Graphical Language with Delayed Trace: Picturing Quantum Computing with Finite Memory

Abstract

Graphical languages, like quantum circuits or ZXcalculus, have been successfully designed to represent (memoryless) quantum computations acting on a finite number of qubits. Meanwhile, delayed traces have been used as a graphical way to represent finite-memory computations on streams, in a classical setting (cartesian data types). We merge those two approaches and describe a general construction that extends any graphical language, equipped with a notion of discarding, to a graphical language of finite memory computations. In order to handle cases like the ZX-calculus, which is complete for post-selected quantum mechanics, we extend the delayed trace formalism beyond the causal case, refining the notion of causality for stream transformers. We design a stream semantics based on stateful morphism sequences and, under some assumptions, show universality and completeness results. Finally, we investigate the links of our framework with previous works on cartesian data types, signal flow graphs, and quantum channels with memories.

Causality in Higher Order Process Theories

“…In this framework, the notions of sequential and parallel composition of processes are placed at the forefront by adopting the mathematical structure of symmetric monoidal category (SMC) [61]. The process theoretic framework, often aided by its easy-to-use graphical language [31,37], has led to categorical formalisation of the notions of entanglement [36], phase [34], complementarity [33,46], causal/temporal structure [38,39,59,68,78,65], information extraction [40,77], postivity [82], dynamics [46], and memory [13], and the interactions between them [80,84,45].…”