We define several models of computation based on permuting distinguishable particles (which we call balls) and characterize their computational complexity. In the quantum setting, we use the representation theory of the symmetric group to find variants of this model which are intermediate between BPP and DQC1 (the class of problems solvable with one clean qubit) and between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an exactly solvable scattering problem of particles moving on a line. Despite the simplicity of this model from the perspective of mathematical physics, we show that if we allow intermediate destructive measurements and specific input states, then the model cannot be efficiently simulated classically up to multiplicative error unless the polynomial hierarchy collapses. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP, and that a nondeterministic version of this model is NP-complete.

We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen such object has the required properties with positive yet tiny probability. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.

This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA = QCMA.We prove three results about this question.First, we give a "quantum oracle separation" between QMA and QCMA.More concretely, we show that any quantum algorithm needs Ω 2 n m+1 queries to find an n-qubit "marked state" |ψ , even if given an m-bit classical description of |ψ together with a quantum black box that recognizes |ψ .Second, we give an explicit QCMA protocol that nearly achieves this lower bound.Third, we show that, in the one previously-known case where quantum proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful.In particular, Watrous gave a QMA protocol for verifying non-membership in finite groups.Under plausible group-theoretic assumptions, we give a QCMA protocol for the same problem.Even with no assumptions, our protocol makes only polynomially many queries to the group oracle.We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.

We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity $2^{O(\sqrt{\log\ N})}$. 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 as usual with a quantum character transform, which in the case of $D_N$ is essentially the abelian quantum Fourier transform. This yields the name of a group representation of $D_N$, which is not by itself useful, and a state in the representation, which is a valuable but indecipherable qubit. The algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of $D_N$. Once the algorithm obtains certain target representations, direct measurements reveal the hidden subgroup.

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 /spl Ascr/ embed in N(1+/spl ogr/(1)) copies of another quantum memory /spl Bscr/. This relationship captures the notion that/spl Bscr/ is as at least as useful as /spl Ascr/ for all purposes in the bulk limit. We show that the embeddings exist if and only if for all p/spl ges/1, the p-norm of the shape of /spl Ascr/ does not exceed the p-norm of the shape of /spl Bscr/. The log of the p-norm of the shape of /spl Ascr/ can be interpreted as the maximum of S(/spl rho/)+H(/spl rho/)/p (quantum entropy plus discounted classical entropy) taken over all mixed states /spl rho/ on /spl Ascr/. We also establish a noiseless coding theorem that justifies these entropies. The noiseless coding theorem and the bulk embedding theorem together say that either /spl Ascr/ blindly bulk-encodes into /spl Bscr/ with perfect fidelity, or A admits a state that does not visibly bulk-encode into/spl Bscr/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.