Improved Soundness for QMA with Multiple Provers

We present three contributions to the understanding of QMA with multiple provers: 1) We give a tight soundness analysis of the protocol of [Blier and Tapp, ICQNM '09], yielding a soundness gap Omega(1/N^2). Our improvement is achieved without the use of an instance with a constant soundness gap (i.e., without using a PCP). 2) We give a tight soundness analysis of the protocol of [Chen and Drucker, ArXiV '10], thereby improving their result from a monolithic protocol where Theta(sqrt(N)) provers are needed in order to have any soundness gap, to a protocol with a smooth trade-off between the number of provers k and a soundness gap Omega(k^2/N), as long as k>=Omega(log N). (And, when k=Theta(sqrt(N)), we recover the original parameters of Chen and Drucker.) 3) We make progress towards an open question of [Aaronson et al., ToC '09] about what kinds of NP-complete problems are amenable to sublinear multiple-prover QMA protocols, by observing that a large class of such examples can easily be derived from results already in the PCP literature - namely, at least the languages recognized by a non-deterministic RAMs in quasilinear time.Comment: 24 pages; comments welcom

Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity

We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-$3$ and depth-$4$ arithmetic circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-$3$ arithmetic circuits for a polynomial in $VNP$. 2. Exponential lower bounds for homogeneous depth-$4$ arithmetic circuits with bounded individual degree for a polynomial in $VNP$. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-$4$ arithmetic circuits for the Permanent imply a separation between ${\#}P$ and $ACC$. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean circuit complexity. Besides, it is known from a recent result of Kumar and Saptharishi [KS15] that over constant sized finite fields, strong enough average case functional lower bounds for homogeneous depth-$4$ circuits imply superpolynomial lower bounds for homogeneous depth-$5$ circuits. Our proofs are based on a family of new complexity measures called shifted evaluation dimension, and might be of independent interest

Extending the baseline: Spitzer Mid-Infrared Photometry of Globular Cluster Systems in the Centaurus A and Sombrero Galaxies

Spitzer IRAC mid-infrared photometry is presented for the globular cluster (GC) systems of the NGC 5128 ("Centaurus A") and NGC 4594 ("Sombrero") galaxies. Existing optical photometric and spectroscopic are combined with this new data in a comprehensive optical to mid-IR colour catalogue of 260 GCs. Empirical colour-metallicity relationships are derived for all optical to mid-IR colour combinations. These colours prove to be very effective quantities to test the photometric predictions of simple stellar population (SSP) models. In general, four SSP models show larger discrepancies between each other and the data at bluer wavelengths, especially at high metallicities. Such differences become very important when attempting to use colour-colour model predictions to constrain the ages of stellar populations. Furthermore, the age-substructure determined from colour-colour diagrams and 91 NGC 5128 GCs with spectroscopic ages from Beasley et al. (2008) are inconsistent, suggesting any apparent GC system age-substructure implied by a colour-colour analysis must be verified independently. Unlike blue wavebands, certain optical to mid-IR colours are insensitive to the flux from hot horizontal branch stars and thus provide an excellent metallicity proxy. The NGC 5128 GC system shows strong bimodality in the optical R-band to mid-IR colour distributions, hence proving it is bimodal in metallicity. In this new colour space, a colour-magnitude trend, a "blue tilt", is found in the NGC 5128 metal-poor GC data. The NGC 5128 young GCs do not contribute to this trend. [abridged]Comment: 16 pages, 12 colour figures. To be published in MNRAS. Catalogue available from the first author. Full resolution copy available here http://lee.spitler.googlepages.com/spitzer_spitler.pd

Resonantly Interacting Fermions In a Box

We use two fundamental theoretical frameworks to study the finite-size (shell) properties of the unitary gas in a periodic box: 1) an ab initio Quantum Monte Carlo (QMC) calculation for boxes containing 4 to 130 particles provides a precise and complete characterization of the finite-size behavior, and 2) a new Density Functional Theory (DFT) fully encapsulates these effects. The DFT predicts vanishing shell structure for systems comprising more than 50 particles, and allows us to extrapolate the QMC results to the thermodynamic limit, providing the tightest bound to date on the ground-state energy of the unitary gas: \xi_S <= 0.383(1). We also apply the new functional to few-particle harmonically trapped systems, comparing with previous calculations.Comment: Updated to correspond with published version: 4+ pages, 2 figures, 2 tables, Palatino and Euler font

UBRI Photometry of Globular Clusters in the Leo Group Galaxy NGC 3379

We present wide area UBRI photometry for globular clusters around the Leo group galaxy NGC 3379. Globular cluster candidates are selected from their B-band magnitudes and their (U-B)o vs (B-I)o colours. A colour-colour selection region was defined from photometry of the Milky Way and M31 globular cluster systems. We detect 133 globular cluster candidates which, supports previous claims of a low specific frequency for NGC 3379. The Milky Way and M31 reveal blue and red subpopulations, with (U-B)o and (B-I)o colours indicating mean metallicities similar to those expected based on previous spectroscopic work. The stellar population models of Maraston (2003) and Brocato etal (2000) are consistent with both subpopulations being old, and with metallicities of [Fe/H] \~ -1.5 and -0.6 for the blue and red subpopulations respectively. The models of Worthey (1994) do not reproduce the (U-B)o colours of the red (metal-rich) subpopulation for any modelled age. For NGC 3379 we detect a blue subpopulation with similar colours and presumably age/metallicity, to that of the Milky Way and M31 globular cluster systems. The red subpopulation is less well defined, perhaps due to increased photometric errors, but indicates a mean metallicity of [Fe/H] ~ -0.6.Comment: 12 pages, Latex, 10 figures, 1 table, submitted for publication in MNRAS, Fig. 11 available in source file or from [email protected]

Dimension Expanders via Rank Condensers

An emerging theory of "linear algebraic pseudorandomness: aims to understand the linear algebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such algebraic objects such as subspace designs, dimension expanders, seeded rank condensers, two-source rank condensers, and rank-metric codes. In particular, with the recent construction of near-optimal subspace designs by Guruswami and Kopparty as a starting point, we construct good (seeded) rank condensers (both lossless and lossy versions), which are a small collection of linear maps F^n to F^t for t<<n such that for every subset of F^n of small rank, its rank is preserved (up to a constant factor in the lossy case) by at least one of the maps. We then compose a tensoring operation with our lossy rank condenser to construct constant-degree dimension expanders over polynomially large fields. That is, we give a constant number of explicit linear maps A_i from F^n to F^n such that for any subspace V of F^n of dimension at most n/2, the dimension of the span of the A_i(V) is at least (1+Omega(1)) times the dimension of V. Previous constructions of such constant-degree dimension expanders were based on Kazhdan\u27s property T (for the case when F has characteristic zero) or monotone expanders (for every field F); in either case the construction was harder than that of usual vertex expanders. Our construction, on the other hand, is simpler. For two-source rank condensers, we observe that the lossless variant (where the output rank is the product of the ranks of the two sources) is equivalent to the notion of a linear rank-metric code. For the lossy case, using our seeded rank condensers, we give a reduction of the general problem to the case when the sources have high (n^Omega(1)) rank. When the sources have constant rank, combining this with an "inner condenser" found by brute-force leads to a two-source rank condenser with output length nearly matching the probabilistic constructions

On Identity Testing of Tensors, Low-rank Recovery and Compressed Sensing

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but has no known such black-box algorithm. We recast this problem as a question of finding a low-dimensional subspace H, spanned by rank 1 tensors, such that any non-zero tensor in the dual space ker(H) has high rank. We obtain explicit constructions of essentially optimal-size hitting sets for tensors of degree 2 (matrices), and obtain quasi-polynomial sized hitting sets for arbitrary tensors (but this second hitting set is less explicit). We also show connections to the task of performing low-rank recovery of matrices, which is studied in the field of compressed sensing. Low-rank recovery asks (say, over the reals) to recover a matrix M from few measurements, under the promise that M is rank <=r. We also give a formal connection between low-rank recovery and the task of sparse (vector) recovery: any sparse-recovery algorithm that exactly recovers vectors of length n and sparsity 2r, using m non-adaptive measurements, yields a low-rank recovery scheme for exactly recovering nxn matrices of rank <=r, making 2nm non-adaptive measurements. Furthermore, if the sparse-recovery algorithm runs in time \tau, then the low-rank recovery algorithm runs in time O(rn^2+n\tau). We obtain this reduction using linear-algebraic techniques, and not using convex optimization, which is more commonly seen in compressed sensing algorithms. By using a dual Reed-Solomon code, we are able to (deterministically) construct low-rank recovery schemes taking 4nr measurements over the reals, such that the measurements can be all rank-1 matrices, or all sparse matrices.Comment: 55 page