Stochastic emulation of quantum algorithms

Quantum algorithms profit from the interference of quantum states in an exponentially large Hilbert space and the fact that unitary transformations on that Hilbert space can be broken down to universal gates that act only on one or two qubits at the same time. The former aspect renders the direct classical simulation of quantum algorithms difficult. Here we introduce higher-order partial derivatives of a probability distribution of particle positions as a new object that shares these basic properties of quantum mechanical states needed for a quantum algorithm. Discretization of the positions allows one to represent the quantum mechanical state of $\nb$ qubits by $2(\nb+1)$ classical stochastic bits. Based on this, we demonstrate many-particle interference and representation of pure entangled quantum states via derivatives of probability distributions and find the universal set of stochastic maps that correspond to the quantum gates in a universal gate set. We prove that the propagation via the stochastic map built from those universal stochastic maps reproduces up to a prefactor exactly the evolution of the quantum mechanical state with the corresponding quantum algorithm, leading to an automated translation of a quantum algorithm to a stochastic classical algorithm. We implement several well-known quantum algorithms, analyse the scaling of the needed number of realizations with the number of qubits, and highlight the role of destructive interference for the cost of the emulation.

Quantum mechanical state of the quantum system, and tunneling effect (a new approach)

A physical model proposed in this paper resolves existing contradictions between wave and corpuscle properties of quantum particles (electrons) in tunneling effect. By means of AFM methods the work shows that the existence of the second electron on the other side of potential barrier is necessary to realize a tunneling effect in a probe-surface system. At this, only quantum mechanical state of the electron — not its corpuscular properties — described by the wave function tunnels (penetrates) through the potential barrier. The presence of the second electron on the other side of the potential barrier is necessary to establish concatenation with the first, so to say, ‘tunneling’ one. At lack of the second electron on the other side of the potential barrier, only tunneling quantum mechanical state of the first electron is possible — without transferring its corpuscular properties.

A self-adjoint decomposition of the radial momentum operator

With acceptance of the Dirac's observation that the canonical quantization entails using Cartesian coordinates, we examine the operator erPr rather than Pr itself and demonstrate that there is a decomposition of erPr into a difference of two self-adjoint but noncommutative operators, in which one is the total momentum and another is the transverse one. This study renders the operator Pr indirectly measurable and physically meaningful, offering an explanation of why the mean value of Pr over a quantum mechanical state makes sense and supporting Dirac's claim that Pr "is real and is the true momentum conjugate to r".