Erasure-Resilient Sublinear-Time Graph Algorithms

We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or ε-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than ε, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least ε, then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter ε. For estimating the average degree, our results provide an “interpolation” between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. ‘06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms ‘08) and Eden et al. (ICALP ‘17). We conclude with a discussion of our model and open questions raised by our work.

Querying a Matrix through Matrix-Vector Products

We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products , namely, the algorithm chooses vectors v 1 , ⃛ , v q , and observes Mv 1 , ⃛ , Mv q . Here the v i can be randomized as well as chosen adaptively as a function of Mv 1 , ⃛ , Mv i-1 . Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.

Property Testing on Graphs and Games

Constant-time algorithms are powerful tools, since they run by reading only a constant-sized part of each input. Property testing is the most popular research framework for constant-time algorithms. In property testing, an algorithm determines whether a given instance satisfies some predetermined property or is far from satisfying the property with high probability by reading a constant-sized part of the input. A property is said to be testable if there is a constant-time testing algorithm for the property. This chapter covers property testing on graphs and games. The fields of graph algorithms and property testing are two of the main streams of research on discrete algorithms and computational complexity. In the section on graphs in this chapter, we present some important results, particularly on the characterization of testable graph properties. At the end of the section, we show results that we published in 2020 on a complete characterization (necessary and sufficient condition) of testable monotone or hereditary properties in the bounded-degree digraphs. In the section on games, we present results that we published in 2019 showing that the generalized chess, Shogi (Japanese chess), and Xiangqi (Chinese chess) are all testable. We believe that this is the first results for testable EXPTIME-complete problems.

Inline Non-Destructive Material Property Testing; The Future of Manufacturing

Across all industries, material specifications are tightening beyond previously understood process capabilities. Slight shifts in material grade, microstructure, heat treatment, or alloy composition can significantly impact long term material integrity. This study examines the feasibility of non-contact, 100% inline magneto-inductive testing on material/components destined for the automotive, aerospace, agricultural, and medical markets to ensure proper material quality standards. To test the hypothesis that material grade, carbon content, density, and alloy composition can be accurately tested in real time during production, an experiment was conducted utilizing magneto-inductive test instrumentation and encircling coil. Throughout this experiment, and proposed future state of manufacturing, 100% of material was tested. Results yielded clear confirmation in accordance with the hypothesis. This data driven subjective approach provided the ability to accurately, efficiently, and autonomously verify proper material grade had been used for the designated product. Ensuring proper material composition and material properties without slowing production using this testing method should be considered when improved quality is desired.

Schur–Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers $$U^{\otimes t}$$ U ⊗ t of all unitaries $$U\in U(d)$$ U ∈ U ( d ) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group. We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well). We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem. We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

Guest Column

Communication complexity studies the amount of communication necessary to compute a function whose value depends on information distributed among several entities. Yao [Yao79] initiated the study of communication complexity more than 40 years ago, and it has since become a central eld in theoretical computer science with many applications in diverse areas such as data structures, streaming algorithms, property testing, approximation algorithms, coding theory, and machine learning. The textbooks [KN06,RY20] provide excellent overviews of the theory and its applications.