Quantum Computation by Adiabatic Evolution

We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.Comment: 24 pages, 12 figures, LaTeX, amssymb,amsmath, BoxedEPS packages; email to [email protected]

How many functions can be distinguished with k quantum queries?

Suppose an oracle is known to hold one of a given set of D two-valued functions. To successfully identify which function the oracle holds with k classical queries, it must be the case that D is at most 2^k. In this paper we derive a bound for how many functions can be distinguished with k quantum queries.Comment: 5 pages. Lower bound on sorting n items improved to (1-epsilon)n quantum queries. Minor changes to text and corrections to reference

A context-free and a 1-counter geodesic language for a Baumslag-Solitar group

We give a language of unique geodesic normal forms for the Baumslag-Solitar group BS(1,2) that is context-free and 1-counter. We discuss the classes of context-free, 1-counter and counter languages, and explain how they are inter-related

Intermediate problems in modular circuits satisfiability

In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite algebras had been introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However the problem for nilpotent (which had not been shown to be NP-hard) but not supernilpotent algebras (which had been shown to be polynomial time) remained open. In this paper we provide a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circuits need exponentially many modular counting gates to produce boolean conjunctions of any arity), satisfiability over these algebras have intermediate complexity between $\Omega(2^{c\log^{h-1} n})$ and $O(2^{c\log^h n})$, where $h$ measures how much a nilpotent algebra fails to be supernilpotent. We also sketch how these examples could be used as paradigms to fill the nilpotent versus supernilpotent gap in general. Our examples are striking in view of the natural strong connections between circuits satisfiability and Constraint Satisfaction Problem for which the dichotomy had been shown by Bulatov and Zhuk