[Paper Review] A Quantum Observable for the Graph Isomorphism Problem

Mark Ettinger, Peter Høyer|ArXiv.org|Jan 13, 1999

Quantum Computing Algorithms and Architecture3 references48 citations

TL;DR

This paper proposes a quantum observable in the Hilbert space ℂ[(Sₙ ≀ S₂)ᵐ] that distinguishes isomorphic from non-isomorphic graphs: it returns 'yes' with certainty if two graphs are isomorphic, and 'no' with probability at least 1 − n!/2ᵐ if they are not. The observable is based on superpositions of coset states and exploits the structure of involutive swaps in the wreath product group, though efficient quantum implementation remains unproven.

ABSTRACT

Suppose we are given two graphs on $n$ vertices. We define an observable in the Hilbert space $\Co[(S_n \wr S_2)^m]$ which returns the answer ``yes'' with certainty if the graphs are isomorphic and ``no'' with probability at least $1-n!/2^m$ if the graphs are not isomorphic. We do not know if this observable is efficiently implementable.

Motivation & Objective

• To develop a quantum observable that determines graph isomorphism using group-theoretic structure.
• To explore whether quantum mechanics can solve the graph isomorphism problem via hidden subgroup techniques.
• To define an observable in a Hilbert space whose measurement outcome reveals isomorphism status with high confidence.
• To examine the feasibility of implementing such an observable efficiently on a quantum computer.
• To relate the graph isomorphism problem to the code equivalence problem via shared group structure.

Proposed method

• The observable is defined on the Hilbert space ℂ[(Sₙ ≀ S₂)ᵐ], where m is a parameter controlling precision.
• It uses projections P₀ and P₁ onto subspaces ℋ₀ and ℋ₁, where ℋ₁ is spanned by k-vectors associated with involutive swaps k ∈ G.
• k-vectors are superpositions of group elements and their images under multiplication by an involutive swap k.
• The observable L = λ₀P₀ + λ₁P₁ measures whether a state lies in ℋ₁ (indicating isomorphism) or its orthogonal complement.
• Coset states |cH⟩ for the automorphism group H of the disjoint union graph are used as input states.
• The method relies on the fact that if graphs are isomorphic, all such coset states lie entirely in ℋ₁, while for nonisomorphic graphs, the overlap with ℋ₁ is bounded by n!/2ᵐ.

Experimental results

Research questions

• RQ1 Can a quantum observable be constructed that decides graph isomorphism with certainty for isomorphic graphs and high probability for nonisomorphic ones?
• RQ2 Is the proposed observable efficiently implementable using quantum circuits?
• RQ3 How does the structure of the wreath product group Sₙ ≀ S₂ relate to the automorphism group of a disjoint graph union?
• RQ4 What is the relationship between the graph isomorphism problem and the code equivalence problem in this quantum framework?
• RQ5 Can the hidden subgroup structure of Sₙ ≀ S₂ be exploited to design a polynomial-time quantum algorithm for graph isomorphism?

Key findings

• If two graphs are isomorphic, the observable returns 'yes' with certainty, as the input state lies entirely within the subspace ℋ₁.
• If the graphs are not isomorphic, the probability of measuring 'yes' is at most n!/2ᵐ, so the probability of measuring 'no' is at least 1 − n!/2ᵐ.
• The dimension of the Hilbert space ℂ[(Sₙ ≀ S₂)ᵐ] grows as 2ᵐ(n!)²ᵐ, which is exponential in m but polynomial in n when m is logarithmic in n.
• The observable is not known to be efficiently implementable, despite its favorable measurement properties.
• The method extends to the code equivalence problem, as shown by Manny Knill, due to shared group-theoretic structure.
• The subgroup G′ of Sₙ ≀ S₂ generated by involutive swaps (index 2) can be used instead of the full group, reducing Hilbert space dimension.