The general stable quantum memory unit is a hybrid consisting of a classical digit with a quantum digit (qudit) assigned to each classical state. The shape of the memory is the vector of sizes of these qudits, which may differ. We determine when N copies of a quantum memory A embed in N(1+o(1)) copies of another quantum memory B. This relationship captures the notion that B is as at least as useful as A for all purposes in the bulk limit. We show that the embeddings exist if and only if for all p >= 1, the p-norm of the shape of A does not exceed the p-norm of the shape of B. The log of the p-norm of the shape of A can be interpreted as the maximum of S(\rho) + H(\rho)/p (quantum entropy plus discounted classical entropy) taken over all mixed states \rho on A. We also establish a noiseless coding theorem that justifies these entropies. The noiseless coding theorem and the bulk embedding theorem together say that either A blindly bulk-encodes into B with perfect fidelity, or A admits a state that does not visibly bulk-encode into B with high fidelity. In conclusion, the utility of a hybrid quantum memory is determined by its simultaneous capacity for classical and quantum entropy, which is not a finite list of numbers, but rather a convex region in the classical-quantum entropy plane.

Fusion categories are fundamental objects in quantum algebra, but their definition is narrow in some respects. By definition a fusion category must be k-linear for some field k, and every simple object V is strongly simple, meaning that (V) = k. We prove that linearity follows automatically from semisimplicity: Every connected, finite, semisimple, rigid, monoidal category \C is k-linear and finite-dimensional for some field k. Barring inseparable extensions, such a category becomes a multifusion category after passing to an algebraic extension of k. The proof depends on a result in Galois theory of independent interest, namely a finiteness theorem for abstract composita.

Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L in S X I, a thickened, compact, oriented surface S, such that the link complement (S X I) - L has no essential vertical cylinder.

We present a definition of an invariant #(M,H), defined for every finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) H and for every closed, framed 3-manifold M. When H is a quantized universal enveloping algebra, #(M,H) is closely related to well-known quantum link invariants such as the HOMFLY polynomial, but it is not a topological quantum field theory.

We present a quantum algorithm for the dihedral hidden subgroup problem with time and query complexity $O(\exp(C\sqrt{\log N}))$. In this problem an oracle computes a function $f$ on the dihedral group $D_N$ which is invariant under a hidden reflection in $D_N$. By contrast the classical query complexity of DHSP is $O(\sqrt{N})$. The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins with the quantum character transform on the group, just as for other hidden subgroup problems. Then it tensors irreducible representations of $D_N$ and extracts summands to obtain target representations. Finally, state tomography on the target representations reveals the hidden subgroup.

We show that there are no non-trivial linear dependencies among p-norms of vectors in finite dimensions that hold for all p. The proof is by complex analytic continuation.

We present a construction for improving numerical cubature formulas with equal weights and a convolution structure, in particular equal-weight product formulas, using linear error-correcting codes. The construction is most effective in low degree with extended BCH codes. Using it, we obtain several sequences of explicit, positive, interior cubature formulas with good asymptotics for each fixed degree $t$ as the dimension $n \to \infty$. Using a special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain an equal-weight $t$-cubature formula on the $n$-cube with $O(n^{\floor{t/2}})$ points, which is within a constant of the Stroud lower bound. We also obtain $t$-cubature formulas on the $n$-sphere, $n$-ball, and Gaussian $\R^n$ with $O(n^{t-2})$ points when $t$ is odd. When $\mu$ is spherically symmetric and $t=5$, we obtain $O(n^2)$ points. For each $t \ge 4$, we also obtain explicit, positive, interior formulas for the $n$-simplex with $O(n^{t-1})$ points; for $t=3$, we obtain O(n) points. These constructions asymptotically improve the non-constructive Tchakaloff bound. Some related results were recently found independently by Victoir, who also noted that the basic construction more directly uses orthogonal arrays.

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group $G$, the invariant counts homomorphisms from the fundamental group of the manifold to $G$. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

Given a tame knot K presented in the form of a knot diagram, we show that the problem of determining whether K is knotted is in the complexity class NP, assuming the generalized Riemann hypothesis (GRH). In other words, there exists a polynomial-length certificate that can be verified in polynomial time to prove that K is non-trivial. GRH is not needed to believe the certificate, but only to find a short certificate. This result complements the result of Hass, Lagarias, and Pippenger that unknottedness is in NP. Our proof is a corollary of major results of others in algebraic geometry and geometric topology.

In previous paper, the author applied the permanent-determinant method of Kasteleyn and its non-bipartite generalization, the Hafnian-Pfaffian method, to obtain a determinant or a Pfaffian that enumerates each of the ten symmetry classes of plane partitions. After a cosmetic generalization of the Kasteleyn method, we identify the matrices in the four determinantal cases (plain plane partitions, cyclically symmetric plane partitions, transpose-complement plane partitions, and the intersection of the last two types) in the representation theory of sl(2,C). The result is a unified proof of the four enumerations.