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 Judges do not cite theoretical articles often; they rarely "apply" theories to particular cases.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.

/pdf/quantum-algorithms-an-overview-3mbngnqis4.pdf

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

We prove tight $\Omega(n^{1/3})$ lower bounds on the quantum query complexity of the Collision and the Set Equality problems, provided that the size of the alphabet is large enough. We do this using the negative-weight adversary method. Thus, we reprove the result by Aaronson and Shi, as well as a more recent development by Zhandry.

Adversary Lower Bounds for the Collision and the Set Equality Problems

In this thesis we investigate complete systems of MUBs and SIC-POVMs These are highly symmetric sets of vectors in Hilbert space, interesting because of their applications in quantum tomography, quantum cryptography and other areas It is known that these objects form complex projective 2-designs, that is, they satisfy Welch bounds for k = 2 with equality Using this fact, we derive a necessary and sufficient condition for a set of vectors to be a complete system of MUBs or a SIC-POVM This condition uses the orthonormality of a specific set of vectors Then we define homogeneous systems, as a special case of systems of vectors for which the condition takes an especially elegant form We show how known results and some new results naturally follow from this construction

https://uwspace.uwaterloo.ca/bitstream/handle/10012/4159/ABelovs.pdf;sequence=1

Welch Bounds and Quantum State Tomography

In this note we construct an explicit optimal (negative-weight) adversary matrix for the element distinctness problem, given that the size of the alphabet is sufficiently large.

Adversary Lower Bound for Element Distinctness

The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems. Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combinatorial Nullstellensatz and to the Chevalley-Warning Theorem over the two elements field GF(2). The input of these problems contain PPA-circuits which are arithmetic circuits with special symmetric properties that assure that the polynomials computed by them have always an even number of zeros. In the proof of the result we relate the multilinear degree of the polynomials to the parity of the maximal parse subcircuits that compute monomials with maximal multilinear degree, and we show that the maximal parse subcircuits of a PPA-circuit can be paired in polynomial time.

On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz

Given a random permutation f : [N] → [N] as a black box and y e [N], we want to output x = f-1(y). Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but not on the input y. Classically, there is a data structure of size O(S) and an algorithm that with the help of the data structure, given f(x), can invert f in time O(T), for every choice of parameters S, T, such that S ċ T ≥ N. We prove a quantum lower bound of T2 ċ S = Ω(eN) for quantum algorithms that invert a random permutation f on an e fraction of inputs, where T is the number of queries to f and S is the amount of advice. This answers an open question of De et al.

We also give a Ω(√N/m) quantum lower bound for the simpler but related Yao's box problem, which is the problem of recovering a bit xj, given the ability to query an N-bit string x at any index except the j-th, and also given m bits of classical advice that depend on x but not on j.

Aleksandrs Belovs

Papers

Adversary Lower Bounds for the Collision and the Set Equality Problems

Welch Bounds and Quantum State Tomography

Adversary Lower Bound for Element Distinctness

On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz

Quantum lower bound for inverting a permutation with advice