Approximate Span Programs

Span programs are a model of computation that have been used to design quantum algorithms, mainly in the query model. For any decision problem, there exists a span program that leads to an algorithm with optimal quantum query complexity, but finding such an algorithm is generally challenging. We consider new ways of designing quantum algorithms using span programs. We show how any span program that decides a problem $f$ can also be used to decide "property testing" versions of $f$, or more generally, approximate the span program witness size, a property of the input related to $f$. For example, using our techniques, the span program for OR, which can be used to design an optimal algorithm for the OR function, can also be used to design optimal algorithms for: threshold functions, in which we want to decide if the Hamming weight of a string is above a threshold or far below, given the promise that one of these is true; and approximate counting, in which we want to estimate the Hamming weight of the input. We achieve these results by relaxing the requirement that 1-inputs hit some target exactly in the span program, which could make design of span programs easier. We also give an exposition of span program structure, which increases the understanding of this important model. One implication is alternative algorithms for estimating the witness size when the phase gap of a certain unitary can be lower bounded. We show how to lower bound this phase gap in some cases. As applications, we give the first upper bounds in the adjacency query model on the quantum time complexity of estimating the effective resistance between $s$ and $t$, $R_{s,t}(G)$, of $\tilde O(\frac{1}{\epsilon^{3/2}}n\sqrt{R_{s,t}(G)})$, and, when $\mu$ is a lower bound on $\lambda_2(G)$, by our phase gap lower bound, we can obtain $\tilde O(\frac{1}{\epsilon}n\sqrt{R_{s,t}(G)/\mu})$, both using $O(\log n)$ space

Lower Bounds for Monotone Span Programs

The model of span programs is a linear algebraic model of computation. Lower bounds for span programs imply lower bounds for contact schemes, symmetric branching programs and for formula size. Monotone span programs correspond also to linear secret-sharing schemes. We present a new technique for proving lower bounds for monotone span programs. The main result proved here yields quadratic lower bounds for the size of monotone span programs, improving on the largest previously known bounds for explicit functions. The bound is asymptotically tight for the function corresponding to a class of 4-cliques

Effect of tail-fin span on stability and control characteristics of a Canard-controlled missile at supersonic Mach numbers

An experimental wind-tunnel investigation was conducted at Mach numbers from 1.60 to 3.50 to obtain the longitudinal and lateral-directional aerodynamic characteristics of a circular, cruciform, canard-controlled missile with variations in tail-fin span. In addition, comparisons were made with the experimental aerodynamic characteristics using three missile aeroprediction programs: MISSILE1, MISSILE2, and NSWCDM. The results of the investigation indicate that for the test Mach number range, canard roll control at low angles of attack is feasible on tail-fin configurations with tail-to-canard span ratios of less than or equal to 0.75. The conards are effective pitch and yaw control devices on each tail-fin span configuration tested. Programs MISSILE1 and MISSILE2 provide very good predictions of longitudinal aerodynamic characteristics and fair predictions of lateral-directional aerodynamic characteristics at low angles of attack, with MISSILE2 predictions generally in better agreement with test data. Program NSWCDM provides good longitudinal and lateral-directional aerodynamic predictions that improve with increases in tail-tin span

Span Programs and Quantum Space Complexity

While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of space-bounded quantum computation. We relate the quantum space complexity of computing a function f with one-sided error to the logarithm of its span program size, a classical quantity that is well-studied in attempts to prove formula size lower bounds. In the more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (two-sided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program size. While no non-trivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first non-trivial lower bound on the approximate span program size of an explicit function

Refactorings of Design Defects using Relational Concept Analysis

Software engineers often need to identify and correct design defects, ıe} recurring design problems that hinder development and maintenance\ud by making programs harder to comprehend and--or evolve. While detection\ud of design defects is an actively researched area, their correction---mainly\ud a manual and time-consuming activity --- is yet to be extensively\ud investigated for automation. In this paper, we propose an automated\ud approach for suggesting defect-correcting refactorings using relational\ud concept analysis (RCA). The added value of RCA consists in exploiting\ud the links between formal objects which abound in a software re-engineering\ud context. We validated our approach on instances of the <span class='textit'></span>Blob\ud design defect taken from four different open-source programs

Span programs and quantum query complexity: The general adversary bound is nearly tight for every boolean function

The general adversary bound is a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a logarithmic factor. In more detail, the proof has two steps, each based on "span programs," a certain linear-algebraic model of computation. First, we give an SDP that outputs for any boolean function a span program computing it that has optimal "witness size." The optimal witness size is shown to coincide with the general adversary lower bound. Second, we give a quantum algorithm for evaluating span programs with only a logarithmic query overhead on the witness size. The first result is motivated by a quantum algorithm for evaluating composed span programs. The algorithm is known to be optimal for evaluating a large class of formulas. The allowed gates include all constant-size functions for which there is an optimal span program. So far, good span programs have been found in an ad hoc manner, and the SDP automates this procedure. Surprisingly, the SDP's value equals the general adversary bound. A corollary is an optimal quantum algorithm for evaluating "balanced" formulas over any finite boolean gate set. The second result extends span programs' applicability beyond the formula evaluation problem. A strong universality result for span programs follows. A good quantum query algorithm for a problem implies a good span program, and vice versa. Although nearly tight, this equivalence is nontrivial. Span programs are a promising model for developing more quantum algorithms.Comment: 70 pages, 2 figure

High Energy Cosmic Neutrinos

While the general principles of high-energy neutrino detection have been understood for many years, the deep, remote geographical locations of suitable detector sites have challenged the ingenuity of experimentalists, who have confronted unusual deployment, calibration, and robustness issues. Two high energy neutrino programs are now operating (Baikal and AMANDA), with the expectation of ushering in an era of multi-messenger astronomy, and two Mediterranean programs have made impressive progress. The detectors are optimized to detect neutrinos with energies of the order of 1-10 TeV, although they are capable of detecting neutrinos with energies of tens of MeV to greater than PeV. This paper outlines the interdisciplinary scientific agenda, which span the fields of astronomy, particle physics, and cosmic ray physics, and describes ongoing worldwide experimental programs to realize these goals.Comment: 15 pages, 9 figures, talk presented at the Nobel Symposium on Particle Physics and the Universe, Sweden, August 199