Scalable Emulation of Sign-Problem−-Free Hamiltonians with Room Temperature p-bits

The growing field of quantum computing is based on the concept of a q-bit which is a delicate superposition of 0 and 1, requiring cryogenic temperatures for its physical realization along with challenging coherent coupling techniques for entangling them. By contrast, a probabilistic bit or a p-bit is a robust classical entity that fluctuates between 0 and 1, and can be implemented at room temperature using present-day technology. Here, we show that a probabilistic coprocessor built out of room temperature p-bits can be used to accelerate simulations of a special class of quantum many-body systems that are sign-problem$-$free or stoquastic, leveraging the well-known Suzuki-Trotter decomposition that maps a $d$-dimensional quantum many body Hamiltonian to a $d$+1-dimensional classical Hamiltonian. This mapping allows an efficient emulation of a quantum system by classical computers and is commonly used in software to perform Quantum Monte Carlo (QMC) algorithms. By contrast, we show that a compact, embedded MTJ-based coprocessor can serve as a highly efficient hardware-accelerator for such QMC algorithms providing several orders of magnitude improvement in speed compared to optimized CPU implementations. Using realistic device-level SPICE simulations we demonstrate that the correct quantum correlations can be obtained using a classical p-circuit built with existing technology and operating at room temperature. The proposed coprocessor can serve as a tool to study stoquastic quantum many-body systems, overcoming challenges associated with physical quantum annealers.Comment: Fixed minor typos and expanded Appendi

The turnpike property in finite-dimensional nonlinear optimal control

Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry. They refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal control problem. We provide in this paper a general version of a turnpike theorem, valuable for nonlinear dynamics without any specific assumption, and for very general terminal conditions. Not only the optimal trajectory is shown to remain exponentially close to a steady-state, but also the corresponding adjoint vector of the Pontryagin maximum principle. The exponential closedness is quantified with the use of appropriate normal forms of Riccati equations. We show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method, or a shooting method. In particular, we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector, not at the initial time, but at the middle of the trajectory

Probabilistic-Bits based on Ferroelectric Field-Effect Transistors for Stochastic Computing

A probabilistic-bit (p-bit) is the fundamental building block in the circuit network of a stochastic computing, and it could produce a continuous random bit-stream with tunable probability. Utilizing the stochasticity in few-domain ferroelectric material(FE), we propose for the first time, the p-bits based on ferroelectric FET. The stochasticity of the FE p-bits stems from the thermal noise-induced lattice vibration, which renders dipole fluctuations and is tunable by an external electric field. The impact of several key FE parameters on p-bits' stochasticity is evaluated, where the domain properties are revealed to play crucial roles. Furthermore, the integer factorization based on FE p-bits circuit network is performed to verify its functionality, and the accuracy is found to depend on FE p-bits' stochasticity. The proposed FE p-bits possess the advantages of both extremely low hardware coast and the compatibility with CMOS-technology, rendering it a promising candidate for stochastic computing applications.Comment: 23 pages, 7 figures and supplementary materials with 3 note

A new class of distribution-free tests for time series models specification

The construction of asymptotically distribution free time series model specification tests using as statistics the estimated residual autocorrelations is considered from a general view point. We focus our attention on Box-Pierce type tests based on the sum of squares of a few estimated residual autocorrelations. This type of tests belong to the class defined by quadratic forms of weighted residual autocorrelations, where weights are suitably transformed resulting in asymptotically distribution free tests. The weights can be optimally chosen to maximize the power function when testing in the direction of local alternatives. The optimal test in this class against MA, AR or Bloomfield alternatives is a Box-Pierce type test based on the sum of squares of a few transformed residual autocorrelations. Such transformations are, in fact, the recursive residuals in the projection of the residual autocorrelations on a certain score function

Understanding Trade-offs in Stellarator Design with Multi-objective Optimization

In designing stellarators, any design decision ultimately comes with a trade-off. Improvements in particle confinement, for instance, may increase the burden on engineers to build more complex coils, and the tightening of financial constraints may simplify the design and worsen some aspects of transport. Understanding trade-offs in stellarator designs is critical in designing high performance devices that satisfy the multitude of physical, engineering, and financial criteria. In this study we show how multi-objective optimization (MOO) can be used to investigate trade-offs and develop insight into the role of design parameters. We discuss the basics of MOO, as well as practical solution methods for solving MOO problems. We apply these methods to bring insight into the selection of two common design parameters: the aspect ratio of an ideal magnetohydrodynamic equilibrium, and the total length of the electromagnetic coils