We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away --- we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.

/pdf/quantum-computing-in-the-nisq-era-and-beyond-4jnwt61yqa.pdf

Quantum Computing in the NISQ era and beyond

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of these methods sprung off from the development of geometric functional analysis since the 1970's. They have applications in several fields, most notably in theoretical computer science, statistics and signal processing. A few basic applications are covered in this text, particularly for the problem of estimating covariance matrices in statistics and for validating probabilistic constructions of measurement matrices in compressed sensing. These notes are written particularly for graduate students and beginning researchers in different areas, including functional analysts, probabilists, theoretical statisticians, electrical engineers, and theoretical computer scientists.

Introduction to the non-asymptotic analysis of random matrices.

The problem 2-LOCAL HAMILTONIAN has been shown to be complete for the quantumcomputational class QMA [1]. In this paper we show that this important problemremains QMA-complete when the interactions of the 2-local Hamiltonian are betweenqubits on a two-dimensional (2-D) square lattice. Our results are partially derived withnovel perturbation gadgets that employ mediator qubits which allow us to manipulatek-local interactions. As a side result, we obtain that quantum adiabatic computationusing 2-local interactions restricted to a 2-D square lattice is equivalent to the circuitmodel of quantum computation. Our perturbation method also shows how any stabilizerspace associated with a k-local stabilizer (for constant k) can be generated as anapproximate ground-space of a 2-local Hamiltonian.

The complexity of quantum spin systems on a two-dimensional square lattice

Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is !(lnn). We prove that the adjacency matrix and the Laplacian of that random graph are concentrated around the corresponding matrices of the weighted graph whose edge weights are the probabilities in the random model. We apply this result to two different settings. In bond percolation, we show that, whenever the minimum expected degree in the random model is not too small, the Laplacian of the percolated graph is typically close to that of the original graph. As a corollary, we improve upon a bound for the spectral gap of the percolated graph due to Chung and Horn.

Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges

We study the complexity of the Local Hamiltonian Problem (denoted as LH-MIN) in the special case when a Hamiltonian obeys the condition that all off-diagonal matrix elements in the standard basis are real and non-positive We will call such Hamiltonians, which are common in the natural world, stoquastic An equivalent characterization of stoquastic Hamiltonians is that they have an entry-wise non-negative Gibbs density matrix for any temperature We prove that LH-MIN for stoquastic Hamiltonians belongs to the complexity class AM -- a probabilistic version of NP with two rounds of communication between the prover and the verifier We also show that 2-local stoquastic LH-MIN is hard for the class MA With the additional promise of having a polynomial spectral gap, we show that stoquastic LH-MIN belongs to the class PostBPP=BPPpath--a generalization of BPP in which a post-selective readout is allowed This last result also shows that any problem solved by adiabatic quantum computation using stoquastic Hamiltonians is in PostBPP

The complexity of stoquastic local Hamiltonian problems

We study the complexity of the Local Hamiltonian Problem (denoted as LH-MIN) in the special case when a Hamiltonian obeys conditions of the Perron-Frobenius theorem: all off-diagonal matrix elements in the standard basis are real and non-positive. We will call such Hamiltonians, which are common in the natural world, stoquastic. An equivalent characterization of stoquastic Hamiltonians is that they have an entry-wise non-negative Gibbs density matrix for any temperature. We prove that LH-MIN for stoquastic Hamiltonians belongs to the complexity class AM -- a probabilistic version of NP with two rounds of communication between the prover and the verifier. We also show that 2-local stoquastic LH-MIN is hard for the class MA. With the additional promise of having a polynomial spectral gap, we show that stoquastic LH-MIN belongs to the class POSTBPP=BPPpath -- a generalization of BPP in which a post-selective readout is allowed. This last result also shows that any problem solved by adiabatic quantum computation using stoquastic Hamiltonians lies in PostBPP.

https://arxiv.org/pdf/quant-ph/0606140.pdf

The Complexity of Stoquastic Local Hamiltonian Problems

We give a new, elementary proof of a key inequality used by Rudelson in the derivation of his well-known bound for random sums of rank-one operators. Our approach is based on Ahlswede and Winter's technique for proving operator Chernoff bounds. We also prove a concentration inequality for sums of random matrices of rank one with explicit constants.

/pdf/sums-of-random-hermitian-matrices-and-an-inequality-by-43xnrwxrv6.pdf

Roberto I. Oliveira

Papers

The complexity of quantum spin systems on a two-dimensional square lattice

Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges

The complexity of stoquastic local Hamiltonian problems

The Complexity of Stoquastic Local Hamiltonian Problems

Sums of random Hermitian matrices and an inequality by Rudelson