# Quantum CPOs

@article{Kornell2021QuantumC, title={Quantum CPOs}, author={Andre Kornell and Bert Lindenhovius and Michael W. Mislove}, journal={ArXiv}, year={2021}, volume={abs/2109.02196} }

We introduce the monoidal closed category qCPO of quantum cpos, whose objects are ‘quantized’ analogs of ω-complete partial orders (cpos). The category qCPO is enriched over the category CPO of cpos, and contains both CPO, and the opposite of the category FdAlg of finite-dimensional von Neumann algebras as monoidal subcategories. We use qCPO to construct a sound model for the quantum programming language Proto-Quipper-M (PQM) extended with term recursion, as well as a sound and computationally…

## References

SHOWING 1-10 OF 31 REFERENCES

### Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory

- Computer Science, MathematicsLog. Methods Comput. Sci.
- 2020

Categorical models of a circuit-based (quantum) functional programming language such that the circuit language is embedded inside the host language, and this structure is also related to linear/non-linear models.

### A categorical model for a quantum circuit description language

- Computer ScienceQPL
- 2017

This paper formalizes a small, but useful fragment of Quipper called Proto-Quipper-M, which is type-safe and has a formal denotational and operational semantics, and defines the programming language to fit the model.

### Quantum sets

- Mathematics
- 2018

A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger…

### Enriching a Linear/Non-linear Lambda Calculus: A Programming Language for String Diagrams

- Computer ScienceLICS
- 2018

This work presents an adequacy result for the diagram-free fragment of the language which corresponds to a modified version of Benton and Wadler's adjoint calculus with recursion, and extends the language with general recursion and proves soundness.

### An axiomatisation of computationally adequate domain theoretic models of FPC

- MathematicsProceedings Ninth Annual IEEE Symposium on Logic in Computer Science
- 1994

Categorical models of the metalanguage FPC (a type theory with sums, products, exponentials and recursive types) are defined. Then, domain-theoretic models of FPC are axiomatised and a wide subclass…

### Applying quantitative semantics to higher-order quantum computing

- Computer SciencePOPL
- 2014

This paper proposes a denotational semantics for a quantum lambda calculus with recursion and an infinite data type, using constructions from quantitative semantics of linear logic.

### Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory

- Computer ScienceFoSSaCS
- 2020

This paper constructs a sound categorical model for the quantum programming language QPL and provides the first detailed semantic treatment of user-defined inductive datatypes in quantum programming, and shows the denotational interpretation is invariant with respect to big-step reduction.

### A lambda calculus for quantum computation with classical control

- Computer ScienceMathematical Structures in Computer Science
- 2006

A functional programming language for quantum computers by extending the simply-typed lambda calculus with quantum types and operations, and gives a type system using affine intuitionistic linear logic.

### QWIRE: a core language for quantum circuits

- Computer SciencePOPL
- 2017

This paper introduces QWIRE (``choir''), a language for defining quantum circuits and an interface for manipulating them inside of an arbitrary classical host language, along with its type system and operational semantics, which it is proved is safe and strongly normalizing whenever the host language is.

### LNL-FPC: The Linear/Non-linear Fixpoint Calculus

- Computer ScienceLog. Methods Comput. Sci.
- 2021

A type system with mixed linear and non-linear recursive types called LNL-FPC (the linear/non-linear fixpoint calculus) and a new technique for solving recursive domain equations within cartesian categories by constructing the solutions over pre-embeddings is described.