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 Judges do not cite theoretical articles often; they rarely "apply" theories to particular cases.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 lower bounds for the following variant of the counting problem considered by Aaronson et al. The task is to distinguish whether an input set $x\subseteq [n]$ has size either $k$ or $k'=(1+\epsilon)k$. We assume the algorithm has access to 
* the membership oracle, which, for each $i\in [n]$, can answer whether $i\in x$, or not; and 
* the uniform superposition $|\psi_x\rangle = \sum_{i\in x} |i\rangle/\sqrt{|x|}$ over the elements of $x$. Moreover, we consider three different ways how the algorithm can access this state: 
** the algorithm can have copies of the state $|\psi_x\rangle$; 
** the algorithm can execute the reflecting oracle which reflects about the state $|\psi_x\rangle$; 
** the algorithm can execute the state-generating oracle (or its inverse) which performs the transformation $|0\rangle\mapsto |\psi_x\rangle$. 
Without the second type of resources (related to $|\psi_x\rangle$), the problem is well-understood, see Brassard et al. The study of the problem with the second type of resources was recently initiated by Aaronson et al. 
We completely resolve the problem for all values of $1/k \le \epsilon\le 1$, giving tight trade-offs between all types of resources available to the algorithm. Thus, we close the main open problems from Aaronson et al. 
The lower bounds are proven using variants of the adversary bound by Belovs and employing analysis closely related to the Johnson association scheme.

Tight Quantum Lower Bound for Approximate Counting with Quantum States

The complexity of infinite words is considered from the point of view of a transformation with a Mealy machine that is the simplest model of a finite automaton transducer. We are mostly interested in algebraic properties of the underlying partially ordered set. Results considered with the existence of supremum, infimum, antichains, chains and density aspects are investigated.

/pdf/some-algebraic-properties-of-machine-poset-of-infinite-words-y381w4w7kq.pdf

Some algebraic properties of machine poset of infinite words

We study space and time efficient quantum algorithms for two graph problems -- deciding whether an $n$-vertex graph is a forest, and whether it is bipartite. Via a reduction to the s-t connectivity problem, we describe quantum algorithms for deciding both properties in $\tilde{O}(n^{3/2})$ time and using $O(\log n)$ classical and quantum bits of storage in the adjacency matrix model. We then present quantum algorithms for deciding the two properties in the adjacency array model, which run in time $\tilde{O}(n\sqrt{d_m})$ and also require $O(\log n)$ space, where $d_m$ is the maximum degree of any vertex in the input graph.

Time and Space Efficient Quantum Algorithms for Detecting Cycles and Testing Bipartiteness

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.

Quantum lower bound for inverting a permutation with advice.

We study how efficiently a k-element set S⊆[n] can be learned from a uniform superposition |S> of its elements. One can think of |S>=∑_{i∈S}|i>/√|S| as the quantum version of a uniformly random sample over S, as in the classical analysis of the "coupon collector problem." We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n-k=O(1) missing elements then O(k) copies of |S> suffice, in contrast to the Θ(k log k) random samples needed by a classical coupon collector. On the other hand, if n-k=Ω(k), then Ω(k log k) quantum samples are necessary.
More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through |S>. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.

https://ir.cwi.nl/pub/30084/LIPIcs-TQC-2020-10.pdf

Aleksandrs Belovs

Papers

Tight Quantum Lower Bound for Approximate Counting with Quantum States

Some algebraic properties of machine poset of infinite words

Time and Space Efficient Quantum Algorithms for Detecting Cycles and Testing Bipartiteness

Quantum lower bound for inverting a permutation with advice.

Quantum Coupon Collector