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Quantum Information Effects

TL;DR: This work studies the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation, and provides universal categorical constructions that semantically interpret this arrow metalanguage with choice.
Abstract: We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguage with choice, starting with any rig groupoid interpreting the reversible base language. Several properties of quantum measurement follow in general, and we translate quantum flow charts into our language. The semantic constructions turn the category of unitaries between Hilbert spaces into the category of completely positive trace-preserving maps, and they turn the category of bijections between finite sets into the category of functions with chosen garbage. Thus they capture the fundamental theorems of classical and quantum reversible computing of Toffoli and Stinespring.
Citations
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Journal ArticleDOI
TL;DR: In this article , a universal construction of finite-dimensional C*-algebras and completely positive trace-nonincreasing maps from the rig category of finite dimensional Hilbert spaces and unitaries is presented.
Abstract: We provide a universal construction of the category of finite-dimensional C*-algebras and completely positive trace-nonincreasing maps from the rig category of finite-dimensional Hilbert spaces and unitaries. This construction, which can be applied to any rig groupoid, is described in three steps, each associated with their own universal property, and draws on results from dilation theory in finite dimension. In this way, we explicitly construct the category that captures hybrid quantum/classical computation with possible nontermination from the category of its reversible foundations. We discuss how this construction can be used in the design and semantics of quantum programming languages.

1 citations

Journal ArticleDOI
TL;DR: This paper introduces Jeopardy, a functional programming language that guarantees program invertibility without imposing local reversibility, and outlines three approaches that can give a partial static guarantee.
Abstract: Algorithms are ways of mapping problems to solutions. An algorithm is invertible precisely when this mapping is injective, such that the initial problem can be uniquely inferred from its solution. While invertible algorithms can be described in general-purpose languages, no guarantees are generally made by such languages as regards invertibility, so ensuring invertibility requires additional (and often non-trivial) proof. On the other hand, while reversible programming languages guarantee that their programs are invertible by restricting the permissi-ble operations to those which are locally invertible, writing programs in the reversible style can be cumbersome, and may differ significantly from conventional implementations even when the implemented algorithm is, in fact, invertible. In this paper we introduce Jeopardy, a functional programming language that guarantees program invertibility without imposing local reversibility. In particular, Jeopardy allows the limited use of uninvertible – and even nondeterministic! – operations, provided that they are used in a way that can be statically determined to be invertible. To this end, we outline an implicitly available arguments analysis and three further approaches that can give a partial static guarantee to the (generally difficult) problem of guaranteeing invertibility.

1 citations

Journal ArticleDOI
TL;DR: In this paper , Hughes' arrows are used to construct a universal quantum programming language from two copies of Pi, the internal language of rig groupoids, which can answer positively the question whether a computational effect exists that turns reversible classical computation into quantum computation.
Abstract: Free categorical constructions characterise quantum computing as the combination of two copies of a reversible classical model, glued by the complementarity equations of classical structures. This recipe effectively constructs a computationally universal quantum programming language from two copies of Pi, the internal language of rig groupoids. The construction consists of Hughes' arrows. Thus answer positively the question whether a computational effect exists that turns reversible classical computation into quantum computation: the quantum effect. Measurements can be added by layering a further effect on top. Our construction also enables some reasoning about quantum programs (with or without measurement) through a combination of classical reasoning and reasoning about complementarity.
References
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Book
01 Jan 2000
TL;DR: In this article, the quantum Fourier transform and its application in quantum information theory is discussed, and distance measures for quantum information are defined. And quantum error-correction and entropy and information are discussed.
Abstract: Part I Fundamental Concepts: 1 Introduction and overview 2 Introduction to quantum mechanics 3 Introduction to computer science Part II Quantum Computation: 4 Quantum circuits 5 The quantum Fourier transform and its application 6 Quantum search algorithms 7 Quantum computers: physical realization Part III Quantum Information: 8 Quantum noise and quantum operations 9 Distance measures for quantum information 10 Quantum error-correction 11 Entropy and information 12 Quantum information theory Appendices References Index

25,929 citations

01 Dec 2010
TL;DR: This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.
Abstract: Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.

14,825 citations


"Quantum Information Effects" refers background in this paper

  • ...While there is a program that inputs a piece of classical data and outputs two copies of that data, no such program exists for quantum data; this is the no cloning theorem [27, 14]....

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  • ..., [27]) of density matrices....

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  • ...For more details we refer to [39, 27]....

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Journal ArticleDOI
TL;DR: This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing, with a focus on entanglement.
Abstract: This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal with entanglement. The paper by R. Mosseri and P. Ribeiro presents a detailed description of the two-and three-qubit geometry in Hilbert space, dealing with the geometry of fibrations and discrete geometry. The paper by J.-G.Luque et al. is more algebraic and considers invariants of pure k-qubit states and their application to entanglement measurement.

14,205 citations

Journal ArticleDOI
TL;DR: Two simple, but representative, models of bistable devices are subjected to a more detailed analysis of switching kinetics to yield the relationship between speed and energy dissipation, and to estimate the effects of errors induced by thermal fluctuations.
Abstract: It is argued that computing machines inevitably involve devices which perform logical functions that do not have a single-valued inverse. This logical irreversibility is associated with physical irreversibility and requires a minimal heat generation, per machine cycle, typically of the order of kT for each irreversible function. This dissipation serves the purpose of standardizing signals and making them independent of their exact logical history. Two simple, but representative, models of bistable devices are subjected to a more detailed analysis of switching kinetics to yield the relationship between speed and energy dissipation, and to estimate the effects of errors induced by thermal fluctuations.

3,629 citations

Journal ArticleDOI
Charles H. Bennett1
TL;DR: This result makes plausible the existence of thermodynamically reversible computers which could perform useful computations at useful speed while dissipating considerably less than kT of energy per logical step.
Abstract: The usual general-purpose computing automaton (e.g.. a Turing machine) is logically irreversible- its transition function lacks a single-valued inverse. Here it is shown that such machines may he made logically reversible at every step, while retainillg their simplicity and their ability to do general computations. This result is of great physical interest because it makes plausible the existence of thermodynamically reversible computers which could perform useful computations at useful speed while dissipating considerably less than kT of energy per logical step. In the first stage of its computation the logically reversible automaton parallels the corresponding irreversible automaton, except that it saves all intermediate results, there by avoiding the irreversible operation of erasure. The second stage consists of printing out the desired output. The third stage then reversibly disposes of all the undesired intermediate results by retracing the steps of the first stage in backward order (a process which is only possible because the first stage has been carried out reversibly), there by restoring the machine (except for the now-written output tape) to its original condition. The final machine configuration thus contains the desired output and a reconstructed copy of the input, but no other undesired data. The foregoing results are demonstrated explicitly using a type of three-tape Turing machine. The biosynthesis of messenger RNA is discussed as a physical example of reversible computation.

3,497 citations


"Quantum Information Effects" refers background in this paper

  • ...However, by the seminal works of Toffoli [Toffoli 1980] and Bennett [Bennett 1973], and more recently by James and Sabry [James and Sabry 2012], we know that it can also be phrased in terms of reversible operations, as long as we consider systems to be open and interact with an environment that is…...

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  • ...However, by the seminal works of Toffoli [35] and Bennett [2], and more recently by James and Sabry [19], we know that it can also be phrased in terms of reversible operations, as long as we consider systems to be open and interact with an environment that is eventually disregarded....

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Trending Questions (1)
What are the fundamental properties of quantum information?

The paper does not explicitly mention the fundamental properties of quantum information.