The Symmetric Group Defies Strong Fourier Sampling

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

Reflection Positivity and Conformal Symmetry

The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory and duality between unitary representations of the euclidean motion group and the Poincare group. On the mathematical side this duality can be made precise as follows. If \g is a Lie algebra with an involutive automorphism $\tau$. Decompose \g = \fh \oplus \fq = \ker(\tau - \1) \oplus \ker(\tau + \1) into $\tau$-eigenspaces and let \g^c := \fh \oplus i \fq. At the core of the notion of reflection positivity is the idea that this duality can sometimes be implemented on the level of unitary representations. The idea is simple on the Lie algebra level: Let (\pi,\cH^0) be a representation of \g where $\pi$ acts by skew-symmetric operators. Assume that there exists a unitary operator $J$ of order two such that $J \pi J = \pi\circ \tau$ and a \g-invariant subspace \cK^0 which is {\it $J$-positive}. Then complex linear extension leads to a representation of \g^c on \cK^0 by operators which are skew-symmetric with respect to $h_J$, so that we obtain a "unitary" representation of \g^c on the pre-Hilbert space \cK_J^0 := \cK_J/{v \: \cK^0 \: h_J(v,v)=0}. The aim of this article is twofold. First we discuss reflection positivity in an abstract setting using {\it reflection positive distributions} on the Lie group $G_\tau =G\rtimes {1,\tau}$ and {\it reflection positive distribution vectors} of a unitary representation of $G_\tau$. Then we apply these ideas to the conformal group \OO_{1,n+1}^+(\R) of the sphere \bS^n as well as the the half-space picture mostly used in physics