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Encodability Criteria for Quantum Based Systems

Anna Schmitt, Kirstin Peters, Yuxin Deng

TL;DR

The paper investigates how to compare expressive power across quantum based process calculi by applying Gorla's classical encodability criteria. It demonstrates a positive encoding from $CQS$ to $OQS$ that preserves compositionality, name invariance, operational correspondence, divergence reflection, and success sensitiveness, while introducing quantum specific invariants such as preserving the size of quantum registers. It further shows a negative separation: there is no encoding from $OQS$ to $CQS$ that is compositional, operationally corresponding, and success sensitive, and extends this hardness to related qCCS to CQP translations. The results suggest that Gorla's framework remains informative in the quantum setting but is likely not exhaustively capturing all quantum aspects, motivating new criteria such as qubit invariance and register size preservation and future probabilistic encodability notions.

Abstract

Quantum based systems are a relatively new research area for that different modelling languages including process calculi are currently under development. Encodings are often used to compare process calculi. Quality criteria are used then to rule out trivial or meaningless encodings. In this new context of quantum based systems, it is necessary to analyse the applicability of these quality criteria and to potentially extend or adapt them. As a first step, we test the suitability of classical criteria for encodings between quantum based languages and discuss new criteria. Concretely, we present an encoding, from a language inspired by CQP into a language inspired by qCCS. We show that this encoding satisfies compositionality, name invariance (for channel and qubit names), operational correspondence, divergence reflection, success sensitiveness, and that it preserves the size of quantum registers. Then we show that there is no encoding from qCCS into CQP that is compositional, operationally corresponding, and success sensitive.

Encodability Criteria for Quantum Based Systems

TL;DR

The paper investigates how to compare expressive power across quantum based process calculi by applying Gorla's classical encodability criteria. It demonstrates a positive encoding from to that preserves compositionality, name invariance, operational correspondence, divergence reflection, and success sensitiveness, while introducing quantum specific invariants such as preserving the size of quantum registers. It further shows a negative separation: there is no encoding from to that is compositional, operationally corresponding, and success sensitive, and extends this hardness to related qCCS to CQP translations. The results suggest that Gorla's framework remains informative in the quantum setting but is likely not exhaustively capturing all quantum aspects, motivating new criteria such as qubit invariance and register size preservation and future probabilistic encodability notions.

Abstract

Quantum based systems are a relatively new research area for that different modelling languages including process calculi are currently under development. Encodings are often used to compare process calculi. Quality criteria are used then to rule out trivial or meaningless encodings. In this new context of quantum based systems, it is necessary to analyse the applicability of these quality criteria and to potentially extend or adapt them. As a first step, we test the suitability of classical criteria for encodings between quantum based languages and discuss new criteria. Concretely, we present an encoding, from a language inspired by CQP into a language inspired by qCCS. We show that this encoding satisfies compositionality, name invariance (for channel and qubit names), operational correspondence, divergence reflection, success sensitiveness, and that it preserves the size of quantum registers. Then we show that there is no encoding from qCCS into CQP that is compositional, operationally corresponding, and success sensitive.
Paper Structure (12 sections, 24 theorems, 39 equations, 3 figures)

This paper contains 12 sections, 24 theorems, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary.
  3. Quantum Based Systems
  4. Process Calculi
  5. A Calculus for Closed Quantum Systems
  6. A Calculus for Open Quantum Systems
  7. Encodings and Quality Criteria
  8. Encoding Quantum Based Systems
  9. Separating Quantum Based Systems
  10. Quality Criteria for Quantum Based Systems
  11. Conclusions
  12. Type System of Closed Quantum Systems

Key Result

Lemma 3.2

If $\Sigma \vdash P$ then $\mathsf{fq}{\left( P \right)} \subseteq \Sigma$.

Figures (3)

  • Figure 1: Semantics of $\mathsf{CQS}$
  • Figure 2: Typing Rules for $\mathsf{CQS}$
  • Figure 3: Semantics of $\mathsf{OQS}$

Theorems & Definitions (65)

  • Definition 2.1: Super-Operator, Operator-Sum Representation, NielsenChuang10
  • Definition 3.1: $\mathsf{CQS}$
  • Lemma 3.2: Free Qubits
  • Lemma 3.3: Preservation
  • Lemma 3.4: Unique Ownership of Qubits
  • Example 3.5: Quantum Teleportation
  • Definition 3.6: $\mathsf{OQS}$
  • Definition 4.1: Correspondence Simulation, petersGlabbeek15
  • Definition 4.2: Compositionality, gorla10
  • Definition 4.3: Name Invariance
  • ...and 55 more