A Quantum Observable for the Graph Isomorphism Problem

Transcription

1 A Quantu Observable for the Graph Isoorphis Proble Mark Ettinger Los Alaos National Laboratory Peter Høyer BRICS Abstract Suppose we are given two graphs on n vertices. We define an observable in the Hilbert space C[(S n S ) ] which returns the answer yes with certainty if the graphs are isoorphic and no with probability at least n! if the graphs are not isoorphic. We do not know if this observable is efficiently ipleentable. Introduction The graph isoorphis proble is to deterine if two graphs Γ, Γ on n vertices are isoorphic. Let Γ be the disjoint union graph of Γ and Γ. Without loss of generality we ay assue that both Γ and Γ are connected. In this case the autoorphis group of Γ is a subgroup of the wreath product S n S (which is itself a subgroup of S n ). Clearly, knowledge of a set of generators for this autoorphis group is sufficient to decide the isoorphis question. This fact has resulted in the suggestion that a quantu coputer ay be able to efficiently find a set of generators for the autoophis group and thus solve the graph isoorphis proble. This idea originates in the hidden subgroup view of quantu algoriths []. The Abelian hidden subgroup proble can be solved in polynoial tie and utilizes the Fourier observable or, equivalently stated, the quantu algorith utilizes the quantu Fourier transfor. We use the terinology NIS 8, MS B30, Los Alaos National Laboratory, Los Alaos, NM 87545, USA. Eail: ettinger@lanl.gov. BRICS, Departent of Coputer Science, University of Aarhus, DK 8000 Aarhus C, Denark. Eail: hoyer@brics.dk. Basic Research in Coputer Science, Centre of the Danish National Research Foundation.

2 Fourier observable to ephasize the particular point of view gerane to the ain result of this paper. A quantu algorith is siply a unitary change-of-basis transforation fro the coputational basis to the basis of the observable. We reark that in this paper Fourier observable refers to the Abelian case. The difficulties of finding hidden subgroups of noncoutative groups have been explored in several papers including [, 6]. For ore inforation on the Abelian hidden subgroup proble, see for exaple the references in [, 6]. There are several iportant differences between the observable presented here and the Fourier observable. The first difference is that the present observable operates on a larger Hilbert space. Recently it was shown in [] that a hidden noncoutative group ay be found in only polynoially any calls to the oracle function, although the algorith given in [] requires exponential tie. This result was proved by showing that the tensor product states corresponding to different possible hidden subgroups are alost orthogonal in the larger Hilbert space C[G ]. In the present paper we work in such a Hilbert space. The second difference is that our observable reveals nothing directly about the autoorphis group other than whether or not it contains an isoorphis between the two graphs. However we ay then find the full autoorphis group using a well known classical reduction [4]. Thirdly and finally, whereas it is known that the Fourier observable is efficiently ipleentable, we have not been able to deonstrate this for the observable presented below. Such an efficient ipleentation would result in a polynoial-tie quantu algorith for the graph isoorphis proble. The Observable Let G = S n S. Since the wreath product is a seidirect product (S n S n ) S we write an eleent as a triple (σ, τ, b). We refer to any eleent of G of the for k =(g, g, ) as an involutive swap. Let H = C[G ]. Note that di(h) = G = (n!). For each k G, we define a k-vector to be a vector of the for: ( ) ( c + c k ) ( c + c k ) for soe c,...,c G. Define H(k) to be the subspace spanned by all k-vectors. Notice that if v and v are unequal k-vectors then they are orthogonal. Therefore di(h(k)) = ( G ). Let H = k H(k)bethe

3 su over all n! involutive swaps. Notice that di(h ) n!( G ). Let H 0 be the orthogonal copleent to H in H. Our observable is defined as L = λ 0 P 0 + λ P where P 0 and P are projections onto H 0 and H respectively, and λ 0,λ C. Let us see what this observable yields when we apply it to the states that we ay easily produce, i.e., tensor products of coset states. Let H G be the autoorphis group of Γ. Let ψ be a tensor product of coset states of H, i.e. ψ = c H c H, where for any non-epty subset X G, X = x. X x X Theore If Γ and Γ are isoorphic then ψ P ψ =. Proof If Γ and Γ are isoorphic via the involutive swap k then k H, and thus any coset state of H ay be written (oitting noralizations): ch = ch + ch k + + ch H + ch H k. It is then easy to see that tensor products of these cosets state can be written as sus of k-vectors. For exaple c H c H =( c h + c h k ) ( c h + c h k )+. Any su of k-vectors is, by definition, in H and the result follows. Theore If Γ and Γ are not isoorphic then ψ P 0 ψ n!. Proof Assue the graphs are nonisoorphic. We show ψ P ψ n!. First, suppose ψ = g g = (g,g,...,g ). This occurs when 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 has the for: ( ) (g,...,g ) + (g k,...,g ) + + (g k,...,g k). Therefore ψ P (k) ψ =,wherep(k) is the projection onto H(k). This iplies ψ P ψ n!. For nontrivial H thearguentisalostidentical except that since ψ is not a basis state we ust su the probability contributions over the support, resulting in identical conclusions. 3

4 3 Conclusion We have described a quantu observable on a Hilbert space for which the logarith of its diension is polynoial in the nuber of vertices of the graphs. This observable decides the isoorphis question with high probability. However we do not know if this observable is efficiently ipleentable. Furtherore, we reark that Manny Knill [3] has observed that this observable suffices to also solve the code equivalence proble. Since linear codes have canonical fors we ay consider the code equivalence proble to be a hidden stabilizer proble over the sae group S n S. See [5] for a discussion of the relationship of the classical coplexities of graph isoorphis and code equivalence. Finally we reark on the group S n S 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 eleents of G of the for (σ, τ, b) whereboth σ and τ are even or both are odd. Thus G has index in G and this allows us to work in a saller Hilbert space. 4 Acknowledgeents We would like to thank Manny Knill and Richard Hughes, Gian-Carlo Rota and Alain Tapp for helpful discussions on this proble. References [] Ettinger, Mark and Peter Høyer, On quantu algoriths for noncoutative hidden subgroups. To appear in Proceedings of the Sixteenth International Syposiu on Theoretical Aspects in Coputer Science, 999. [] Ettinger, Mark,Peter Høyer and Manny Knill, Hidden subgroups states are alost orthogonal. In preparation, January 999. [3] Knill, Manny. Personal counication, Noveber 998. [4] Mathon, Rudolf, A note on the graph isoorphis proble, Inforation Processing Letters, Vol. 8, March 979, pp [5] Petrank, ErezandRonM.Roth, Is code equivalence easy to decide?, IEEE Transactions on Inforation Theory, Vol. 43, Septeber 997, pp

5 [6] Rötteler, Martin and Thoas Beth, Polynoial-tie solution to the hidden subgroup proble for a class of non-abelian groups. Available on Los Alaos e-print archive ( asquantph/98070 (Deceber 4, 998). 5