## Geometry of abstraction in quantum computation

Citations: | 2 - 2 self |

### BibTeX

@MISC{Pavlovic_geometryof,

author = {Dusko Pavlovic},

title = {Geometry of abstraction in quantum computation},

year = {}

}

### OpenURL

### Abstract

Quantum algorithms are sequences of abstract operations, performed on non-existent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum algorithms provide feasible solutions of important hard problems, such as factoring and discrete log (which are the building blocks of modern cryptography). It is of a great practical interest to precisely characterize the computational resources needed to execute such quantum algorithms. There are many ideas how to build a quantum computer. Can we prove some necessary conditions? Categorical semantics help with such questions. We show how to implement an important family of quantum algorithms using just abelian groups and relations.

### Citations

1372 | Quantum Computation and Quantum Information - Nielsen, Chuang - 2000 |

882 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ...rs do? They do a variety of things, but there is a ”design pattern” that they often follow, based on the Hidden Subgroup Problem (HSP) [26, 28, sec. 5.4]. Shor’s factoring and discrete log algorithms =-=[37]-=- are examples of this pattern, as well as Hallgren’s algorithm for the Pell equation [15]. They all provide an exponential speedup with respect to the best known classical algorithms. The simplest mem... |

852 |
A formulation of the simple theory of types
- Church
- 1940
(Show Context)
Citation Context ...the fundamental lemma of recursion theory: the s-m-n theorem [19]. Church, finally3 proposed the formal operations of function abstraction and data application as the driving force of all computation =-=[6]-=-. This proposal became the foundation of functional programming. Lawvere’s observation that Church’s λ-abstraction could be interpreted as an adjunction transposition [24] was a critical step towards ... |

427 |
Introduction to higher order categorical logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...ntics of polynomial constructions, and of the abstraction and substitution operations. In the framework of cartesian (closed) categories, such a treatment goes back to Lambek and Scott’s seminal work =-=[22, 23]-=-. It was extended to monoidal categories in [30]. Here we extend it to dagger-monoidal categories. 3.1 Polynomial constructions Adjoining an indeterminate x to a ring R leads to the ring of polynomial... |

387 |
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte der
- Gödel
- 1931
(Show Context)
Citation Context ...x) −−−−−→B X Notion of abstraction. Function abstraction is what makes programming possible. The first example of program abstraction were probably Gödel’s numberings of primitive recursive functions =-=[14]-=-. Gödel’s construction demonstrated that recursive programs, specifying entire families of computations (of the values of a function for all its inputs), can be stored as data. Von Neumann later expli... |

379 | Basic Concepts of Enriched Category Theory
- Kelly
- 1982
(Show Context)
Citation Context ... 2 Preliminaries 2.1 Monoidal categories We assume that the reader has some understanding of the basic categorical concepts and terminology [27], and work with symmetric monoidal categories (C, ⊗, I) =-=[17, 16]-=-. Strictness. For simplicity, and without loss of generality, we tacitly assume that each of our monoidal categories is strictly associative and unitary, i.e. that the objects form a monoid in the usu... |

355 | On the power of quantum computation
- Simon
- 1997
(Show Context)
Citation Context ...ithm for the Pell equation [15]. They all provide an exponential speedup with respect to the best known classical algorithms. The simplest member of the family is Simon’s algorithm for period finding =-=[38]-=-, which we use as the running example. The other HSP algorithms only differ in ”domain specific” details, but yield to the same semantics. The input for Simon’s algorithm is an arbitrary function f : ... |

223 |
Spinors and Space-time I
- Penrose, Rindler
- 1986
(Show Context)
Citation Context ...1.1 String diagrams Calculations in monoidal categories are supported by a simple and intuitive graphical language: the string diagrams. This language has its roots in Penrose’s diagrammatic notation =-=[33]-=-, and it has been formally developed in categorical coherence theory, and in particular in Joyal and Street’s geometry of tensor calculus [16]. The objects are drawn as strings, and the morphisms as b... |

150 | A categorical semantics of quantum protocols
- Abramsky, Coecke
- 2004
(Show Context)
Citation Context ...trace operators, given above, require a monoidal structure in C. The interactions between the dagger with the monoidal structure, and in particular with the duals, has been recognized and analyzed in =-=[1, 35, 36]-=-. A dagger-monoidal category (C, ⊗, I, ‡) is a dagger-category with a monoidal structure where all canonical isomorphisms, that form the monoidal structure, are unitary. When the monoidal structure is... |

145 |
The geometry of tensor calculus
- JOYAL, STREET
- 1991
(Show Context)
Citation Context ... 2 Preliminaries 2.1 Monoidal categories We assume that the reader has some understanding of the basic categorical concepts and terminology [27], and work with symmetric monoidal categories (C, ⊗, I) =-=[17, 16]-=-. Strictness. For simplicity, and without loss of generality, we tacitly assume that each of our monoidal categories is strictly associative and unitary, i.e. that the objects form a monoid in the usu... |

138 |
A single quantum cannot be cloned
- Wootters, Zurek
- 1982
(Show Context)
Citation Context ...not occur in it. The first problem with quantum programming is that quantum data cannot be manipulated in this way: it is a fundamental property of quantum states that they generally cannot be copied =-=[41, 11]-=-, or even deleted [29, 2]. So how do we write quantum programs? In particular, given a program f(x) for a function f, what kind of a program transformation leads to the quantum program Uf |x, y〉, that... |

107 |
Coherence for compact closed categories
- Kelly, Laplaza
- 1980
(Show Context)
Citation Context ...C, the forgetful functor CB −→ C is couniversal for the conservative functors among them. The exactness properties of CB, induced by the various properties of C, were analyzed in [4]. If C is compact =-=[18]-=- and right exact with biproducts, then CB turns out to be a pretopos. In any case, if C represents a quantum universe, CB can be thought of as the category of classical data. 4.2.1 Orthonormality of b... |

99 | Premonoidal categories and notions of computation
- Power, Robinson
- 1997
(Show Context)
Citation Context ...an algebras. Since Boolean algebras are primal, every function between them can be expressed as a polynomial. 7 The tensor m ⊗ n = m × n is functorial in each argument, but it is not a bifunctor. See =-=[34]-=- for a discussion about such structures. This has no repercussions for us, since the definition of the functor Ξ ⊗(−) , spelled out explicitly below, makes no use of the arrow part of ⊗. 20sBoth side ... |

82 | Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem
- Hallgren
- 2007
(Show Context)
Citation Context ...w, based on the Hidden Subgroup Problem (HSP) [26, 28, sec. 5.4]. Shor’s factoring and discrete log algorithms [37] are examples of this pattern, as well as Hallgren’s algorithm for the Pell equation =-=[15]-=-. They all provide an exponential speedup with respect to the best known classical algorithms. The simplest member of the family is Simon’s algorithm for period finding [38], which we use as the runni... |

77 |
Frobenius Algebras and 2D Topological Quantum Field Theories
- Kock
- 2003
(Show Context)
Citation Context ...(ii) (X ⊗ ∇) ◦ (η ⊗ X) = ∆ = (∇ ⊗ X) ◦ (X ⊗ η) ∇ = ∆ = ∇ ∆ ∆ ⊥ ⊥ (iii) (X ⊗ ∇) ◦ (∆ ⊗ X) = ∆ ◦ ∇ = (∇ ⊗ X) ◦ (X ⊗ ∆) ∇ = ∆ = ∇ ∆ ∇ ∆ 15Remark. Condition (iii) is the Frobenius condition, analyzed in =-=[5, 4, 20, 7]-=-. Condition (ii) is Lawvere’s earlier version of the same [25]. In each of the last three conditions, the commutativity assumption makes one of the equations redundant. The equivalence of (i-iii), how... |

75 |
Cartesian Bicategories I
- Carboni, Walters
- 1987
(Show Context)
Citation Context ...(ii) (X ⊗ ∇) ◦ (η ⊗ X) = ∆ = (∇ ⊗ X) ◦ (X ⊗ η) ∇ = ∆ = ∇ ∆ ∆ ⊥ ⊥ (iii) (X ⊗ ∇) ◦ (∆ ⊗ X) = ∆ ◦ ∇ = (∇ ⊗ X) ◦ (X ⊗ ∆) ∇ = ∆ = ∇ ∆ ∇ ∆ 15Remark. Condition (iii) is the Frobenius condition, analyzed in =-=[5, 4, 20, 7]-=-. Condition (ii) is Lawvere’s earlier version of the same [25]. In each of the last three conditions, the commutativity assumption makes one of the equations redundant. The equivalence of (i-iii), how... |

51 |
Communication by EPR Devices
- Dieks
- 1982
(Show Context)
Citation Context ...not occur in it. The first problem with quantum programming is that quantum data cannot be manipulated in this way: it is a fundamental property of quantum states that they generally cannot be copied =-=[41, 11]-=-, or even deleted [29, 2]. So how do we write quantum programs? In particular, given a program f(x) for a function f, what kind of a program transformation leads to the quantum program Uf |x, y〉, that... |

44 |
Recursive predicates and quantifiers
- Kleene
- 1943
(Show Context)
Citation Context ...icated this as the fundamental principle of computer architecture. Kleene, on the other side, refined the idea of program abstraction into the fundamental lemma of recursion theory: the s-m-n theorem =-=[19]-=-. Church, finally3 proposed the formal operations of function abstraction and data application as the driving force of all computation [6]. This proposal became the foundation of functional programmin... |

42 |
Categories for the Working Mathematician. Number 5
- Lane
- 1971
(Show Context)
Citation Context ...e basic semantical prerequisites, notations and terminology. 2 Preliminaries 2.1 Monoidal categories We assume that the reader has some understanding of the basic categorical concepts and terminology =-=[27]-=-, and work with symmetric monoidal categories (C, ⊗, I) [17, 16]. Strictness. For simplicity, and without loss of generality, we tacitly assume that each of our monoidal categories is strictly associa... |

29 |
Ordinal sums and equational doctrines
- Lawvere
- 1969
(Show Context)
Citation Context ...(X ⊗ ∇) ◦ (∆ ⊗ X) = ∆ ◦ ∇ = (∇ ⊗ X) ◦ (X ⊗ ∆) ∇ = ∆ = ∇ ∆ ∇ ∆ 15Remark. Condition (iii) is the Frobenius condition, analyzed in [5, 4, 20, 7]. Condition (ii) is Lawvere’s earlier version of the same =-=[25]-=-. In each of the last three conditions, the commutativity assumption makes one of the equations redundant. The equivalence of (i-iii), however, holds without this commutativity. Proof. (a⇒b) Using the... |

25 |
Impossibility of deleting an unknown quantum state”, Nature 404
- Pati, Braunstein
- 2000
(Show Context)
Citation Context ... problem with quantum programming is that quantum data cannot be manipulated in this way: it is a fundamental property of quantum states that they generally cannot be copied [41, 11], or even deleted =-=[29, 2]-=-. So how do we write quantum programs? In particular, given a program f(x) for a function f, what kind of a program transformation leads to the quantum program Uf |x, y〉, that we used to specify the u... |

23 | 2007) Quantum measurements without sums
- Coecke, Pavlovic
(Show Context)
Citation Context ...mputer, a structure that supports copying, deleting and abstraction can be construed as its classical interface. This is what we call a classical structure. An early analysis of this structure was in =-=[7]-=-. In the meantime, there are several versions, and many applications [8, 31, 12]. In recent work, Coecke [9] uses the term basis structures for the same concept, because a classical structure over a f... |

22 | R.: Interacting quantum observables - Coecke, Duncan - 2008 |

21 | Categorical logic of names and abstraction in action calculi
- Pavlovic
- 1997
(Show Context)
Citation Context ...ction and substitution operations. In the framework of cartesian (closed) categories, such a treatment goes back to Lambek and Scott’s seminal work [22, 23]. It was extended to monoidal categories in =-=[30]-=-. Here we extend it to dagger-monoidal categories. 3.1 Polynomial constructions Adjoining an indeterminate x to a ring R leads to the ring of polynomials R[x]. Its universal property is that every rin... |

20 | Complementary observables and uncertainty relations - Kraus - 1987 |

17 |
Toposes, Triples, and Theories. Number 278 in Grundlehren der mathematischen Wissenschaften
- Barr, Wells
- 1985
(Show Context)
Citation Context ...fined as a functor T : C −→ C together with a monoid structure TT m −→ T h ←− Id in the category of endofunctors on C. With the corresponding monoid homomorphisms, monads form a category on their own =-=[3]-=-. Dually, comonads on C can be defined as comonoids in the category of endofunctors over C, and accomodate similar developments. The categories of algebras for a monad and coalgebras for a comonad, an... |

13 | Quantum measurements and finite geometry - Wootters - 2004 |

11 | A New Description of Orthogonal Bases
- Coecke, Pavlovic, et al.
- 2008
(Show Context)
Citation Context ...2]. In recent work, Coecke [9] uses the term basis structures for the same concept, because a classical structure over a finitely dimensional Hilbert space precisely correspond to a choice of a basis =-=[10]-=-, and can be viewed as a purely categorical, element-free version of this notion. While the simple basis intuitions are attractive, I stick 1 E.g., Laplace’s transform maps a differential equation int... |

8 | Complementarity in categorical quantum mechanics
- Heunen
- 2010
(Show Context)
Citation Context ... problem with quantum programming is that quantum data cannot be manipulated in this way: it is a fundamental property of quantum states that they generally cannot be copied [41, 11], or even deleted =-=[29, 2]-=-. So how do we write quantum programs? In particular, given a program f(x) for a function f, what kind of a program transformation leads to the quantum program Uf |x, y〉, that we used to specify the u... |

8 |
group representations
- Matrices
- 1991
(Show Context)
Citation Context ...(ii) (X ⊗ ∇) ◦ (η ⊗ X) = ∆ = (∇ ⊗ X) ◦ (X ⊗ η) ∇ = ∆ = ∇ ∆ ∆ ⊥ ⊥ (iii) (X ⊗ ∇) ◦ (∆ ⊗ X) = ∆ ◦ ∇ = (∇ ⊗ X) ◦ (X ⊗ ∆) ∇ = ∆ = ∇ ∆ ∇ ∆ 15Remark. Condition (iii) is the Frobenius condition, analyzed in =-=[5, 4, 20, 7]-=-. Condition (ii) is Lawvere’s earlier version of the same [25]. In each of the last three conditions, the commutativity assumption makes one of the equations redundant. The equivalence of (i-iii), how... |

8 | Quantum and classical structures in nondeterministic computation
- Pavlović
- 2009
(Show Context)
Citation Context ...be construed as its classical interface. This is what we call a classical structure. An early analysis of this structure was in [7]. In the meantime, there are several versions, and many applications =-=[8, 31, 12]-=-. In recent work, Coecke [9] uses the term basis structures for the same concept, because a classical structure over a finitely dimensional Hilbert space precisely correspond to a choice of a basis [1... |

7 |
Toy quantum categories
- Coecke, Edwards
- 2008
(Show Context)
Citation Context ...ace. This is what we call a classical structure. An early analysis of this structure was in [7]. In the meantime, there are several versions, and many applications [8, 31, 12]. In recent work, Coecke =-=[9]-=- uses the term basis structures for the same concept, because a classical structure over a finitely dimensional Hilbert space precisely correspond to a choice of a basis [10], and can be viewed as a p... |

6 | Quantum hidden subgroup algorithms: An algorithmic toolkit - Lomonaco, Kauffman - 2007 |

4 |
Lawvere. Adjointness in foundations
- William
- 2006
(Show Context)
Citation Context ...ving force of all computation [6]. This proposal became the foundation of functional programming. Lawvere’s observation that Church’s λ-abstraction could be interpreted as an adjunction transposition =-=[24]-=- was a critical step towards categorical semantics of computation. Theorem 3.1 spells out this observation in terms of polynomial categories. Besides the familiar λ-abstraction, which uses the right a... |

3 | Idempotents in dagger categories: (extended abstract
- Selinger
(Show Context)
Citation Context ...trace operators, given above, require a monoidal structure in C. The interactions between the dagger with the monoidal structure, and in particular with the duals, has been recognized and analyzed in =-=[1, 35, 36]-=-. A dagger-monoidal category (C, ⊗, I, ‡) is a dagger-category with a monoidal structure where all canonical isomorphisms, that form the monoidal structure, are unitary. When the monoidal structure is... |

2 |
Coecke Éric Oliver Paquette and Dusko Pavlovic. Classical and quantum structuralism
- Bob
- 2008
(Show Context)
Citation Context ...be construed as its classical interface. This is what we call a classical structure. An early analysis of this structure was in [7]. In the meantime, there are several versions, and many applications =-=[8, 31, 12]-=-. In recent work, Coecke [9] uses the term basis structures for the same concept, because a classical structure over a finitely dimensional Hilbert space precisely correspond to a choice of a basis [1... |

2 |
From types to sets
- Lambek
- 1980
(Show Context)
Citation Context ...ntics of polynomial constructions, and of the abstraction and substitution operations. In the framework of cartesian (closed) categories, such a treatment goes back to Lambek and Scott’s seminal work =-=[22, 23]-=-. It was extended to monoidal categories in [30]. Here we extend it to dagger-monoidal categories. 3.1 Polynomial constructions Adjoining an indeterminate x to a ring R leads to the ring of polynomial... |

1 |
An Introduction to Transform Theory, volume 42 of Pure and Applied Mathematics
- Widder
- 1971
(Show Context)
Citation Context ... n. The values of the function f are recovered from Uf|x, 0〉 = |x, f(x)〉. The other conceptual component of Simon’s algorithm, and of all HSP-algorithms, is a standard application of transform theory =-=[39]-=-: transform the inputs into another domain, where the computation is easier, compute the outputs there, and then transform them back 1 . In our special case, Uf is thus precomposed and postcomposed wi... |