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

TL;DR

This paper introduces a unified graphical language framework that combines quantum circuit representations with finite-memory stream processing, extending existing formalisms to handle quantum computations with memory effects.

Contribution

It develops a general construction to extend graphical languages with discarding to include finite-memory computations, and refines causality notions for stream transformers in quantum settings.

Findings

Extended graphical language for finite-memory quantum computations

Proved universality and completeness under certain conditions

Linked framework to classical data types and quantum channels with memory

Abstract

Graphical languages, like quantum circuits or ZX-calculus, 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.