Tradeoffs for reliable quantum information storage in surface codes and color codes

The family of hyperbolic surface codes is one of the rare families of quantum LDPC codes with non-zero rate and unbounded minimum distance. First, we introduce a family of hyperbolic color codes. This produces a new family of quantum LDPC codes with non-zero rate and with minimum distance logarithmic in the blocklength. Second, we study the tradeoff between the length n, the number of encoded qubits k and the distance d of surface codes and color codes. We prove that kd^2 is upper bounded by C(log k)^2n, where C is a constant that depends only on the row weight of the parity-check matrix. Our results prove that the best asymptotic minimum distance of LDPC surface codes and color codes with non-zero rate is logarithmic in the length.Comment: 10 page

Efficient color code decoders in d≥2d\geq 2 dimensions from toric code decoders

We introduce an efficient decoder of the color code in $d\geq 2$ dimensions, the Restriction Decoder, which uses any $d$-dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-filp and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold $p_{\textrm{2D}} \approx 10.2\%$ on the square-octagon lattice is on a par with the toric code threshold on the square lattice.Comment: 28 pages, 8 figure

Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel

Using combinatorial arguments, we determine an upper bound on achievable rates of stabilizer codes used over the quantum erasure channel. This allows us to recover the no-cloning bound on the capacity of the quantum erasure channel, R is below 1-2p, for stabilizer codes: we also derive an improved upper bound of the form : R is below 1-2p-D(p) with a function D(p) that stays positive for 0 < p < 1/2 and for any family of stabilizer codes whose generators have weights bounded from above by a constant - low density stabilizer codes. We obtain an application to percolation theory for a family of self-dual tilings of the hyperbolic plane. We associate a family of low density stabilizer codes with appropriate finite quotients of these tilings. We then relate the probability of percolation to the probability of a decoding error for these codes on the quantum erasure channel. The application of our upper bound on achievable rates of low density stabilizer codes gives rise to an upper bound on the critical probability for these tilings.Comment: 32 page

A Construction of Quantum LDPC Codes from Cayley Graphs

We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi. It is based on the Cayley graph of Fn together with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the Quantum code in $\mathcal{O}(dn^2)$ where d is the minimum distance of the classical code. When the classical code is the $[n, 1, n]$ repetition code, we are able to compute the exact parameters of the associated Quantum code which are $[[2^n, 2^{\frac{n+1}{2}}, 2^{\frac{n-1}{2}}]]$.Comment: The material in this paper was presented in part at ISIT 2011. This article is published in IEEE Transactions on Information Theory. We point out that the second step of the proof of Proposition VI.2 in the published version (Proposition 25 in the present version and Proposition 18 in the ISIT extended abstract) is not strictly correct. This issue is addressed in the present versio

The First Direct Measurements of Magnetic Fields on Very Low-Mass Stars

We present the first direct magnetic field measurements on M dwarfs cooler than spectral class M4.5. Utilizing a new method based on the effects of a field on the FeH band near 1 micron, we obtain information on whether the integrated surface magnetic flux (Bf) is low (well under 1 kilogauss), intermediate (between 1 and about 2.5 kG), or strong (greater than about 3 kG) on a set of stars ranging from M2 down to M9. We also measure the rotational broadening (vsini) and Halpha emission for more than 20 stars. Our goal is to advance the understanding of how dynamo field production varies with stellar parameters for very low-mass stars, how the field and emission activity are related, and whether there is a connection between the rotation and magnetic flux. We find that fields are produced throughout the M-dwarfs. Among the early M stars we have too few targets to yield conclusive results. In the mid-M stars, there is a clear connection between slow rotation and weak fields. In the late-M stars, rotation is always measureable, and the strongest fields go with the most rapid rotators. These very cool rapid rotators have the largest magnetic flux in the whole sample. Halpha emission is found to be a good general proxy for magnetic fields. The drop-off in fractional emission near the bottom of the main sequence is not accompanied by a drop-off in magnetic flux, lending credence to the hypothesis that it is due to atmospheric coupling to the field rather than changes in the field itself. It is clear that the methodology we have developed can be further applied to discover more about the behavior of magnetic dynamos and magnetic activity in cool and fully convective objects.Comment: 33 pages, accepted by ApJ, abstract abbreviated for astro-p

The Galactic disk mass function: reconciliation of the HST and nearby determinations

We derive and parametrize the Galactic mass function (MF) below 1 \msol characteristic of both single objects and binary systems. We resolve the long standing discrepancy between the MFs derived from the HST and from the nearby luminosity functions, respectively. We show that this discrepancy stemmed from {\it two} cumulative effects, namely (i) incorrect color-magnitude determined distances, due a substantial fraction of M dwarfs in the HST sample belonging to the metal-depleted, thick-disk population, as corrected recently by Zheng et al. and (ii) unresolved binaries. We show that both the nearby and HST MF for unresolved systems are consistent with a fraction $\sim$50% of M-dwarf binaries, with the mass of both the primaries and the companions originating from the same underlying single MF. This implies that $\sim$30% of M dwarfs should have an M dwarf companion and $\sim$20% should have a brown dwarf companion, in agreement with recent determinations. The present calculations show that the so-called "brown-dwarf desert" should be reinterpreted as a lack of high mass-ratio (m_2/m_1\la 0.1) systems, and does not preclude a substantial fraction of brown dwarfs as companions of M dwarfs or for other brown dwarfs.Comment: 16 pages, Latex file, uses aasms4.sty, to appear in ApJ Letter

Determination of jet fuel thermal deposit rate using a modified JFTOT

Three fuels having different breakpoint temperatures were studied in the modified jet fuel thermal oxidation tester. The lower stability fuel with a breakpoint of 240 C was first stressed at a constant temperature. After repeating this procedure at several different temperatures, an Arrehenius plot was drawn from the data. The correlation coefficient and the energy of activation were calculated to be 0.97 and 8 kcal/mole respectively. Two other fuels having breakpoint temperatures of 271 C and 285 C were also studied in a similar manner. A straight line was drawn through the data at a slope equivalent to the slope of the lower stability fuel. The deposit formation rates for the three fuels were determined at 260 C, and a relative deposit formation rate at this temperature was calculated and plotted as a function of the individual fuel's breakpoint temperatures