Concrete resource analysis of the quantum linear system algorithm used to compute the electromagnetic scattering cross section of a 2D target

We provide a detailed estimate for the logical resource requirements of the quantum linear system algorithm (QLSA) [Phys. Rev. Lett. 103, 150502 (2009)] including the recently described elaborations [Phys. Rev. Lett. 110, 250504 (2013)]. Our resource estimates are based on the standard quantum-circuit model of quantum computation; they comprise circuit width, circuit depth, the number of qubits and ancilla qubits employed, and the overall number of elementary quantum gate operations as well as more specific gate counts for each elementary fault-tolerant gate from the standard set {X, Y, Z, H, S, T, CNOT}. To perform these estimates, we used an approach that combines manual analysis with automated estimates generated via the Quipper quantum programming language and compiler. Our estimates pertain to the example problem size N=332,020,680 beyond which, according to a crude big-O complexity comparison, QLSA is expected to run faster than the best known classical linear-system solving algorithm. For this problem size, a desired calculation accuracy 0.01 requires an approximate circuit width 340 and circuit depth of order $10^{25}$ if oracle costs are excluded, and a circuit width and depth of order $10^8$ and $10^{29}$, respectively, if oracle costs are included, indicating that the commonly ignored oracle resources are considerable. In addition to providing detailed logical resource estimates, it is also the purpose of this paper to demonstrate explicitly how these impressively large numbers arise with an actual circuit implementation of a quantum algorithm. While our estimates may prove to be conservative as more efficient advanced quantum-computation techniques are developed, they nevertheless provide a valid baseline for research targeting a reduction of the resource requirements, implying that a reduction by many orders of magnitude is necessary for the algorithm to become practical.Comment: 37 pages, 40 figure

Resource Evaluation of Quantum Linear Systems Algorithm for Application to Electromagnetic Scattering Problems

Current limitations in quantum computing technology does not allow for very large applications of quantum algorithms, and it is the nature of quantum algorithms not only to be able to solve problems of interest much more quickly than classical means but also to do so with less resources which makes them so promising! One such problem of interest is the application of the Quantum Linear Systems Algorithm, along with a few other subroutines, to the calculation of an electromagnetic scattering cross-section via finite element methods. This work composes a resource analysis of the algorithm as well as required subroutines and details the primary contributors to the resources involved as well as methods to decrease these resource requirements

Programming Quantum Computers Using Design Automation

Recent developments in quantum hardware indicate that systems featuring more than 50 physical qubits are within reach. At this scale, classical simulation will no longer be feasible and there is a possibility that such quantum devices may outperform even classical supercomputers at certain tasks. With the rapid growth of qubit numbers and coherence times comes the increasingly difficult challenge of quantum program compilation. This entails the translation of a high-level description of a quantum algorithm to hardware-specific low-level operations which can be carried out by the quantum device. Some parts of the calculation may still be performed manually due to the lack of efficient methods. This, in turn, may lead to a design gap, which will prevent the programming of a quantum computer. In this paper, we discuss the challenges in fully-automatic quantum compilation. We motivate directions for future research to tackle these challenges. Yet, with the algorithms and approaches that exist today, we demonstrate how to automatically perform the quantum programming flow from algorithm to a physical quantum computer for a simple algorithmic benchmark, namely the hidden shift problem. We present and use two tool flows which invoke RevKit. One which is based on ProjectQ and which targets the IBM Quantum Experience or a local simulator, and one which is based on Microsoft's quantum programming language Q$\#$.Comment: 10 pages, 10 figures. To appear in: Proceedings of Design, Automation and Test in Europe (DATE 2018

Quantum attacks on Bitcoin, and how to protect against them

The key cryptographic protocols used to secure the internet and financial transactions of today are all susceptible to attack by the development of a sufficiently large quantum computer. One particular area at risk are cryptocurrencies, a market currently worth over 150 billion USD. We investigate the risk of Bitcoin, and other cryptocurrencies, to attacks by quantum computers. We find that the proof-of-work used by Bitcoin is relatively resistant to substantial speedup by quantum computers in the next 10 years, mainly because specialized ASIC miners are extremely fast compared to the estimated clock speed of near-term quantum computers. On the other hand, the elliptic curve signature scheme used by Bitcoin is much more at risk, and could be completely broken by a quantum computer as early as 2027, by the most optimistic estimates. We analyze an alternative proof-of-work called Momentum, based on finding collisions in a hash function, that is even more resistant to speedup by a quantum computer. We also review the available post-quantum signature schemes to see which one would best meet the security and efficiency requirements of blockchain applications.Comment: 21 pages, 6 figures. For a rough update on the progress of Quantum devices and prognostications on time from now to break Digital signatures, see https://www.quantumcryptopocalypse.com/quantum-moores-law

Quantum Algorithms for Fluid Simulations

This chapter describes results of a recent investigation aiming to assess the potential of quantum computing and suitably designed algorithms for future computational fluid dynamics applications. For quantum computers becoming available in the near future, it can be expected that applications of quantum computing follow the quantum coprocessor model, where selected parts of the computational task for which efficient quantum algorithms exist are executed on the quantum hardware. For example, in computational fluid dynamics algorithm, this hybrid quantum/classical approach is discussed, and in particular it is shown how the approximate quantum Fourier transform (AQFT) can be used in the Poisson solvers of the considered method for the incompressible-flow Navier-Stokes equations. The analysis shows that despite the inevitable errors introduced by applying AQFT, the method produces meaningful results for three-dimensional example problems. A second example of a quantum algorithm for flow simulations is then described. This method based on kinetic modeling of the flow was developed to reduce the information transfer between quantum and classical hardware in the quantum coprocessor model. It is shown that this quantum algorithm can be executed fully on quantum hardware during a simulation. The conclusion summarizes further challenges for algorithm developments and future work

Computational Scattering Models for Elastic and Electromagnetic Waves in Particulate Media

Numerical models were developed to simulate the propagation of elastic and electromagnetic waves in an arbitrary, dense dispersion of spherical particles. The scattering interactions were modeled with vector multipole fields using pure-orbital vector spherical harmonics, and solved using the full vector form of the boundary conditions. Multiple scattering was simulated by translating the scattered wave fields from one particle to another with the use of translational addition theorems, summing the multiple-scattering contributions, and recalculating the scattering in an iterative fashion to a convergent solution. The addition theorems were rederived in this work using an integral method, and were shown to be numerically equivalent to previously published theorems. Both ordered and disordered collections of up to 5,000 spherical particles were used to demonstrate the ability of the scattering models to predict the spatial and frequency distributions of the transmitted waves. The results of the models show they are qualitatively correct for many particle configurations and material properties, displaying predictable phenomena such as refractive focusing, mode conversion, and photonic band gaps. However, the elastic wave models failed to converge for specific frequency regions, possibly due to resonance effects. Additionally, comparison of the multiple-scattering simulations with those using only single-particle scattering showed the multiple-scattering computations are quantitatively inaccurate. The inaccuracies arise from nonconvergence of the translational addition theorems, introducing errors into the translated fields, which minimize the multiple-scattering contributions and bias the field amplitudes towards single-scattering contributions. The addition theorems are shown to converge very slowly, and to exhibit plateaus in convergence behavior that can lead to false indications of convergence. The theory and algorithms developed for the models are broad-based, and can accommodate a variety of structures, compositions, and wave modes. The generality of the approach also lends itself to the modeling of static fields and currents. Suggestions are presented for improving and implementing the models, including extension to nonspherical particles, efficiency improvements for the algorithms, and specific applications in a variety of fields