A Lambda Calculus for Quantum Computation

@article{Tonder2004ALC,
  title={A Lambda Calculus for Quantum Computation},
  author={Andr{\'e} van Tonder},
  journal={SIAM J. Comput.},
  year={2004},
  volume={33},
  pages={1109-1135}
}
  • A. V. Tonder
  • Published 21 July 2003
  • Computer Science
  • SIAM J. Comput.
The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine and continues to be of enormous benefit in the classical theory of computation. We propose that quantum computation, like its classical counterpart, may benefit from a version of the lambda calculus suitable for expressing and reasoning about quantum algorithms. In this paper we develop a… 
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195 Citations
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195 Citations

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References

SHOWING 1-10 OF 82 REFERENCES
Quantum complexity theory
TLDR
This dissertation proves that relative to an oracle chosen uniformly at random, the class NP cannot be solved on a quantum Turing machine in time $o(2\sp{n/2}).$ and gives evidence suggesting that quantum Turing Machines cannot efficiently solve all of NP.
Extending the Lambda Calculus to Express Randomized and Quantumized Algorithms
TLDR
A formal metalanguage called the lambda-p calculus is introduced for the specification of programming languages that allow true random number generation and it is demonstrated how selected randomized algorithms can be programmed directly in the Lambda-q calculus.
The lambda-q calculus can efficiently simulate quantum computers
TLDR
The lambda-q calculus may be strictly stronger than quantum computers because NP-complete problems such as satisfiability are efficiently solvable in the lambda-Q calculus but there is a widespread doubt that they are efficientlysolvable by quantum computers.
Quantum Computation, Categorical Semantics and Linear Logic
TLDR
A type theory and denotational semantics are developed and provided for a simple fragment of the quantum lambda calculus, a formal language for quantum computation based on linear logic.
Quantum theory, the Church–Turing principle and the universal quantum computer
  • D. Deutsch
  • Physics, Philosophy
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1985
It is argued that underlying the Church–Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical principle: ‘every finitely realizible…
Towards a quantum programming language
  • P. Selinger
  • Computer Science
    Mathematical Structures in Computer Science
  • 2004
TLDR
This paper describes the syntax and semantics of a simple quantum programming language with high-level features such as loops, recursive procedures, and structured data types, and has an interesting denotational semantics in terms of complete partial orders of superoperators.
Toward an architecture for quantum programming
TLDR
A template high level quantum language is presented which complements a generic general purpose classical language with a set of quantum primitives, and easily lends itself to automatic, hardware independent, circuit simplification.
Quantum Circuit Complexity
  • A. Yao
  • Computer Science
    FOCS
  • 1993
TLDR
It is shown that any function computable in polynomial time by a quantum Turing machine has aPolynomial-size quantum circuit, and this result enables us to construct a universal quantum computer which can simulate a broader class of quantum machines than that considered by E. Bernstein and U. Vazirani (1993), thus answering an open question raised by them.
A Procedural Formalism for Quantum Computing
TLDR
QCL (Quantum Computation Language) is a hight level, architecture independent programming language for quantum computers, with a syntax derived from classical procedural languages like C or Pascal, which allows for the complete implementation and simulation of quantum algorithms in one consistent formalism.
Computational Interpretations of Linear Logic
...
...