Unitary property testing lower bounds by polynomials

We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between $\mathsf{QMA}$ and $\mathsf{QMA(2)}$, a long standing question in quantum complexity theory.Comment: 58 pages, v2: typos corrected, Section 6.1-6.3 revised, added some new result

Unitary Property Testing Lower Bounds by Polynomials

We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between QMA and QMA(2), a long standing question in quantum complexity theory

On the Algebraic Proof Complexity of Tensor Isomorphism

The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz & Grohe, SODA \u2717). Our main results are an ?(n) lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank. We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices (which is essentially the same as 2-TI), or deriving BA = I from AB = I. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication AB = I ? BA = I. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI

Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn\u27t

Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Stanley, 1999], in Schubert calculus [Ledoux and Malham, 2010] as well as in Enumerative combinatorics [Gasharov, 1996; Stanley, 1984; Stanley, 1999]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [Hallgren et al., 2000; Ryan O\u27Donnell and John Wright, 2015] and Geometric complexity theory [Ikenmeyer and Panova, 2017]. However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational complexity of syntactically computing Schur polynomials has not been studied much. In particular, it is not known whether Schur polynomials can be computed efficiently by algebraic formulas. In this work, we address this question, and show that unless every polynomial with a small algebraic branching program (ABP) has a small algebraic formula, there are Schur polynomials that cannot be computed by algebraic formula of polynomial size. In other words, unless the algebraic complexity class VBP is equal to the complexity class VF, there exist Schur polynomials which do not have polynomial size algebraic formulas. As a consequence of our proof, we also show that computing the determinant of certain generalized Vandermonde matrices is essentially as hard as computing the general symbolic determinant. To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not multilinear. A key ingredient of our proof is the study of composition of well behaved algebraically independent polynomials with a homogeneous polynomial, and might be of independent interest

Molecular epidemiology, drug susceptibility and economic aspects of tuberculosis in mubende district, Uganda

<div><p>Background</p><p>Tuberculosis (TB) remains a global public health problem whose effects have major impact in developing countries like Uganda. This study aimed at investigating genotypic characteristics and drug resistance profiles of <i>Mycobacterium tuberculosis</i> isolated from suspected TB patients. Furthermore, risk factors and economic burdens that could affect the current control strategies were studied.</p><p>Methods</p><p>TB suspected patients were examined in a cross-sectional study at the Mubende regional referral hospital between February and July 2011. A questionnaire was administered to each patient to obtain information associated with TB prevalence. Isolates of <i>M. tuberculosis</i> recovered during sampling were examined for drug resistance to first line anti-TB drugs using the BACTEC-MGIT960^TMsystem. All isolates were further characterized using deletion analysis, spoligotyping and MIRU-VNTR analysis. Data were analyzed using different software; MIRU-VNTR <i>plus</i>, SITVITWEB, BioNumerics and multivariable regression models.</p><p>Results</p><p><i>M. tuberculosis</i> was isolated from 74 out of 344 patients, 48 of these were co-infected with HIV. Results from the questionnaire showed that previously treated TB, co-infection with HIV, cigarette smoking, and overcrowding were risk factors associated with TB, while high medical related transport bills were identified as an economic burden. Out of the 67 isolates that gave interpretable results, 23 different spoligopatterns were detected, nine of which were novel patterns. T2 with the sub types Uganda-I and Uganda-II was the most predominant lineage detected. Antibiotic resistance was detected in 19% and multidrug resistance was detected in 3% of the isolates.</p><p>Conclusion</p><p>The study detected <i>M. tuberculosis</i> from 21% of examined TB patients, 62% of whom were also HIV positive. There is a heterogeneous pool of genotypes that circulate in this area, with the T2 lineage being the most predominant. High medical related transport bills and drug resistance could undermine the usefulness of the current TB strategic interventions.</p></div