Routed quantum circuits

Abstract

We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces. We therefore propose an extension to the framework of quantum circuits, given by routed linear maps and routed quantum circuits. We prove that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries. We show that our framework encompasses the `extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)], which we derive as a special case, endowing them with a sound semantics.

Composable constraints

“…As a first and focused outcome, providing a simple notion which encompasses both the constructions introduced by [14] to describe sectorial constraints in FHilb and the study of signalling conditions in unitary channels [17], the theory of composable constraints gives a formal backing to their intuitive similarity. In addition, the abstract phrasing of the framework leaves room for developing a language which can be used to study and understand their essential differences, for instance by expanding the observations of [17] in which links are noted between important features of constraints and their time-symmetry.…”

“…The use of finite relations to encode sectorial constraints was originally developed for the case C = FHilb in [14], in order to model more accurately some quantum-theoretical scenarios, giving rise to so-called routed maps; this in turn was the main motivation for the generalisation to arbitrary composable constraints presented here.…”

“…from physical constraints, rules of a game, or technological limitations. This is for instance the case for the study of superpositions of channels in quantum theory [1][2][3] -and more generally for that of the coherent control of gates and channels [4][5][6][7][8][9][10][11][12][13] -, a notion whose formal definition is a subtle matter, and for which a recently proposed formalism [14,15] makes a crucial use of so-called sectorial constraints on morphisms. In another line of research [16][17][18][19], it has been proposed to describe the causal structure of unitary channels using sets of 'no-influence relations' used to encode constraints which forbid certain input factors of channels from signalling to certain output factors.…”

“…diagrammatic decompositions of unitary channels that are equivalent to these channels' causal structure [16][17][18][19] (see in particular [18]). Indeed, some causal decompositions cannot be written in terms of standard circuits, but only using more elaborate circuits (later called index-matching circuits [14]), which relied on constraints and whose exact semantics remained unclear 1 .…”

“…The point of Ref. [14] is, in the context of the category FHilb of linear maps on finite-dimensional Hilbert spaces, to build a theory encompassing sectorial constraints on these linear maps. Sectorial constraints express the fact that a given linear map is forbidden to relate some sectors (i.e.…”