arXiv subjects

Amr Sabry

Publications and source records attributed to Amr Sabry.

29 records · Page 2Linked to original sources

arXiv2018

From Symmetric Pattern-Matching to Quantum Control (Extended Version)

One perspective on quantum algorithms is that they are classical algorithms having access to a special kind of memory with exotic properties. This perspective suggests that, even in the case of quantum algorithms, the control flow notions of sequencing, conditionals, loops, and recursion are entirely classical. There is however, another notion of control flow, that is itself quantum. The notion of quantum conditional expression is reasonably well-understood: the execution of the two expressions becomes itself a superposition of executions. The quantum counterpart of loops and recursion is however not believed to be meaningful in its most general form. In this paper, we argue that, under the right circumstances, a reasonable notion of quantum loops and recursion is possible. To this aim, we first propose a classical, typed, reversible language with lists and fixpoints. We then extend this language to the closed quantum domain (without measurements) by allowing linear combinations of terms and restricting fixpoints to structurally recursive fixpoints whose termination proofs match the proofs of convergence of sequences in infinite-dimensional Hilbert spaces. We additionally give an operational semantics for the quantum language in the spirit of algebraic lambda-calculi and illustrate its expressiveness by modeling several common unitary operations.

cs.LO↗

arXiv2017

From Reversible Programs to Univalent Universes and Back

We establish a close connection between a reversible programming language based on type isomorphisms and a formally presented univalent universe. The correspondence relates combinators witnessing type isomorphisms in the programming language to paths in the univalent universe; and combinator optimizations in the programming language to 2-paths in the univalent universe. The result suggests a simple computational interpretation of paths and of univalence in terms of familiar programming constructs whenever the universe in question is computable.

cs.PL↗

arXiv2016

Reversible Communicating Processes

Reversible distributed programs have the ability to abort unproductive computation paths and backtrack, while unwinding communication that occurred in the aborted paths. While it is natural to assume that reversibility implies full state recovery (as with traditional roll-back recovery protocols), an interesting alternative is to separate backtracking from local state recovery. For example, such a model could be used to create complex transactions out of nested compensable transactions where a programmer-supplied compensation defines the work required to "unwind" a transaction. Reversible distributed computing has received considerable theoretical attention, but little reduction to practice; the few published implementations of languages supporting reversibility depend upon a high degree of central control. The objective of this paper is to demonstrate that a practical reversible distributed language can be efficiently implemented in a fully distributed manner. We discuss such a language, supporting CSP-style synchronous communication, embedded in Scala. While this language provided the motivation for the work described in this paper, our focus is upon the distributed implementation. In particular, we demonstrate that a "high-level" semantic model can be implemented using a simple point-to-point protocol.

cs.PL↗

arXiv2013

Geometry of Discrete Quantum Computing

Conventional quantum computing entails a geometry based on the description of an n-qubit state using 2^{n} infinite precision complex numbers denoting a vector in a Hilbert space. Such numbers are in general uncomputable using any real-world resources, and, if we have the idea of physical law as some kind of computational algorithm of the universe, we would be compelled to alter our descriptions of physics to be consistent with computable numbers. Our purpose here is to examine the geometric implications of using finite fields Fp and finite complexified fields Fp^2 (based on primes p congruent to 3 mod{4}) as the basis for computations in a theory of discrete quantum computing, which would therefore become a computable theory. Because the states of a discrete n-qubit system are in principle enumerable, we are able to determine the proportions of entangled and unentangled states. In particular, we extend the Hopf fibration that defines the irreducible state space of conventional continuous n-qubit theories (which is the complex projective space CP{2^{n}-1}) to an analogous discrete geometry in which the Hopf circle for any n is found to be a discrete set of p+1 points. The tally of unit-length n-qubit states is given, and reduced via the generalized Hopf fibration to DCP{2^{n}-1}, the discrete analog of the complex projective space, which has p^{2^{n}-1} (p-1)\prod_{k=1}^{n-1} (p^{2^{k}}+1) irreducible states. Using a measure of entanglement, the purity, we explore the entanglement features of discrete quantum states and find that the n-qubit states based on the complexified field Fp^2 have p^{n} (p-1)^{n} unentangled states (the product of the tally for a single qubit) with purity 1, and they have p^{n+1}(p-1)(p+1)^{n-1} maximally entangled states with purity zero.

quant-ph↗

arXiv2013

Discrete Quantum Theories

We explore finite-field frameworks for quantum theory and quantum computation. The simplest theory, defined over unrestricted finite fields, is unnaturally strong. A second framework employs only finite fields with no solution to x^2+1=0, and thus permits an elegant complex representation of the extended field by adjoining i=\sqrt{-1}. Quantum theories over these fields recover much of the structure of conventional quantum theory except for the condition that vanishing inner products arise only from null states; unnaturally strong computational power may still occur. Finally, we are led to consider one more framework, with further restrictions on the finite fields, that recovers a local transitive order and a locally-consistent notion of inner product with a new notion of cardinal probability. In this framework, conventional quantum mechanics and quantum computation emerge locally (though not globally) as the size of the underlying field increases. Interestingly, the framework allows one to choose separate finite fields for system description and for measurement: the size of the first field quantifies the resources needed to describe the system and the size of the second quantifies the resources used by the observer. This resource-based perspective potentially provides insights into quantitative measures for actual computational power, the complexity of quantum system definition and evolution, and the independent question of the cost of the measurement process.

quant-ph↗

arXiv2011

The Power of Discrete Quantum Theories

We explore the implications of restricting the framework of quantum theory and quantum computation to finite fields. The simplest proposed theory is defined over arbitrary finite fields and loses the notion of unitaries. This makes such theories unnaturally strong, permitting the search of unstructured databases faster than asymptotically possible in conventional quantum computing. The next most general approach chooses finite fields with no solution to x^2+1=0, and thus permits an elegant complex-like representation of the extended field by adjoining i=sqrt(-1). Quantum theories over these fields retain the notion of unitaries and --- for particular problem sizes --- allow the same algorithms as conventional quantum theory. These theories, however, still support unnaturally strong computations for certain problem sizes, but the possibility of such phenomena decreases as the size of the field increases.

quant-ph↗

arXiv2011

Solving UNIQUE-SAT in a Modal Quantum Theory

In recent work, Benjamin Schumacher and Michael D. Westmoreland investigate a version of quantum mechanics which they call modal quantum theory. This theory is obtained by instantiating the mathematical framework of Hilbert spaces with a finite field instead of the field of complex numbers. This instantiation collapses much the structure of actual quantum mechanics but retains several of its distinguishing characteristics including the notions of superposition, interference, and entanglement. Furthermore, modal quantum theory excludes local hidden variable models, has a no-cloning theorem, and can express natural counterparts of quantum information protocols such as superdense coding and teleportation. We show that the problem of UNIQUE-SAT --- which decides whether a given Boolean formula is unsatisfiable or has exactly one satisfying assignment --- is deterministically solvable in any modal quantum theory in constant time. The solution exploits the lack of orthogonality in modal quantum theories and is not directly applicable to actual quantum theory.

quant-ph↗

arXiv2011

Quantum Computing over Finite Fields

In recent work, Benjamin Schumacher and Michael~D. Westmoreland investigate a version of quantum mechanics which they call "modal quantum theory" but which we prefer to call "discrete quantum theory". This theory is obtained by instantiating the mathematical framework of Hilbert spaces with a finite field instead of the field of complex numbers. This instantiation collapses much the structure of actual quantum mechanics but retains several of its distinguishing characteristics including the notions of superposition, interference, and entanglement. Furthermore, discrete quantum theory excludes local hidden variable models, has a no-cloning theorem, and can express natural counterparts of quantum information protocols such as superdense coding and teleportation. Our first result is to distill a model of discrete quantum computing from this quantum theory. The model is expressed using a monadic metalanguage built on top of a universal reversible language for finite computations, and hence is directly implementable in a language like Haskell. In addition to superpositions and invertible linear maps, the model includes conventional programming constructs including pairs, sums, higher-order functions, and recursion. Our second result is to relate this programming model to relational programming, e.g., a pure version of Prolog over finite relations. Surprisingly discrete quantum computing is identical to conventional logic programming except for a small twist that is responsible for all the ``quantum-ness.'' The twist occurs when merging sets of answers computed by several alternatives: the answers are combined using an "exclusive" version of logical disjunction. In other words, the two branches of a choice junction exhibit an "interference" effect: an answer is produced from the junction if it occurs in one or the other branch but not both.

quant-ph↗

arXiv2010

Lazy Evaluation and Delimited Control

The call-by-need lambda calculus provides an equational framework for reasoning syntactically about lazy evaluation. This paper examines its operational characteristics. By a series of reasoning steps, we systematically unpack the standard-order reduction relation of the calculus and discover a novel abstract machine definition which, like the calculus, goes "under lambdas." We prove that machine evaluation is equivalent to standard-order evaluation. Unlike traditional abstract machines, delimited control plays a significant role in the machine's behavior. In particular, the machine replaces the manipulation of a heap using store-based effects with disciplined management of the evaluation stack using control-based effects. In short, state is replaced with control. To further articulate this observation, we present a simulation of call-by-need in a call-by-value language using delimited control operations.

cs.PL↗

arXiv2009

The Arrow Calculus as a Quantum Programming Language

We express quantum computations (with measurements) using the arrow calculus extended with monadic constructions. This framework expresses quantum programming using well-understood and familiar classical patterns for programming in the presence of computational effects. In addition, the five laws of the arrow calculus provide a convenient framework for equational reasoning about quantum computations that include measurements.

quant-ph↗

arXiv2005

An Algebra of Pure Quantum Programming

We develop a sound and complete equational theory for the functional quantum programming language QML. The soundness and completeness of the theory are with respect to the previously-developed denotational semantics of QML. The completeness proof also gives rise to a normalisation algorithm following the normalisation by evaluation approach. The current work focuses on the pure fragment of QML omitting measurements.

quant-ph↗