[Paper Review] Tight Results on Multiregister Fourier Sampling: Quantum Measurements for Graph Isomorphism Require Entanglement

Cristopher Moore, Alexander Russell|ArXiv.org|Nov 15, 2005

Quantum Computing Algorithms and Architecture32 references3 citations

TL;DR

This paper establishes that efficient quantum algorithms for Graph Isomorphism require entangled measurements on at least Ω(n log n) coset states of the symmetric group S_n, using a general method to bound information extraction from entangled measurements on tensor products of hidden subgroup states. The result shows that individual or unentangled measurements cannot extract sufficient information, and the bound is tight to within a constant factor for the symmetric group case relevant to Graph Isomorphism.

ABSTRACT

We establish a general method for proving bounds on the information that can be extracted via arbitrary entangled measurements on tensor products of hidden subgroup coset states. When applied to the symmetric group, the method yields an Omega(n log n) lower bound on the number of coset states over which we must perform an entangled measurement in order to obtain non-negligible information about a hidden involution. These results are tight to within a multiplicative constant and apply, in particular, to the case relevant for the Graph Isomorphism problem. Part of our proof was obtained after learning from Hallgren, Roetteler, and Sen that they had obtained similar results.

Motivation & Objective

• To determine the minimum number of coset states required for entangled measurements to extract non-negligible information about a hidden subgroup in the symmetric group S_n.
• To close the gap between information-theoretic and computational feasibility in the hidden subgroup problem for non-abelian groups.
• To prove that individual or unentangled measurements on coset states are insufficient for solving Graph Isomorphism efficiently.
• To establish a general method for bounding information gain via arbitrary entangled measurements on tensor products of coset states.
• To show that the symmetric group case relevant to Graph Isomorphism requires Ω(n log n) registers for any entangled measurement to yield useful information.

Proposed method

• Develops a general framework to bound the information extractable via arbitrary entangled measurements on tensor products of hidden subgroup coset states.
• Applies the method to the symmetric group S_n, focusing on the case relevant to Graph Isomorphism via the wreath product K = S_n ≀ ℤ₂.
• Uses representation theory, including Plancherel measure and irreducible representations, to characterize the structure of coset states.
• Defines 'bad' representations Λ as those induced from ρ⊗ρ with d_ρ < n^{n/5}, and bounds their contribution using probabilistic and concentration arguments.
• Employs the total variation distance between the measurement outcome distribution and uniform to quantify information gain.
• Uses Markov's inequality and tail bounds to show that with high probability, the measurement outcome distribution is close to uniform unless k = Ω(n log n).

Experimental results

Research questions

• RQ1 What is the minimum number of coset states required for entangled measurements to extract non-negligible information about a hidden involution in the symmetric group S_n?
• RQ2 Can individual or unentangled measurements on coset states suffice to solve the Graph Isomorphism problem in the hidden subgroup framework?
• RQ3 Is the information-theoretic bound on the number of registers required for entangled measurements tight for the symmetric group?
• RQ4 How does the Plancherel measure on high-dimensional representations affect the feasibility of solving the hidden subgroup problem?
• RQ5 Can the method be generalized to other groups where a large fraction of the Plancherel measure lies on high-dimensional representations?

Key findings

• The paper proves a lower bound of Ω(n log n) on the number of coset states required for entangled measurements to extract non-negligible information about a hidden involution in the symmetric group S_n.
• The bound is tight to within a multiplicative constant, meaning that no fewer than Ω(n log n) registers can yield useful information via entangled measurements.
• For k < C n log₂ n with C < 1/10, the expected total variation distance between the measurement outcome distribution and uniform is n^{-Ω(n)}, implying negligible information gain.
• The analysis shows that individual or unentangled measurements on coset states cannot solve Graph Isomorphism efficiently, as they fail to extract sufficient information.
• The method applies generally to groups where a large fraction of the Plancherel measure is on high-dimensional representations, and such groups require Θ(n log n) coset states for entangled measurements.
• The result implies that any efficient quantum algorithm for Graph Isomorphism must use entangled measurements on at least Ω(n log n) copies of the coset state.