Orthogonal Vector Computations

Abstract

Quantum computation is the suitable orthogonal encoding of possibly holistic functional properties into state vectors, followed by a projective measurement.Keywords: quantum theory; probability theory; quantum logic; quantum algorithms; parity PACS: 03.65.Ca; 03.65.Aa; 03.65.Aa; 03.67.Ac Identifying Quantum Physical Means for ComputationThe hypothesis pursued in this paper is that the power of quantum computation solely resides in a proper "translation" of "holistic" properties of functions-manifesting themselves in the relational values for different elements of their domain, or of their entire image-into orthogonal subspaces and their associated perpendicular projections. This is usually facilitated by quantum parallelism-the possibility to co-represent and co-encode classically distinct and mutually exclusive clauses into simultaneous coherent superpositions thereof. However, in order to take advantage of parallelism, it is necessary to be able to analyse the resulting quantum state by suitably chosen orthogonal projections. In what follows we shall therefore attempt to enumerate conditions under which a given algorithmic task can be quantum mechanically encoded into orthogonal subspaces, thereby identifying criteria for potential quantum speedups. This paper is organized as follows: after a brief and somewhat iconoclastic exposition of the standard quantum formalism (which may be amusing to some but considered to be superfluous by others) strategies to encode properties of computable functions into orthogonal subspaces are discussed. This involves the possibility to orthogonalise non-orthogonal vectors of some initial Hilbert space by interpreting them as orthogonal projections of mutually orthogonal vectors in a Hilbert space of greater dimension. Locating Quantum ResourcesStarting with Plack's [1] (p. 31) "act of despair" it may not be totally unreasonable to state that the newly discovered quantum capacities, unheard of in classical continuum mechanics, caused shock and awe among some of the most prominent creators of quantum mechanics. Most notably Schrödinger struggled with quantum coherence, today known as quantum parallelism, throughout his entire life, bringing forward seemingly absurd consequences of the formalism, such as the cat paradox, or quantum jellification [2]. In the same series of papers [3] he also mentioned the capacity that multiple quanta may be entangled; a resource related to quantum non-locality, which was put forward by Einstein on various occasions [4,5] to question the consistency of quantum and relativity theory.On the formal side, Hilbert space quantum mechanics [6] and, in particular, quantum logic have been conceived [7] as attempts to systematically identify and study the valid, feasible, operational resources and capacities of quantized physical systems in terms of algebraic structures arising

Quantum hocus-pocus

“…Nevertheless, with all these provisos, a potential quantum advantage resides in the possibility to encode certain suitable relational functional properties representable by (equi-)partitions of the image of the function [21,22] into suitable orthogonal projections [23]. Unfortunately this is not ubiquitous, as for certain tasks such as parity effective speedups are impossible [24].…”

Dimensional Lifting through the Generalized Gram–Schmidt Process

“…Some quantum computation tasks require the orthogonalization of previously non-orthogonal vectors. This might be best understood in terms of mutually exclusive outcomes of generalized beam splitter experiments, where the entire array of output ports corresponds to an ensemble of mutually orthogonal subspaces, or, equivalently, mutually orthogonal perpendicular projection operators [ 2 ].…”

“…Let us, for the sake of a physical example, study configurations associated with decision problems that can be efficiently (that is, with some speedup with respect to purely classical means [ 2 ]) encoded quantum mechanically. The inverse problem is the projection of orthogonal systems of vectors onto lower dimensions.…”

“…One instance of such a quantum computation involving the reduction to ensembles of orthogonal vectors (and their associated span or projection operators) is the Deutsch–Jozsa algorithm, as reviewed in Ref. [ 2 ]. Another, somewhat contrived, problem can be constructed in three dimensions from a eutactic star which is the projection onto the plane formed by the first two coordinates of a three-dimensional orthormal basis which, together with the Cartesian standard basis, forms a pair of unbiased bases [ 11 ].…”

“…One instance of such a quantum computation involving the reduction to ensembles of orthogonal vectors (and their associated span or projection operators) is the Deutsch-Jozsa algorithm, as reviewed in Ref. [2]. Another, somewhat contrived, problem can be constructed in three dimensions from an eutactic star…”