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arXiv:2311.06116 · Proceedings of the ACM on Programming Languages

Quantum Bisimilarity via Barbs and Contexts: Curbing the Power of Non-deterministic Observers

Lorenzo Ceragioli, Fabio Gadducci, Giuseppe Lomurno, Gabriele Tedeschi

Published Jan 2, 2024• 11 citations
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Abstract

Past years have seen the development of a few proposals for quantum extensions of process calculi. The rationale is clear: with the development of quantum communication protocols, there is a need to abstract and focus on the basic features of quantum concurrent systems, like CCS and CSP have done for their classical counterparts. So far, though, no accepted standard has emerged, neither for the syntax nor for the behavioural semantics. Indeed, the various proposals do not agree on what should be the observational properties of quantum values, and as a matter of fact, the soundness of such properties has never been validated against the prescriptions of quantum theory. To this aim, we introduce a new calculus, Linear Quantum CCS (lqCCS), and investigate the features of behavioural equivalences based on barbs and contexts. Our calculus can be thought of as an asynchronous, linear version of qCCS, which is in turn based on value-passing CCS. The combination of linearity and asynchronous communication fits well with the properties of quantum systems (e.g. the no-cloning theorem), since it ensures that each qubit is sent exactly once, precisely specifying which qubits of a process interact with the context. We exploit contexts to examine how bisimilarities relate to quantum theory. We show that the observational power of general contexts is incompatible with quantum theory: roughly, they can perform non-deterministic moves depending on quantum values without measuring (hence perturbing) them. Therefore, we refine the operational semantics in order to prevent contexts from performing unfeasible non-deterministic choices. This induces a coarser bisimilarity that better fits the quantum setting: ( i ) it lifts the indistinguishability of quantum states to the distributions of processes and, despite the additional constraints, ( i i ) it preserves the expressiveness of non-deterministic choices based on classical information. To the best of our knowledge, our semantics is the first one that satisfies the two properties above.

DOI
10.1145/3632885

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Past years have seen the development of a few proposals for quantum extensions of process calculi. The rationale is clear: with the development of quantum communication protocols, there is a need to abstract and focus on the basic features of quantum concurrent systems, like CCS and CSP have done for their classical counterparts. So far, though, no accepted standard has emerged, neither for the syntax nor for the behavioural semantics. Indeed, the various proposals do not agree on what should be the observational properties of quantum values, and as a matter of fact, the soundness of such properties has never been validated against the prescriptions of quantum theory. To this aim, we introduce a new calculus, Linear Quantum CCS (lqCCS), and investigate the features of behavioural equivalences based on barbs and contexts. Our calculus can be thought of as an asynchronous, linear version of qCCS, which is in turn based on value-passing CCS. The combination of linearity and asynchronous communication fits well with the properties of quantum systems (e.g. the no-cloning theorem), since it ensures that each qubit is sent exactly once, precisely specifying which qubits of a process interact with the context. We exploit contexts to examine how bisimilarities relate to quantum theory. We show that the observational power of general contexts is incompatible with quantum theory: roughly, they can perform non-deterministic moves depending on quantum values without measuring (hence perturbing) them. Therefore, we refine the operational semantics in order to prevent contexts from performing unfeasible non-deterministic choices. This induces a coarser bisimilarity that better fits the quantum setting: ( i ) it lifts the indistinguishability of quantum states to the distributions of processes and, despite the additional constraints, ( i i ) it preserves the expressiveness of non-deterministic choices based on classical information. To the best of our knowledge, our semantics is the first one that satisfies the two properties above.
Recent developments in the field of quantum communication demonstrate that secure communication protocols based on quantum features are already practical. Reliable verification techniques are of paramount importance for these technologies, given their high implementation cost and critical contexts of application. Extensions of process calculi such as CCS and 𝛑-calculus have been proposed in the literature, together with various notions of behavioural equivalence. However, their standard probabilistic models turn out to introduce some non-deterministic capabilities that are not aligned with the observational properties of physical quantum systems, leading to bisimilarity notions that distinguish processes that should be physically equivalent. Nonetheless, we argue that non-deterministic features are fundamental to account for inputs, environments and adversarial behaviour. To address this issue, we propose lqCCS, a process calculus that integrates concurrency, non-determinism and quantum capabilities. The calculus is enriched with a linear type system that enforces the no-cloning principle and resolves ambiguities in the visibility of ancillary qubits. We introduce a novel semantics in terms of distributions, where explicit physically admissible schedulers constrain probabilistic composition and forbid ill-defined non-deterministic moves, while preserving the expressivity needed to model real-world protocols. We investigate a scheduled version of saturated bisimilarity, deeming two lqCCS processes behaviourally equivalent if no observer can tell them apart. The adequacy of the approach is verified by lifting a known result from quantum mechanics to lqCCS, i.e., that equivalent processes acting on indistinguishable mixtures of quantum states are correctly recognized as bisimilar. Finally, we give an alternative semantics and a labelled bisimilarity based on a quantum generalization of probability distributions. This provides an equivalent characterization of our behavioural equivalence that is a congruence with respect to the parallel operator, enabling compositional reasoning without the need to explicitly check all possible contexts. We describe a rich class of lqCCS processes for which equivalence is decidable using standard techniques, and we analyse real-world quantum communication protocols.
Reliable verification techniques for quantum communication protocols are of paramount importance, given their high implementation cost and critical contexts of application. Extensions of process calculi have been proposed, together with various notions of behavioural equivalence. However, their standard probabilistic models turn out to introduce some non-deterministic capabilities not aligned with the observational properties of physical quantum systems, leading to bisimilarity notions that distinguish physically equivalent processes. Nonetheless, non-deterministic features are fundamental to account for inputs, environments and adversarial behaviour. To address this issue, we propose lqCCS, a process calculus that integrates concurrency, non-determinism and quantum capabilities. We introduce a novel semantics in terms of distributions, where explicit physically admissible schedulers constrain probabilistic composition and forbid ill-defined non-deterministic moves, while preserving the expressivity needed to model real-world protocols. We investigate a scheduled version of saturated bisimilarity, pairing two processes if no observer can tell them apart, and we verify its adequacy by lifting a known result from quantum mechanics to lqCCS: equivalent processes acting on indistinguishable mixtures of quantum states are correctly recognized as bisimilar. Finally, we give an alternative semantics and a labelled bisimilarity based on a quantum generalization of probability distributions. This characterizes our behavioural equivalence as a congruence with respect to the parallel operator, enabling compositional reasoning without the need to explicitly check all possible contexts. We describe a rich class of lqCCS processes for which equivalence is decidable using standard techniques, and we analyse real-world quantum communication protocols.
Modeling and reasoning about concurrent quantum systems is very important both for distributed quantum computing and for quantum protocol verification. As a consequence, a general framework describing formally the communication and concurrency in complex quantum systems is necessary. For this purpose, we propose a model qCCS which is a natural quantum extension of classical value-passing CCS with the input and output of quantum states, and unitary transformations and measurements on quantum systems. The operational semantics of qCCS is given based on probabilistic labeled transition system. This semantics has many different features compared with the proposals in literature in order to describe input and output of quantum systems which are possibly correlated with other components. Based on this operational semantics, we introduce the notions of strong probabilistic bisimilarity and weak probabilistic bisimilarity between quantum processes and discuss some properties of them, such as congruence under various combinators.
The development of quantum communication protocols sparked the interest in quantum extensions of process calculi and behavioural equivalences, but defining a bisimilarity that matches the observational properties of a quantum-capable system is a surprisingly difficult task. The two proposals explicitly addressing this issue, qCCS and lqCCS, do not define an algorithmic verification scheme: the bisimilarity of two processes is proven by comparing their behaviour under all input states. We introduce a new semantic model based on effects, i.e. probabilistic predicates on quantum states that represent their observable properties. We define and investigate the properties of effect distributions and effect labelled transition systems (eLTSs), generalizing probability distributions and probabilistic labelled transition systems (pLTSs), respectively. As a proof of concept, we provide an eLTS-based semantics for a minimal quantum process algebra, which we prove sound and complete with respect to the observable probabilistic behaviour of quantum processes. To the best of our knowledge, ours is the first algorithmically verifiable proposal that abides to the properties of quantum theory.
Modeling and reasoning about concurrent quantum systems is very important for both distributed quantum computing and quantum protocol verification. As a consequence, a general framework formally describing communication and concurrency in complex quantum systems is necessary. For this purpose, we propose a model named qCCS. It is a natural quantum extension of classical value-passing CCS which can deal with input and output of quantum states, and unitary transformations and measurements on quantum systems. The operational semantics of qCCS is given in terms of probabilistic labeled transition system. This semantics has many different features compared with the proposals in the available literature in order to describe the input and output of quantum systems which are possibly correlated with other components. Based on this operational semantics, the notions of strong probabilistic bisimulation and weak probabilistic bisimulation between quantum processes are introduced. Furthermore, some properties of these two probabilistic bisimulations, such as congruence under various combinators, are examined.
We introduce an algebra qCCS of pure quantum processes in which communications by moving quantum states physically are allowed and computations are modeled by super-operators, but no classical data is explicitly involved. An operational semantics of qCCS is presented in terms of (nonprobabilistic) labeled transition systems. Strong bisimulation between processes modeled in qCCS is defined, and its fundamental algebraic properties are established, including uniqueness of the solutions of recursive equations. To model sequential computation in qCCS, a reduction relation between processes is defined. By combining reduction relation and strong bisimulation we introduce the notion of strong reduction-bisimulation, which is a device for observing interaction of computation and communication in quantum systems. Finally, a notion of strong approximate bisimulation (equivalently, strong bisimulation distance) and its reduction counterpart are introduced. It is proved that both approximate bisimilarity and approximate reduction-bisimilarity are preserved by various constructors of quantum processes. This provides us with a formal tool for observing robustness of quantum processes against inaccuracy in the implementation of its elementary gates.