A methods comprises: receiving, by a pseudo random number generator module, an instruction to generate pseudo random numbers from a security application; determining, by the pseudo random number generator module, at least one algebraic input parameter value for a transcendental equation from a randomness library in memory of the device, wherein the transcendental equation comprises a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs; calculating, by the pseudo random number generator module, a solution to the transcendental equation based on the at least one algebraic input parameter value; determining, by the pseudo random number generator module, pseudo random number(s) based on the solution; and storing, by the pseudo random number generator module, the pseudo random number(s) in a randomness library for use as seeds for keys by the security application and as subsequent input parameter values for the pseudo random number generator module.

A methods comprises: receiving, by a pseudo random number generator module, an instruction to generate pseudo random numbers from a security application; determining, by the pseudo random number generator module, at least one algebraic input parameter value for a transcendental equation from a randomness library in memory of the device, wherein the transcendental equation comprises a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs; calculating, by the pseudo random number generator module, a solution to the transcendental equation based on the at least one algebraic input parameter value; determining, by the pseudo random number generator module, pseudo random number(s) based on the solution; and storing, by the pseudo random number generator module, the pseudo random number(s) in a randomness library for use as seeds for keys by the security application and as subsequent input parameter values for the pseudo random number generator module.

Systems and methods for estimating a property of an error in a circuit implemented on an n-qubit quantum system are provided, where the circuit comprises a gate set that comprises a first subset () and a second subset () of elementary gates. The first subset comprises a third subset () of elementary gates each of which consists of an n-fold tensor product of a plurality of single qubit gates. A first procedure is executed that comprises preparing the system in a state ψ and then applying D.sub.1=T.sub.1 to the system. The procedure further comprises, for each respective clock cycle t in clock cycles t∈{2, . . . , m+1}, (a) applying H to the system, where H is an elementary gate in the second subset, and then (b) applying a gate D.sub.t=T.sub.tGHT.sub.t−1.sup.†H.sup.† to the system, where D.sub.t is an element of the first subset. The procedure further comprises performing a measurement readout R. The procedure is repeated for one or more values of {right arrow over (T)} or one or more states ψ or one or more measurement readout procedures R, where m is a positive integer greater than 1, G is an element of the first subset of elementary gates, {right arrow over (T)}=(T.sub.1, . . . , T.sub.m, T.sub.m+1=I), and T.sub.1, . . . , T.sub.m are elements of , with the proviso that n>2.

Systems and methods for estimating a property of an error in a circuit implemented on an n-qubit quantum system are provided, where the circuit comprises a gate set that comprises a first subset () and a second subset () of elementary gates. The first subset comprises a third subset () of elementary gates each of which consists of an n-fold tensor product of a plurality of single qubit gates. A first procedure is executed that comprises preparing the system in a state ψ and then applying D.sub.1=T.sub.1 to the system. The procedure further comprises, for each respective clock cycle t in clock cycles t∈{2, . . . , m+1}, (a) applying H to the system, where H is an elementary gate in the second subset, and then (b) applying a gate D.sub.t=T.sub.tGHT.sub.t−1.sup.†H.sup.† to the system, where D.sub.t is an element of the first subset. The procedure further comprises performing a measurement readout R. The procedure is repeated for one or more values of {right arrow over (T)} or one or more states ψ or one or more measurement readout procedures R, where m is a positive integer greater than 1, G is an element of the first subset of elementary gates, {right arrow over (T)}=(T.sub.1, . . . , T.sub.m, T.sub.m+1=I), and T.sub.1, . . . , T.sub.m are elements of , with the proviso that n>2.

Computer-implemented methods and apparatus are provided for predicting/estimating (i) a non-equilibrium viscosity for at least one given time point in a given temperature profile for a given glass composition, (ii) at least one temperature profile that will provide a given non-equilibrium viscosity for a given glass composition, or (iii) at least one glass composition that will provide a given non-equilibrium viscosity for a given time point in a given temperature profile. The methods and apparatus can be used to predict/estimate stress relaxation in a glass article during forming as well as compaction, stress relaxation, and/or thermal sag or thermal creep of a glass article when the article is subjected to one or more post-forming thermal treatments.

Computer-implemented methods and apparatus are provided for predicting/estimating (i) a non-equilibrium viscosity for at least one given time point in a given temperature profile for a given glass composition, (ii) at least one temperature profile that will provide a given non-equilibrium viscosity for a given glass composition, or (iii) at least one glass composition that will provide a given non-equilibrium viscosity for a given time point in a given temperature profile. The methods and apparatus can be used to predict/estimate stress relaxation in a glass article during forming as well as compaction, stress relaxation, and/or thermal sag or thermal creep of a glass article when the article is subjected to one or more post-forming thermal treatments.

The computer implemented method of solving a Hamiltonian can include performing, in a tensor network contracting a plurality of tensors in the network, a Lanczos method acting on the uncontracted tensors, the Lanczos method including evaluating a recursive relation of an equation including using the equation at least two times, forming a block tridiagonal matrix having a block size greater than one, based on the recursive relation, and diagonalizing the block tridiagonal matrix to obtain new tensors and energy levels of the tensor network, wherein at least one of the uncontracted tensors of the network has an index for the group of excitations; and solving for the rest of the tensor network, yielding an energy level solution of the Hamiltonian, outputting the energy level solution.

A method of determining the center of loading of a rolling element includes providing a rolling element body and at least three load sensors. The sensors are each positioned within a bore of the rolling element body at a separate distance from a reference position. Load measurements are taken with each one of the sensors at various positions about the circumference of the bearing and the center of loading is calculated at each one of the positions to determine the variation in axial loading about the bearing circumference.

Systems and methods for imitation learning in accordance with embodiments of the invention are illustrated. One embodiment includes a method for imitation learning. The method includes steps for initializing a Q-function, training the Q-function using a non-adversarial objective based on a set of one or more expert trajectories, and determining a policy based on the trained Q-function.

Systems and methods for imitation learning in accordance with embodiments of the invention are illustrated. One embodiment includes a method for imitation learning. The method includes steps for initializing a Q-function, training the Q-function using a non-adversarial objective based on a set of one or more expert trajectories, and determining a policy based on the trained Q-function.