Zero-noise extrapolation for quantum-gate error mitigation with identity insertions

Quantum-gate errors are a significant challenge for achieving precision measurements on noisy intermediate-scale quantum (NISQ) computers. This paper focuses on zero-noise extrapolation (ZNE), a technique that can be implemented on existing hardware, studying it in detail and proposing modifications to existing approaches. In particular, we consider identity insertion methods for amplifying noise because they are hardware agnostic. We build a mathematical formalism for studying existing ZNE techniques and show how higher order polynomial extrapolations can be used to systematically reduce depolarizing errors. Furthermore, we introduce a method for amplifying noise that uses far fewer gates than traditional methods. This approach is compared with existing methods for simulated quantum circuits. Comparable or smaller errors are possible with fewer gates, which illustrates the potential for empowering an entirely new class of moderate-depth circuits on near term hardware

On similarity solutions of a boundary layer problem with an upstream moving wall

The problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream is examined. It was previously shown that the solution of this boundary value problem is crucially dependent on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream. In particular, it was proved that a solution exists only if this parameter does not exceed a certain critical value, and numerical evidence was adduced to show that this solution is nonunique. Using Crocco formulation the present work proves this nonuniqueness. Also considered are the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter

Unfolding Quantum Computer Readout Noise

In the current era of noisy intermediate-scale quantum (NISQ) computers, noisy qubits can result in biased results for early quantum algorithm applications. This is a significant challenge for interpreting results from quantum computer simulations for quantum chemistry, nuclear physics, high energy physics, and other emerging scientific applications. An important class of qubit errors are readout errors. The most basic method to correct readout errors is matrix inversion, using a response matrix built from simple operations to probe the rate of transitions from known initial quantum states to readout outcomes. One challenge with inverting matrices with large off-diagonal components is that the results are sensitive to statistical fluctuations. This challenge is familiar to high energy physics, where prior-independent regularized matrix inversion techniques (`unfolding') have been developed for years to correct for acceptance and detector effects when performing differential cross section measurements. We study various unfolding methods in the context of universal gate-based quantum computers with the goal of connecting the fields of quantum information science and high energy physics and providing a reference for future work. The method known as iterative Bayesian unfolding is shown to avoid pathologies from commonly used matrix inversion and least squares methods.Comment: 13 pages, 16 figures; v2 has a typo fixed in Eq. 3 and a series of minor modification

Identification of nonlinearity in conductivity equation via Dirichlet-to-Neumann map

We prove that the linear term and quadratic nonlinear term entering a nonlinear elliptic equation of divergence type can be uniquely identified by the Dirichlet to Neuman map. The unique identifiability is proved using the complex geometrical optics solutions and singular solutions

Leveraging Epidemiology to Improve Risk Assessment.

The field of environmental public health is at an important crossroad. Our current biomonitoring efforts document widespread exposure to a host of chemicals for which toxicity information is lacking. At the same time, advances in the fields of genomics, proteomics, metabolomics, genetics and epigenetics are yielding volumes of data at a rapid pace. Our ability to detect chemicals in biological and environmental media has far outpaced our ability to interpret their health relevance, and as a result, the environmental risk paradigm, in its current state, is antiquated and ill-equipped to make the best use of these new data. In light of new scientific developments and the pressing need to characterize the public health burdens of chemicals, it is imperative to reinvigorate the use of environmental epidemiology in chemical risk assessment. Two case studies of chemical assessments from the Environmental Protection Agency Integrated Risk Information System database are presented to illustrate opportunities where epidemiologic data could have been used in place of experimental animal data in dose-response assessment, or where different approaches, techniques, or studies could have been employed to better utilize existing epidemiologic evidence. Based on the case studies and what can be learned from recent scientific advances and improved approaches to utilizing human data for dose-response estimation, recommendations are provided for the disciplines of epidemiology and risk assessment for enhancing the role of epidemiologic data in hazard identification and dose-response assessment

A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data

We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $\Omega$ from interior Current Density Imaging data obtainable using MRI measurements. We only require knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partial\Omega$. As previously shown, the corresponding voltage potential u in $\Omega$ is a minimizer of the weighted least gradient problem $$u=\hbox{argmin} \{\int_{\Omega}a(x)|\nabla u|: u \in H^{1}(\Omega), \ \ u|_{\partial \Omega}=f\},$$ with $a(x)= |J(x)|$. In this paper we present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(\Omega)$ non-negative and $f\in H^{1/2}(\partial \Omega)$. We give a detailed convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. The dual problem also turns out to yield a novel method to recover the full vector field $J$ from knowledge of its magnitude, and of the voltage $f$ on the boundary. We then present several numerical experiments that illustrate the convergence behavior of the proposed algorithm

Formulas and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential

For the Schrodinger equation at fixed energy with a potential supported in a bounded domain we give formulas and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential. For the case of zero background potential these results were obtained in [R.G.Novikov, Multidimensional inverse spectral problem for the equation -\Delta\psi+(v(x)-Eu(x))\psi=0, Funkt. Anal. i Ego Prilozhen 22(4), pp.11-22, (1988)]

Full-wave invisibility of active devices at all frequencies

There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or "cloaking") from observation by electromagnetic (EM) waves. Here, we prove invisibility, with respect to solutions of the Helmholtz and Maxwell's equations, for several constructions of cloaking devices. Previous results have either been on the level of ray tracing [Le,PSS] or at zero frequency [GLU2,GLU3], but recent numerical [CPSSP] and experimental [SMJCPSS] work has provided evidence for invisibility at frequency $k\ne 0$. We give two basic constructions for cloaking a region $D$ contained in a domain $\Omega$ from measurements of Cauchy data of waves at \p \Omega; we pay particular attention to cloaking not just a passive object, but an active device within $D$, interpreted as a collection of sources and sinks or an internal current.Comment: Final revision; to appear in Commun. in Math. Physic