An algebra of automata which includes both classical and quantum entities

We describe an algebra for composing automata which includes both classical and quantum entities and their communications. We illustrate by describing in detail a quantum protocol.

💡 Research Summary

The paper introduces a unified algebraic framework for modelling and composing both classical and quantum automata, thereby bridging the gap between traditional automata theory and quantum information processing. An automaton is defined as a tuple (S, I, O, δ) where S is the state space, I and O are sets of input and output ports, and δ is a transition map. For classical automata S is a finite set and δ is a stochastic matrix; for quantum automata S is a finite‑dimensional Hilbert space and δ is a completely positive trace‑preserving (CPTP) map (or a set of Kraus operators).

Two fundamental composition operators are provided. The direct sum (⊕) merges independent automata by taking the disjoint union of their state spaces, while the tensor product (⊗) builds a composite system that runs subsystems in parallel. The tensor product is the key to representing quantum entanglement, because the joint state space becomes the tensor product of the individual Hilbert spaces, and the joint transition map is the tensor product of the individual CPTP maps.

A third operator, Connect, specifies how ports of different automata are wired together. A port on the output side of one automaton can be identified with a port on the input side of another, and the resulting global transition map is obtained by appropriately reshaping and contracting the component maps. Crucially, Connect handles both classical bits and qubits uniformly: a measurement performed by a quantum automaton yields a classical outcome that is automatically routed to a classical port of another automaton via the Kraus representation.

The authors demonstrate the expressive power of the algebra by giving a fully detailed model of the quantum teleportation protocol. The protocol is decomposed into four automata: (1) state‑preparation, (2) entanglement generation, (3) Bell‑basis measurement, and (4) correction at the receiver. Each automaton has its own transition map; the measurement automaton outputs a classical two‑bit result, which is then fed as classical input to the correction automaton. The overall protocol is expressed as a composition of tensor products (to represent parallel preparation of the entangled pair) and Connect operations (to route measurement outcomes). The resulting global transition matrix is precisely the product of the individual matrices, arranged according to the wiring diagram, and reproduces the standard teleportation fidelity analysis.

Beyond teleportation, the paper sketches how the same algebra can model quantum key distribution (BB84), quantum error‑correcting codes, and hybrid classical‑quantum circuits. Because the construction forms a monoidal category, notions such as isomorphism, functorial mapping, and diagrammatic reasoning become available, enabling formal verification and compositional reasoning that are difficult in conventional circuit descriptions.

On the implementation side, the authors present a prototype domain‑specific language (DSL) for declaring automata, ports, and connections. A compiler translates a DSL program into a tensor network representation, which can be handed off to existing quantum simulators (e.g., Qiskit, Cirq) for numerical evaluation. This toolchain allows designers to work at a high level of abstraction while still obtaining executable quantum circuits, thereby supporting rapid prototyping of hybrid systems.

In summary, the paper offers a mathematically rigorous yet practically applicable algebra that unifies classical and quantum automata. By treating both kinds of systems as linear maps on finite‑dimensional spaces and by providing systematic composition operators, the framework facilitates modular design, formal analysis, and automated code generation for complex quantum communication and computation protocols.