* Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index

An introduction to classical real analysis, by Karl R. Stromberg. Pp 575. $29·95. 1981. ISBN 0-534-98012-0 (Wadsworth)

Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive description of basic lower bound arguments, covering many of the gems of this complexity Waterloo that have been discovered over the past several decades, right up to results from the last year or two. Many open problems, marked as Research Problems, are mentioned along the way. The problems are mainly of combinatorial flavor but their solutions could have great consequences in circuit complexity and computer science. The book will be of interest to graduate students and researchers in the fields of computer science and discrete mathematics.

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Boolean Function Complexity: Advances and Frontiers

From the Publisher:
Your book fills the gap which all of us felt existed too long. Congratulations on this excellent contribution to our field."
--Jan van Leeuwen, Utrecht University 

"This is an impressive book. The subject has been thoroughly researched and carefully presented. All the machine models central to the modern theory of computation are covered in depth; many for the first time in textbook form. Readers will learn a great deal from the wealth of interesting material presented."
--Andrew C. Yao, Professor of Computer Science, Princeton University

"Models of Computation" is an excellent new book that thoroughly covers the theory of computation including significant recent material and presents it all with insightful new approaches. This long-awaited book will serve as a milestone for the theory community."
--Akira Maruoka, Professor of Information Sciences, Tohoku University

"This is computer science."
--Elliot Winard, Student, Brown University

In Models of Computation: Exploring the Power of Computing, John Savage re-examines theoretical computer science, offering a fresh approach that gives priority to resource tradeoffs and complexity classifications over the structure of machines and their relationships to languages. This viewpoint reflects a pedagogy motivated by the growing importance of computational models that are more realistic than the abstract ones studied in the 1950s, '60s and early '70s. 

Assuming onlysome background in computer organization, Models of Computation uses circuits to simulate machines with memory, thereby making possible an early discussion of P-complete and NP-complete problems. Circuits are also used to demonstrate that tradeoffs between parameters of computation, such as space and time, regulate all computations by machines with memory. Full coverage of formal languages and automata is included along with a substantive treatment of computability. Topics such as space-time tradeoffs, memory hierarchies, parallel computation, and circuit complexity, are integrated throughout the text with an emphasis on finite problems and concrete computational models 
FEATURES:
Includes introductory material for a first course on theoretical computer science. Builds on computer organization to provide an early introduction to P-complete and NP-complete problems. Includes a concise, modern presentation of regular, context-free and phrase-structure grammars, parsing, finite automata, pushdown automata, and computability. Includes an extensive, modern coverage of complexity classes. Provides an introduction to the advanced topics of space-time tradeoffs, memory hierarchies, parallel computation, the VLSI model, and circuit complexity, with parallelism integrated throughout. Contains over 200 figures and over 400 exercises along with an extensive bibliography. 

** Instructor's materials are available from your sales rep. If you do not know your local sales representative, please call 1-800-552-2499 for assistance, or use the Addison Wesley Longman rep-locator at ...

Models of Computation: Exploring the Power of Computing

The spin states of single electrons in gate-defined quantum dots satisfy crucial requirements for a practical quantum computer. These include extremely long coherence times, high-fidelity quantum operation, and the ability to shuttle electrons as a mechanism for on-chip flying qubits. To increase the number of qubits to the thousands or millions of qubits needed for practical quantum information, we present an architecture based on shared control and a scalable number of lines. Crucially, the control lines define the qubit grid, such that no local components are required. Our design enables qubit coupling beyond nearest neighbors, providing prospects for nonplanar quantum error correction protocols. Fabrication is based on a three-layer design to define qubit and tunnel barrier gates. We show that a double stripline on top of the structure can drive high-fidelity single-qubit rotations. Self-aligned inhomogeneous magnetic fields induced by direct currents through superconducting gates enable qubit addressability and readout. Qubit coupling is based on the exchange interaction, and we show that parallel two-qubit gates can be performed at the detuning-noise insensitive point. While the architecture requires a high level of uniformity in the materials and critical dimensions to enable shared control, it stands out for its simplicity and provides prospects for large-scale quantum computation in the near future.

https://advances.sciencemag.org/content/advances/4/7/eaar3960.full.pdf

A crossbar network for silicon quantum dot qubits.

The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome.

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Achieving quantum supremacy with sparse and noisy commuting quantum computations

We provide algorithms for efficiently moving and addressing quantum memory in parallel. These imply that the standard circuit model can be simulated with a low overhead by a more realistic model of a distributed quantum computer. As a result, the circuit model can be used by algorithm designers without worrying whether the underlying architecture supports the connectivity of the circuit. In addition, we apply our results to existing memory-intensive quantum algorithms. We present a parallel quantum search algorithm and improve the time–space trade-off for the element distinctness and collision finding problems.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2012.0686

Efficient distributed quantum computing

We consider the solvability of the equation and generalizations, where the A{sub i} and B are given commuting matrices over an algebraic number field F. In the semigroup membership problem, the variables x{sub i} are constrained to be nonnegative integers. While this problem is NP-complete for variable k, we give a polynomial time algorithm if k is fixed. In the group membership problem, the matrices are assumed to be invertible, and the variables x{sub i} may take on negative values. In this case we give a polynomial time algorithm for variable k and give an explicit description of the set of all solutions (as an affine lattice). The special case of 1 x 1 matrices was recently solved by Guoqiang Ge; we heavily rely on his results.

Multiplicative equations over commuting matrices

We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic problems (factoring and discrete log); in fact, constructive membership testing in the case of 1 × 1 matrices is precisely the discrete log problem. So the reasonable question is whether these problems are solvable in randomized polynomial time using number theory oracles. Building on 25 years of work, including remarkable recent developments by several groups of authors, we are now able to determine the order of a matrix group over a finite field of odd characteristic, and to perform constructive membership testing in such groups, in randomized polynomial time, using oracles for factoring and discrete log. One of the new ingredients of this result is the following. A group is called semisimple if it has no abelian normal subgroups. For matrix groups over finite fields, we show that the order of the largest semisimple quotient can be determined in randomized polynomial time (no number theory oracles required and no restriction on parity). As a by-product, we obtain a natural problem that belongs to BPP and is not known to belong either to RP or to coRP. No such problem outside the area of matrix groups appears to be known. The problem is the decision version of the above: Given a list A of nonsingular d × d matrices over a finite field and an integer N, does the group generated by A have a semisimple quotient of order > N? We also make progress in the area of constructive recognition of simple groups, with the corollary that for a large class of matrix groups, our algorithms become Las Vegas.

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Polynomial-time theory of matrix groups

Let G be a group of matrices with entries over an algebraic number field F (given symbolically). The group G is given by a list of generators. In this paper we give several algorithms, both deterministic and randomized, which can decide in polynomial time whether or not G is jinzte. It is easy to reduce the problem to the case F = Q. As a next step, we present a polynomial time algorithm which transforms G into a group of integral matrices whenever possible. Having done so, the main results of the paper are several polynomial time algorithms to handle the case of integral matrices. We give both randomized and deterministic algorithms to decide finiteness for finitely generated integral matrix groups. Although we are able to prove much better upper bounds for the complexity of the deterministic algorithms, in practice, the randomized algorithms support a much more efficient implementation. Thus, both kinds of algorithms are presented here but only the implementation of the randomized algorithm is explored. All algorithms in some way depend on a characterization (effectively due to Burnside) c)f finite integral matrix groups as those subgroups G s GL(n, Z) which preserve some positive definite quadratic form. The randomized (Monte Carlo) algorithms use recent random walk techniques to obtain nearly uniformly 1Department of Computer Science, Universilt y of Chicago, 1100 E 5t3th St, Chicago, IL 60637 2Eot vbs University, Budapest, Hungary 3Department of Mathematics and Computer Science, Dartmouth College, Hanover, NH 03755 4Partially supported by NSF Grant CCR 9014562 and OTKA Grant 2581. 5Partially supported by NSF Mathematical Sciences Postdoctoral Fellowship. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of tha publication and ita date appear, and notice is given that copying ia by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires s fee and/or specific permission. ACM-ISSAC ‘93-7/93/Kiev, Ukraine 01993 ACM 0-89791-604-219310007101 17...$1.50 distributed random elements in G (in a precise sense, whether G is finite or not). The subsequent analysis rests on a new estimate of the norm of the matrices in finite groups of integral matrices in terms of the norms of the generators. If G is infinite, one of our Monte Carlo algorithms is likely to produce an element with larger norm, thus certifying infiniteness. If G is finite, our other Monte Carlo algorithm is likely to produce a positive definite G-invariant quadratic form, thus certifying finiteness. Consequently, the combination of these two give an eficient and very simple Las Vegas algorithm (randomized algorithm that never errs) which we have successfully implemented. The deterministic algorithms are of two kinds. One algorithm uses some basic facts from the representation theory of finite groups to construct a G-invariant quadratic form (if it exists), by efficiently averaging the matrices A~A. The other deterministic algorithm uses the ellipsoid method to search for a positive definite quadratic form in a linear space U of quadratic forms. The new estimate on the norm of elements of G is used to ensure that if G is finite, then 1( contains a sizable ball of positive definite forms, as required for the ellipsoid method to be applicable.

Deciding finiteness of matrix groups in deterministic polynomial time

We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when the degree n of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of n is not known in advance. As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of S n : the conditional probability that a random element σ ∈ S n is an n-cycle, given that σ n = 1, is at least 1/10.

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Robert Beals

Papers

Efficient distributed quantum computing

Multiplicative equations over commuting matrices

Polynomial-time theory of matrix groups

Deciding finiteness of matrix groups in deterministic polynomial time

A black-box group algorithm for recognizing finite symmetric and alternating groups, I