On the Effect of Quantum Interaction Distance on Quantum Addition Circuits

We investigate the theoretical limits of the effect of the quantum interaction distance on the speed of exact quantum addition circuits. For this study, we exploit graph embedding for quantum circuit analysis. We study a logical mapping of qubits and gates of any $\Omega(\log n)$-depth quantum adder circuit for two $n$-qubit registers onto a practical architecture, which limits interaction distance to the nearest neighbors only and supports only one- and two-qubit logical gates. Unfortunately, on the chosen $k$-dimensional practical architecture, we prove that the depth lower bound of any exact quantum addition circuits is no longer $\Omega(\log {n})$, but $\Omega(\sqrt[k]{n})$. This result, the first application of graph embedding to quantum circuits and devices, provides a new tool for compiler development, emphasizes the impact of quantum computer architecture on performance, and acts as a cautionary note when evaluating the time performance of quantum algorithms.Comment: accepted for ACM Journal on Emerging Technologies in Computing System

Reducing reflections from mesh refinement interfaces in numerical relativity

Full interpretation of data from gravitational wave observations will require accurate numerical simulations of source systems, particularly binary black hole mergers. A leading approach to improving accuracy in numerical relativity simulations of black hole systems is through fixed or adaptive mesh refinement techniques. We describe a manifestation of numerical interface truncation error which appears as slowly converging, artificial reflections from refinement boundaries in a broad class of mesh refinement implementations, potentially compromising the effectiveness of mesh refinement techniques for some numerical relativity applications if left untreated. We elucidate this numerical effect by presenting a model problem which exhibits the phenomenon, but which is simple enough that its numerical error can be understood analytically. Our analysis shows that the effect is caused by variations in finite differencing error generated across low and high resolution regions, and that its slow convergence is caused by the presence of dramatic speed differences among propagation modes typical of 3+1 relativity. Lastly, we resolve the problem, presenting a class of finite differencing stencil modifications, termed mesh-adapted differencing (MAD), which eliminate this pathology in both our model problem and in numerical relativity examples.Comment: 7 page

Consistency of post-Newtonian waveforms with numerical relativity

General relativity predicts the gravitational wave signatures of coalescing binary black holes. Explicit waveform predictions for such systems, required for optimal analysis of observational data, have so far been achieved using the post-Newtonian (PN) approximation. The quality of this treatment is unclear, however, for the important late-inspiral portion. We derive late-inspiral waveforms via a complementary approach, direct numerical simulation of Einstein's equations. We compare waveform phasing from simulations of the last $\sim 14$ cycles of gravitational radiation from equal-mass, nonspinning black holes with the corresponding 2.5PN, 3PN, and 3.5PN orbital phasing. We find phasing agreement consistent with internal error estimates based on either approach, suggesting that PN waveforms for this system are effective until the last orbit prior to final merger.Comment: Replaced with published version -- one figure removed, text and other figures updated for clarity of discussio

Quantum Estimation of Parameters of Classical Spacetimes

We describe a quantum limit to measurement of classical spacetimes. Specifically, we formulate a quantum Cramer-Rao lower bound for estimating the single parameter in any one-parameter family of spacetime metrics. We employ the locally covariant formulation of quantum field theory in curved spacetime, which allows for a manifestly background-independent derivation. The result is an uncertainty relation that applies to all globally hyperbolic spacetimes. Among other examples, we apply our method to detection of gravitational waves using the electromagnetic field as a probe, as in laser-interferometric gravitational-wave detectors. Other applications are discussed, from terrestrial gravimetry to cosmology.Comment: 23 pages. This article supersedes arXiv:1108.522

Arithmetic on a Distributed-Memory Quantum Multicomputer

We evaluate the performance of quantum arithmetic algorithms run on a distributed quantum computer (a quantum multicomputer). We vary the node capacity and I/O capabilities, and the network topology. The tradeoff of choosing between gates executed remotely, through ``teleported gates'' on entangled pairs of qubits (telegate), versus exchanging the relevant qubits via quantum teleportation, then executing the algorithm using local gates (teledata), is examined. We show that the teledata approach performs better, and that carry-ripple adders perform well when the teleportation block is decomposed so that the key quantum operations can be parallelized. A node size of only a few logical qubits performs adequately provided that the nodes have two transceiver qubits. A linear network topology performs acceptably for a broad range of system sizes and performance parameters. We therefore recommend pursuing small, high-I/O bandwidth nodes and a simple network. Such a machine will run Shor's algorithm for factoring large numbers efficiently.Comment: 24 pages, 10 figures, ACM transactions format. Extended version of Int. Symp. on Comp. Architecture (ISCA) paper; v2, correct one circuit error, numerous small changes for clarity, add reference

Toward faithful templates for non-spinning binary black holes using the effective-one-body approach

We present an accurate approximation of the full gravitational radiation waveforms generated in the merger of non-eccentric systems of two non-spinning black holes. Utilizing information from recent numerical relativity simulations and the natural flexibility of the effective-one-body (EOB) model, we extend the latter so that it can successfully match the numerical relativity waveforms during the last stages of inspiral, merger and ringdown. By ``successfully'' here, we mean with phase differences < 8% of a gravitational-wave cycle accumulated by the end of the ringdown phase, maximizing only over time of arrival and initial phase. We obtain this result by simply adding a 4-post-Newtonian order correction in the EOB radial potential and determining the (constant) coefficient by imposing high-matching performances with numerical waveforms of mass ratios m1/m2 = 1, 3/2, 2 and 4, m1 and m2 being the individual black-hole masses. The final black-hole mass and spin predicted by the numerical simulations are used to determine the ringdown frequency and decay time of three quasi-normal-mode damped sinusoids that are attached to the EOB inspiral-(plunge) waveform at the EOB light-ring. The EOB waveforms might be tested and further improved in the future by comparison with extremely long and accurate inspiral numerical-relativity waveforms. They may already be employed for coherent searches and parameter estimation of gravitational waves emitted by non-spinning coalescing binary black holes with ground-based laser-interferometer detectors.Comment: 15 pages, 9 figure

Binary black hole late inspiral: Simulations for gravitational wave observations

Coalescing binary black hole mergers are expected to be the strongest gravitational wave sources for ground-based interferometers, such as the LIGO, VIRGO, and GEO600, as well as the space-based interferometer LISA. Until recently it has been impossible to reliably derive the predictions of General Relativity for the final merger stage, which takes place in the strong-field regime. Recent progress in numerical relativity simulations is, however, revolutionizing our understanding of these systems. We examine here the specific case of merging equal-mass Schwarzschild black holes in detail, presenting new simulations in which the black holes start in the late inspiral stage on orbits with very low eccentricity and evolve for ~1200M through ~7 orbits before merging. We study the accuracy and consistency of our simulations and the resulting gravitational waveforms, which encompass ~14 cycles before merger, and highlight the importance of using frequency (rather than time) to set the physical reference when comparing models. Matching our results to PN calculations for the earlier parts of the inspiral provides a combined waveform with less than half a cycle of accumulated phase error through the entire coalescence. Using this waveform, we calculate signal-to-noise ratios (SNRs) for iLIGO, adLIGO, and LISA, highlighting the contributions from the late-inspiral and merger-ringdown parts of the waveform which can now be simulated numerically. Contour plots of SNR as a function of z and M show that adLIGO can achieve SNR >~ 10 for some intermediate-mass binary black holes (IMBBHs) out to z ~ 1, and that LISA can see massive binary black holes (MBBHs) in the range 3x10^4 100 out to the earliest epochs of structure formation at z > 15.Comment: 17 pages, 20 figures. Final published versio

Modeling kicks from the merger of generic black-hole binaries

Recent numerical relativistic results demonstrate that the merger of comparable-mass spinning black holes has a maximum ``recoil kick'' of up to \sim 4000 \kms. However the scaling of these recoil velocities with mass ratio is poorly understood. We present new runs showing that the maximum possible kick perpendicular to the orbital plane does not scale as $\sim\eta^2$ (where $\eta$ is the symmetric mass ratio), as previously proposed, but is more consistent with $\sim\eta^3$, at least for systems with low orbital precession. We discuss the effect of this dependence on galactic ejection scenarios and retention of intermediate-mass black holes in globular clusters.Comment: 5 pages, 1 figure, 3 tables. Version published in Astrophys. J. Let