- Computer Science, Mathematics, Programming Languages, Logic And Foundations Of Mathematics, Quantum Computing, Algebra, and 9 moreTheory Of Computation, Compilers, Category Theory, Group Theory, Program Semantics, Universal Algebra, Group Ring Theory, Quantum Circuits, and Information Technologyedit
- Thesis is done! "An investigation of some theoretical aspects of reversible computing"
or...
Everything you wanted to know about inverse categories but were afraid to ask.edit
Research Interests:
"This thesis deals with the problem of describing the unit group of specific group rings over the integers. Some results are given in an expository manner as the proofs of the results are normally very dependent on the particular... more
"This thesis deals with the problem of describing the unit group of specific group rings over the integers. Some results are given in an expository manner as the proofs of the results are normally very dependent on the particular group. The ones presented by this method later on are S3,D4,D6 and A4. Next, we present the method for groups of order p3, where p is an odd prime. These come from a paper by Ritter and Sehgal. I consider both non-abelian groups of order p3 and descriptions of the unit groups of both of their respective group rings are presented.I present the method as applied to the groups of order 27. The last theoretical results are on determining the unit group of the group ring over a group of order pq where p ≡ 1( mod q). These results come from a paper by Luthar [3]. No practical examples are done of this method. The next part deals with presenting actual groups and determining the unit structure of their integral group ring. The first two, S3 and D4 are from previous authors. The first was done by Hughes and Pearson [2], the second by C. Polcino- Milies [4]. I also present, D6, which is new result of the thesis."
We prove that a unitary matrix has an exact representation over the Clifford+T gate set with local ancillas if and only if its entries are in the ring Z[1/sqrt(2),i]. Moreover, we show that one ancilla always suffices. These facts were... more
We prove that a unitary matrix has an exact representation over the Clifford+T gate set with local ancillas if and only if its entries are in the ring Z[1/sqrt(2),i]. Moreover, we show that one ancilla always suffices. These facts were conjectured by Kliuchnikov, Maslov, and Mosca. We obtain an algorithm for synthesizing a exact Clifford+T circuit from any such n-qubit operator. We also characterize the Clifford+T operators that can be represented without ancillas.
Research Interests:
This paper consists of two parts. The first part discusses the formal specification of the CASA library. CASA is a java library implementing agent societies and communica-tion under the umbrella of social commitments. The second part... more
This paper consists of two parts. The first part discusses the formal specification of the CASA library. CASA is a java library implementing agent societies and communica-tion under the umbrella of social commitments. The second part explores the possibility of describing social commitment based agent communication via polycategories. Categories and Subject Descriptors
The categorical semantics of reversible computing must be a category which combines the concepts of partiality and the ability to reverse any map in the category. Inverse cate-gories, restriction categories in which each map is a partial... more
The categorical semantics of reversible computing must be a category which combines the concepts of partiality and the ability to reverse any map in the category. Inverse cate-gories, restriction categories in which each map is a partial isomorphism, provide exactly this structure. This thesis explores inverse categories and relates them to both quantum computing and standard non-reversible computing. The former is achieved by showing that
The categorical semantics of reversible computing must be a category which combines the concepts of partiality and the ability to reverse any map in the category. Inverse categories, restriction categories in which each map is a partial... more
The categorical semantics of reversible computing must be a category which combines the concepts of partiality and the ability to reverse any map in the category. Inverse categories, restriction categories in which each map is a partial isomorphism, provide exactly this structure. This thesis explores inverse categories and relates them to both quantum computing and standard non-reversible computing. The former is achieved by showing that commutative Frobenius algebras form an inverse category. The latter is by establishing the equivalence of the category of discrete inverse categories to the category of discrete Cartesian restriction categories — this is the main result of this thesis. This allows one to transfer the formulation of computability given by Turing categories onto discrete inverse categories.
Research Interests:
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined... more
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano's results, simplifying some proofs along the way. We also point out some corollaries to Matsumoto and Amano's work, including an intrinsic characterization of the Clifford+T subgroup of SO(3), which also yields an efficient T-optimal exact single-qubit synthesis algorithm. Interestingly, this also gives an alternative proof of Kliuchnikov, Maslov, and Mosca's exact synthesis result for the Clifford+T subgroup of U(2).
Research Interests:
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined... more
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano’s results, simplifying some proofs along the way. We also point out some corollaries to Matsumoto and Amano’s work, including an intrinsic characterization of the Clifford+T subgroup of SO(3), which also yields an efficientT-optimal exact single-qubit synthesis algorithm. Interestingly, this also gives an alternative proof of Kliuchnikov, Maslov, and Mosca’s exact synthesis result for the Clifford+T subgroup of U(2).
Research Interests:
This paper consists of two parts. The first part discusses the formal specification of the CASA library. CASA is a java library implementing agent societies and communication under the umbrella of social commitments. The second part... more
This paper consists of two parts. The first part discusses the formal specification of the CASA library. CASA is a java library implementing agent societies and communication under the umbrella of social commitments. The second part explores the possibility of describing social commitment based agent communication via polycategories.
This paper consists of two parts. The first part discusses the formal specification of the CASA library. CASA is a java library implementing agent societies and communica- tion under the umbrella of social commitments. The second part... more
This paper consists of two parts. The first part discusses the formal specification of the CASA library. CASA is a java library implementing agent societies and communica- tion under the umbrella of social commitments. The second part explores the possibility of describing social commitment based agent communication via polycategories.
Thesis (M. Sc.)--University of Alberta, 1981. This thesis deals with the problem of describing the unit group of specific group rings over the integers.
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined... more
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano's results, simplifying some proofs along the way. We also point out some corollaries to Matsumoto and Amano's work, including an intrinsic characterization of the Clifford+T subgroup of SO(3), which also yields an efficient T-optimal exact single-qubit synthesis algorithm. Interestingly, this also gives an alternative proof of Kliuchnikov, Maslov, and Mosca's exact synthesis result for the Clifford+T subgroup of U(2).
This paper consists of two parts. The first part discusses the formal specification of the CASA library. CASA is a java library implementing agent societies and communica-tion under the umbrella of social commitments. The second part... more
This paper consists of two parts. The first part discusses the formal specification of the CASA library. CASA is a java library implementing agent societies and communica-tion under the umbrella of social commitments. The second part explores the possibility of ...
Thesis (M. Sc.)--University of Alberta, 1981. This thesis deals with the problem of describing the unit group of specific group rings over the integers.
The categorical semantics of reversible computing must be a category which combines the concepts of partiality and the ability to reverse any map in the category. Inverse categories, restriction categories in which each map is a partial... more
The categorical semantics of reversible computing must be a category which combines the concepts of partiality and the ability to reverse any map in the category. Inverse categories, restriction categories in which each map is a partial isomorphism, provide exactly this structure. This thesis explores inverse categories and relates them to both quantum computing and standard non-reversible computing. The former is achieved by showing that commutative Frobenius algebras form an inverse category. The latter is by establishing the equivalence of the category of discrete inverse categories to the category of discrete Cartesian restriction categories — this is the main result of this thesis. This allows one to transfer the formulation of computability given by Turing categories onto discrete inverse categories.
Research Interests:
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined... more
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano's results, simplifying some proofs along the way. We also point out some corollaries to Matsumoto and Amano's work, including an intrinsic characterization of the Clifford+T subgroup of SO(3), which also yields an efficient T-optimal exact single-qubit synthesis algorithm. Interestingly, this also gives an alternative proof of Kliuchnikov, Maslov, and Mosca's exact synthesis result for the Clifford+T subgroup of U(2).
Research Interests:
This thesis deals with the problem of describing the unit group of specific group rings over the integers. Some results are given in an expository manner as the proofs of the results are normally very dependent on the particular group.... more
This thesis deals with the problem of describing the unit group of specific group rings over the integers. Some results are given in an expository manner as the proofs of the results are normally very dependent on the particular group. The ones presented by this method later on are S3,D4,D6 and A4.
Next, we present the method for groups of order p3, where p is an odd prime.
These come from a paper by Ritter and Sehgal. I consider both non-abelian groups of order p3 and descriptions of the unit groups of both of their respective group rings are presented.I present the method as applied to the groups of order 27.
The last theoretical results are on determining the unit group of the group ring over a group of order pq where p ≡ 1( mod q). These results come from a paper by Luthar [3]. No practical examples are done of this method.
The next part deals with presenting actual groups and determining the unit structure of their integral group ring. The first two, S3 and D4 are from previous authors. The first was done by Hughes and Pearson [2], the second by C. Polcino-
Milies [4]. I also present, D6, which is new result of the thesis.
Next, we present the method for groups of order p3, where p is an odd prime.
These come from a paper by Ritter and Sehgal. I consider both non-abelian groups of order p3 and descriptions of the unit groups of both of their respective group rings are presented.I present the method as applied to the groups of order 27.
The last theoretical results are on determining the unit group of the group ring over a group of order pq where p ≡ 1( mod q). These results come from a paper by Luthar [3]. No practical examples are done of this method.
The next part deals with presenting actual groups and determining the unit structure of their integral group ring. The first two, S3 and D4 are from previous authors. The first was done by Hughes and Pearson [2], the second by C. Polcino-
Milies [4]. I also present, D6, which is new result of the thesis.
This thesis presents the semantics of quantum stacks and a functional quantum programming language, L-QPL. An operational semantics for L-QPL based on quantum stacks in the form of a term logic is developed and used as an interpretation... more
This thesis presents the semantics of quantum stacks and a functional quantum programming language, L-QPL. An operational semantics for L-QPL based on quantum stacks in the form of a term logic is developed and used as an interpretation of quantum circuits. The operational semantics is then extended to handle recursion and algebraic datatypes. Recursion and datatypes are not concepts found in quantum circuits, but both are generally required for modern programming languages.
The language L-QPL is introduced in a discussion and example format. Various example programs using both classical and quantum algorithms are used to illustrate features of the language. Details of the language, including handling of qubits, general data types and classical data are covered.
The quantum stack machine is then presented. Supporting data for operation of the machine are introduced and the transitions induced by the machine’s instructions are given.
The language L-QPL is introduced in a discussion and example format. Various example programs using both classical and quantum algorithms are used to illustrate features of the language. Details of the language, including handling of qubits, general data types and classical data are covered.
The quantum stack machine is then presented. Supporting data for operation of the machine are introduced and the transitions induced by the machine’s instructions are given.
Continuation of previous talk, see below.
This series of two talks will be an introduction to quantum computation, starting with a quick review of the required knowledge of Linear Algebra. Then, I will present quantum computation and discuss how it is different from "standard"... more
This series of two talks will be an introduction to quantum computation, starting with a quick review of the required knowledge of Linear Algebra.
Then, I will present quantum computation and discuss how it is different from "standard" computation. Quantum circuits, which are the most common tool currently used to work with quantum computation and algorithms, will be introduced and the circuits for some common quantum algorithms will be shown. This will be followed by definitions and facts about dagger-categories, which are of interest in the modelling
the semantics of quantum computation.
In the second talk, I will continue with dagger-categories and give an overview of some of the recent research into quantum semantics. The last half will introduce the quantum programming language Linear-QPL, discuss its implementation and conclude with a short demonstration of running a quantum program on the Linear-QPL emulator.
Then, I will present quantum computation and discuss how it is different from "standard" computation. Quantum circuits, which are the most common tool currently used to work with quantum computation and algorithms, will be introduced and the circuits for some common quantum algorithms will be shown. This will be followed by definitions and facts about dagger-categories, which are of interest in the modelling
the semantics of quantum computation.
In the second talk, I will continue with dagger-categories and give an overview of some of the recent research into quantum semantics. The last half will introduce the quantum programming language Linear-QPL, discuss its implementation and conclude with a short demonstration of running a quantum program on the Linear-QPL emulator.
We present the categorical equivalence of reversible and standard computation and begin to explore the links between reversible and quantum computation via the use of Frobenius Algebras.
Description of constructs and programming in L-QPL, a linear Quantum Programming Language