Much has been written about theory and practice in the law, and the tension between practitioners and theorists. Judges do not cite theoretical articles often; they rarely "apply" theories to particular cases. These arguments are not revisited. Instead the Essay explores the working and interaction of theory and practice, practitioners and theorists. The Essay starts with a story about solving a legal issue using our intellectual tools - theory, practice, and their progenies: experience and "gut." Next the Essay elaborates on the nature of theory, practice, experience and "gut." The third part of the Essay discusses theories that are helpful to practitioners and those that are less helpful. The Essay concludes that practitioners theorize, and theorists practice. They use these intellectual tools differently because the goals and orientations of theorists and practitioners, and the constraints under which they act, differ. Theory, practice, experience and "gut" help us think, remember, decide and create. They complement each other like the two sides of the same coin: distinct but inseparable.

Of Theory and Practice

In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

https://chaos.if.uj.edu.pl/~karol/erratum10.pdf

Geometry of Quantum States

Quantum computers are designed to outperform standard computers by running quantum algorithms. Areas in which quantum algorithms can be applied include cryptography, search and optimisation, simulation of quantum systems and solving large systems of linear equations. Here we briefly survey some known quantum algorithms, with an emphasis on a broad overview of their applications rather than their technical details. We include a discussion of recent developments and near-term applications of quantum algorithms.

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Quantum algorithms: an overview

Property testing is concerned with the design of super-fast algorithms for the structural analysis of large quantities of data. The aim is to unveil global features of the data, such as determining whether the data has a particular property or estimating global parameters. Remarkably, it is possible for decisions to be made by accessing only a small portion of the data. Property testing focuses on properties and parameters that go beyond simple statistics. This book provides an extensive and authoritative introduction to property testing. It provides a wide range of algorithmic techniques for the design and analysis of tests for algebraic properties, properties of Boolean functions, graph properties, and properties of distributions.

Introduction to Property Testing

In this note, we develop a bounded-error quantum algorithm that makes ˜ O(n 1=4 e 1=2 ) queries to a function f :f0; 1g n !f0; 1g, accepts when f is monotone, and rejects when f is e-far from being monotone. This result gives a super-quadratic improve- ment compared to the best known randomized algorithm for all e = o(1). The improvement is cubic when e = 1= p n.

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Quantum Algorithm for Monotonicity Testing on the Hypercube

In this paper, we study the following variant of the junta learning problem. We are given oracle access to a Boolean function f on n variables that only depends on k variables, and, when restricted to them, equals some predefined function h. The task is to identify the variables the function depends on.When h is the XOR or the OR function, this gives a restricted variant of the Bernstein---Vazirani or the combinatorial group testing problem, respectively.

We analyze the general case using the adversary bound and give an alternative formulation for the quantum query complexity of this problem. We construct optimal quantum query algorithms for the cases when h is the OR function (complexity is $${\Theta(\sqrt{k})}$$?(k)) or the exact-half function (complexity is $${\Theta(k^{1/4})}$$?(k1/4)). The first algorithm resolves an open problem from Ambainis & Montanaro (Quantum Inf Comput 14(5&6): 439---453, 2014). For the case when h is the majority function, we prove an upper bound of $${O(k^{1/4})}$$O(k1/4). All these algorithms can be made exact. We obtain a quartic improvement when compared to the randomized complexity (if h is the exact-half or the majority function), and a quadratic one when compared to the non-adaptive quantum complexity (for all functions considered in the paper).

Quantum Algorithms for Learning Symmetric Juntas via the Adversary Bound

In the $k$-junta testing problem, a tester has to efficiently decide whether a given function $f:\{0,1\}^n\rightarrow \{0,1\}$ is a $k$-junta (i.e., depends on at most $k$ of its input bits) or is $\epsilon$-far from any $k$-junta. Our main result is a quantum algorithm for this problem with query complexity $\tilde O(\sqrt{k/\epsilon})$ and time complexity $\tilde O(n\sqrt{k/\epsilon})$. This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atici and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of $\Omega(k^{1/3})$ queries for junta-testing (for constant $\epsilon$).

Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

The (negative-weighted) quantum adversary bound is a tight characterisation of the quantum query complexity for any partial function [28, 38]. We analyse the extent to which this bound can be generalised. Ambainis et al. [3] and Lee et al. [34] generalised this bound to the state generation and state conversion problems, respectively. Using the ideas by Lee et al., we get even further generalisations of the bound. We obtain a version of the bound for general input oracles, which are just arbitrary unitaries. We also generalise the bound to the problem of implementing arbitrary unitary transformations. Similarly to the bound by Lee et al., our bound is a lower bound for exact transformation and an upper bound for approximate transformation. This version of the bound possesses the tight composition property. Using this construction, we also obtain lower bounds on the quantum query complexity of functions and relations with general input oracles.

Variations on Quantum Adversary

We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of the disposition of possible certificates in the input string, not the precise values therein. 
Next, we derive a dual formulation of the complexity of a non-adaptive learning graph, and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure and such that a learning graph gives an optimal quantum query algorithm for it. 
For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays, and generalizes the quantum query lower bound for the $k$-sum problem derived recently in arXiv:1206.6528. 
Finally, we use these results to show that the learning graph for the triangle problem from arXiv:1210.1014 is almost optimal in these settings. This also gives a quantum query lower bound for the triangle-sum problem.

Aleksandrs Belovs

Papers

Quantum Algorithm for Monotonicity Testing on the Hypercube

Quantum Algorithms for Learning Symmetric Juntas via the Adversary Bound

Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

Variations on Quantum Adversary

On the Power of Non-Adaptive Learning Graphs