The quantum detection of projectors in finite-dimensional algebras and holography

Abstract

We define the computational task of detecting projectors in finite dimensional associative algebras with a combinatorial basis, labelled by representation theory data, using combinatorial central elements in the algebra. In the first example, the projectors belong to the centre of a symmetric group algebra and are labelled by Young diagrams with a fixed number of boxes n. We describe a quantum algorithm for the task based on quantum phase estimation (QPE) and obtain estimates of the complexity as a function of n. We compare to a classical algorithm related to the projector identification problem by the AdS/CFT correspondence. This gives a concrete proof of concept for classical/quantum comparisons of the complexity of a detection task, based in holographic correspondences. A second example involves projectors labelled by triples of Young diagrams, all having n boxes, with non-vanishing Kronecker coefficient. The task takes as input the projector, and consists of identifying the triple of Young diagrams. In both of the above cases the standard QPE complexities are polynomial in n. A third example of quantum projector detection involves projectors labelled by a triple of Young diagrams, with m, n and m + n boxes respectively, such that the associated Littlewood-Richardson coefficient is non-zero. The projector detection task is to identify the triple of Young diagrams associated with the projector which is given as input. This is motivated by a two-matrix model, related via the AdS/CFT correspondence, to systems of strings attached to giant gravitons. The QPE complexity in this case is polynomial in m and n.

String theory for builders: New mathematical perspectives on algorithms and complexity in quantum physics

“…In a series of recent papers, Dr Sanjaye Ramgoolam of Queen Mary University of London and a team of international collaborators started a research programme to take a fresh look at simple mathematical models of string theory and duality [1][2][3][4][5][6] with a view to wider gravity. One study in the early 1990s, for example, identified certain mathematical model universes within string theory which exhibited a striking 'duality' with ordinary quantum theories, with no explicit strings or gravity [10].…”

“…It is precisely this kind of large particle limit in quantum field theories that features in the AdS/CFT correspondence. In their papers, Dr Ramgoolam and Dr Ben Geloun developed quantum mechanical models where the quantum states are combinations of ribbon graphs [1,2,6]. In this framework, they found new algorithms for computing group theoretic quantities, called 'Kronecker coefficients', which are of particular interest in complexity theory.…”

“…The quantum mechanical models based on ribbon graphs, as well as other such combinatorial structures, allow the formulation of new quantum algorithms associated with finite groups [2]. The complexity of these algorithms can be calculated using quantum computation techniques, and owing to the AdS/CFT correspondence, some of them have classical counterparts which researchers had not previously expected.…”