Atomicity in Distributed Quantum Computing

TL;DR

This work provides a rigorous framework for studying atomicity in distributed quantum computing by distinguishing system dynamics from observable dynamics. It introduces a formal model of non-atomic distributed quantum systems, defines actions, processes, and parallel composition with quantum environments, and uses measure-theoretic observable dynamics to relate quantum evolution to classically observable events. The main result proves that local actions are atomic up to observable dynamics, establishing a principled basis for simplifying reasoning about concurrency in distributed quantum settings. The framework lays groundwork for verification and design of distributed quantum algorithms, while highlighting open questions about non-local atomicity, quantum control flow, and quantum-concurrency concepts beyond the local-atomicity guarantee.

Abstract

Atomicity is a ubiquitous assumption in distributed computing, under which actions are indivisible and appear sequential. In classical computing, this assumption has several theoretical and practical guarantees. In quantum computing, although atomicity is still commonly assumed, it has not been seriously studied, and a rigorous basis for it is missing. Classical results on atomicity do not directly carry over to distributed quantum computing, due to new challenges caused by quantum entanglement and the measurement problem from the underlying quantum mechanics. In this paper, we initiate the study of atomicity in distributed quantum computing. A formal model of (non-atomic) distributed quantum system is established. Based on the Dijkstra-Lamport condition, the system dynamics and observable dynamics of a distributed quantum system are defined, which correspond to the quantum state of and classically observable events in the system, respectively. Within this framework, we prove that local actions can be regarded as if they were atomic, up to the observable dynamics of the system.

Figures (5)

• Figure 1: Proving the synchronous system $S_1$ is equivalent to asynchronous $S_1'$. Both can be thought of as space-time diagrams. Each wire corresponds to a qubit, all together spanning the space. Each box corresponds to an action, which on the space axis specifies the qubits it acts on, and on the time axis specifies the time interval it spans. The arrow of time is from the left to the right.
• Figure 2: Proving the synchronous system $S_2$ is equivalent to asynchronous $S_2'$.
• Figure 3: Proving the synchronous system $S_3$ is equivalent to asynchronous $S_3'$.
• Figure 4: Appending systems in \ref{['fig:egmot-3']} with extra single-qubit quantum measurement actions.
• Figure 5: An example of distributed quantum system $A\parallel B$ of two processes $A$ and $B$: Each process has a tree structure. Every segment corresponds to an action, whose projection onto the time axis is the time interval of this action. Every dotted segment represents that there is no action during the time interval. Every branching corresponds to a set of measurement actions. Two partial processes $A/a$ and $B/b$ are highlighted in bold. For illustration, $\lparen*\rparen{A/a}\mathord{\upharpoonright}_{[0,t_2]}$ has no branching, and $B/b$ is trace-preserving after time $t_1$.

Theorems & Definitions (59)

• Example 1
• Remark 1
• Remark 2: Real-time states vs. discrete abstract states
• Example 2
• Remark 3
• Example 3
• Theorem 1: Informal version of \ref{['thm:local-atom']}
• Definition 1: Semiring
• Definition 2: $\sigma$-algebra
• Definition 3