 C. ∈ Let us see what this observable yields when we apply it to the states that we may easily produce, i.e., tensor products of coset states. Let H 6 G be the automorphism group of Γ. Let ψ be a tensor product of coset states | i of H, i.e. ψ = c1H cmH , | i | i⊗···⊗| i where for any non-empty subset X G, ⊆ 1 X = x . | i X X | i p| | x∈X

Theorem 1 If Γ1 and Γ2 are isomorphic then ψ P1 ψ = 1. h | | i Proof If Γ1 and Γ2 are isomorphic via the involutive swap k then k H, ∈ and thus any coset state of H may be written (omitting normalizations):

cH = ch1 + ch1k + + ch |H| + ch |H| k . | i | i | i · · · | 2 i | 2 i It is then easy to see that tensor products of these cosets state can be written as sums of k-vectors. For example

c1H c2H = ( c1h1 + c1h1k ) ( c2h1 + c2h1k )+ . | i ⊗ | i | i | i ⊗ | i | i · · · Any sum of k-vectors is, by deﬁnition, in 1 and the result follows. H ⊓⊔ n! Theorem 2 If Γ1 and Γ2 are not isomorphic then ψ P0 ψ 1 m . h | | i≥ − 2 n! Proof Assume the graphs are nonisomorphic. We show ψ P1 ψ m . h | | i ≤ 2 First, suppose ψ = g1 gm = (g1, g2,...,gm) . This occurs when | i | i⊗···⊗| i | i both graphs are rigid and H is trivial. For each involutive swap k there exists exactly one k-vector which is not orthogonal to ψ and this k-vector | i has the form: 1 (g1,...,gm) + (g1k,...,gm) + + (g1k,...,gmk) . √2m | i | i · · · | i 1 Therefore ψ P (k) ψ = 2m , where P (k) is the projection onto (k). This h | | in! H implies ψ P1 ψ m . For nontrivial H the argument is almost identical h | | i ≤ 2 except that since ψ is not a basis state we must sum the probability con- | i tributions over the support, resulting in identical conclusions. ⊓⊔

3 3 Conclusion

We have described a quantum observable on a Hilbert space for which the logarithm of its dimension is polynomial in the number of vertices of the graphs. This observable decides the isomorphism question with high probability. However we do not know if this observable is efficiently implementable. Furthermore, we remark that Manny Knill [3] has observed that this observable suffices to also solve the code equivalence problem. Since linear codes have canonical forms we may consider the code equivalence problem to be a hidden stabilizer problem over the same group Sn S2. ≀ See [5] for a discussion of the relationship of the classical complexities of graph isomorphism and code equivalence. Finally we remark on the group Sn S2 with which we have been working. ≀ We could equally well work over the subgroup G′ which is generated by the involutive swaps. It is not difficult to show that G′ consists of all elements of G of the form (σ, τ, b) where both σ and τ are even or both are odd. Thus G′ has index 2 in G and this allows us to work in a smaller Hilbert space.

4 Acknowledgements

We would like to thank Manny Knill and Richard Hughes, Gian-Carlo Rota and Alain Tapp for helpful discussions on this problem.

References

[1] Ettinger, Mark and Peter Høyer, “On quantum algorithms for noncommutative hidden subgroups”. To appear in Proceedings of the Six- teenth International Symposium on Theoretical Aspects in Computer Sci- ence, 1999.

[2] Ettinger, Mark, Peter Høyer and Manny Knill, “Hidden subgroups states are almost orthogonal”. In preparation, January 1999.

[3] Knill, Manny. Personal communication, November 1998.

[4] Mathon, Rudolf, “A note on the graph isomorphism problem”, Infor- mation Processing Letters, Vol. 8, March 1979, pp. 131 – 132.

[5] Petrank, Erez and Ron M. Roth, “Is code equivalence easy to de- cide?”, IEEE Transactions on Information Theory, Vol. 43, September 1997, pp. 1602 – 1604.

4 [6] Rotteler¨ , Martin and Thomas Beth, “Polynomial-time solution to the hidden subgroup problem for a class of non-Abelian groups.” Avail- able on Los Alamos e-Print archive (http://xxx.lanl.gov) as quant- ph/9812070 (December 24, 1998).