Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes

Diagonalization in the spirit of Cantor's diagonal arguments is a widely used tool in theoretical computer sciences to obtain structural results about computational problems and complexity classes by indirect proofs. The Uniform Diagonalization Theorem allows the construction of problems outside complexity classes while still being reducible to a specific decision problem. This paper provides a generalization of the Uniform Diagonalization Theorem by extending it to promise problems and the complexity classes they form, e.g. randomized and quantum complexity classes. The theorem requires from the underlying computing model not only the decidability of its acceptance and rejection behaviour but also of its promise-contradicting indifferent behaviour - a property that we will introduce as "total decidability" of promise problems. Implications of the Uniform Diagonalization Theorem are mainly of two kinds: 1. Existence of intermediate problems (e.g. between BQP and QMA) - also known as Ladner's Theorem - and 2. Undecidability if a problem of a complexity class is contained in a subclass (e.g. membership of a QMA-problem in BQP). Like the original Uniform Diagonalization Theorem the extension applies besides BQP and QMA to a large variety of complexity class pairs, including combinations from deterministic, randomized and quantum classes.Comment: 15 page

A parallel algorithm for Hamiltonian matrix construction in electron-molecule collision calculations: MPI-SCATCI

Construction and diagonalization of the Hamiltonian matrix is the rate-limiting step in most low-energy electron -- molecule collision calculations. Tennyson (J Phys B, 29 (1996) 1817) implemented a novel algorithm for Hamiltonian construction which took advantage of the structure of the wavefunction in such calculations. This algorithm is re-engineered to make use of modern computer architectures and the use of appropriate diagonalizers is considered. Test calculations demonstrate that significant speed-ups can be gained using multiple CPUs. This opens the way to calculations which consider higher collision energies, larger molecules and / or more target states. The methodology, which is implemented as part of the UK molecular R-matrix codes (UKRMol and UKRMol+) can also be used for studies of bound molecular Rydberg states, photoionisation and positron-molecule collisions.Comment: Write up of a computer program MPI-SCATCI Computer Physics Communications, in pres

Simultaneous Matrix Diagonalization for Structural Brain Networks Classification

This paper considers the problem of brain disease classification based on connectome data. A connectome is a network representation of a human brain. The typical connectome classification problem is very challenging because of the small sample size and high dimensionality of the data. We propose to use simultaneous approximate diagonalization of adjacency matrices in order to compute their eigenstructures in more stable way. The obtained approximate eigenvalues are further used as features for classification. The proposed approach is demonstrated to be efficient for detection of Alzheimer's disease, outperforming simple baselines and competing with state-of-the-art approaches to brain disease classification

Complexity of equivalence relations and preorders from computability theory

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. Here $f$ is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and $f$ must be computable. We show that there is a $\Pi_1$-complete equivalence relation, but no $\Pi k$-complete for $k \ge 2$. We show that $\Sigma k$ preorders arising naturally in the above-mentioned areas are $\Sigma k$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $\Sigma 2$, almost inclusion on r.e.\ sets, which is $\Sigma 3$, and Turing reducibility on r.e.\ sets, which is $\Sigma 4$.Comment: To appear in J. Symb. Logi

Object-oriented construction of a multigrid electronic-structure code with Fortran 90

We describe the object-oriented implementation of a higher-order finite-difference density-functional code in Fortran 90. Object-oriented models of grid and related objects are constructed and employed for the implementation of an efficient one-way multigrid method we have recently proposed for the density-functional electronic-structure calculations. Detailed analysis of performance and strategy of the one-way multigrid scheme will be presented.Comment: 24 pages, 6 figures, to appear in Comput. Phys. Com