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arXiv:quant-ph/0511149

Tight Results on Multiregister Fourier Sampling: Quantum Measurements for Graph Isomorphism Require Entanglement

Cristopher Moore, Alexander C. Russell

Published Nov 15, 2005• 3 citations
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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.

DOI
10.48550/arxiv.quant-ph/0511149

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Journal of the ACM•Oct 1, 2010•31 citations
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.
Part I of this paper showed that the hidden subgroup problem over the symmetric group--including the special case relevant to Graph Isomorphism--cannot be efficiently solved by strong Fourier sampling, even if one may perform an arbitrary POVM on the coset state. In this paper, we extend these results to entangled measurements. Specifically, we show that the hidden subgroup problem on the symmetric group cannot be solved by any POVM applied to pairs of coset states. In particular, these hidden subgroups cannot be determined by any polynomial number of one- or two-register experiments on coset states.
The dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of Fourier sampling, introduced by Bernstein and Vazirani and developed by Simon and Shor into an approach to the hidden subgroup problem. This approach has proved successful for abelian groups, leading to efficient algorithms for factoring, extracting discrete logarithms, and other number-theoretic problems. We show, however, that this method cannot resolve the hidden subgroup problem in the symmetric groups, even in the weakest, information-theoretic sense. In particular, we show that the Graph Isomorphism problem cannot be solved by this approach. Our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states.
We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses k = log 2 |G | registers, and each of the 2 k subsets of the registers contributes some information. While this does not, in general, yield an efficient algorithm, it generalizes the relationship between Subset Sum and the HSP in the dihedral group, and sheds some light on how quantum algorithms for Graph Isomorphism might work. 1
It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman [30] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because highly entangled measurements seem hard to implement in general. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n,Fpm) and Gn where G is finite and satisfies a suitable property.
It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because highly entangled measurements seem hard to implement in general. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n, F_{p^m}) and G^n where G is finite and satisfies a suitable property.
It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore et al. [2005] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω( n log n ) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL( n ,F p m ) and G n where G is finite and satisfies a suitable property.