Еugene Grichuk, Margarita Kuzmina, Eduard Manykin NETWORK OF COUPLED STOCHASTIC OSCILLATORS AND ONE-WAY QUANTUM COMPUTATIONS Еugene Grichuk, Margarita Kuzmina, Eduard Manykin National Research Nuclear University, Moscow, Russia Keldysh Institute of Applied Mathematics RAS, Moscow, Russia Russian Research Center “Kurchatov Institute”, Moscow, Russia

Classical and quantum computations Programmable computers perform computational tasks by algorithmic means. Universal Turing Machine is the underlying model for all programmable computations (Alan Turing, 1936) Since 1985 it was recognized (David Deutsch) that all computing devices would ultimately be physical systems, obeying laws of physics rather than the laws of classical logics. It gave rise to the concept of a Universal quantum computer. Quantum computation algorithms are based on evolution of some quantum system and exploitataion of quantum physics laws for computation performance. In classical computations the basic piece of the yes – no information is the bit The quantum analogue of classical bit is qubit, a two-level quantum system in the sate of quantum superposition of and After measurement Quantum computing can put much more computational power than classical computing.

Logical quantum networks Gubit - a two-level quantum-mechanical system - is single processing unit of logical quantum network One-qubit gates specify unitary transformations of single qubit states Two-qubit gates specify qubit interactions Diagram of logical quantum network Quantum computations permit to realize a specific type of computation parallelization that is not inherent to traditional parallel computation algorithms. Examples of created highly efficient quantum algorithms: Quantum factoring algorithm (Peter Shor, 1994) Quantum search algorithm (Lov Grover, 1996)

One-way quantum computations One-way quantum computation schemes (or cluster quantum computations, CQC ) are significantly different of quantum computation algorithms based on logical quantum networks. Computational resource of CQC is a cluster of entangled qubits; information processing and information readout are realized via specially organized sequence of оne-qubit measurements; the choice of measurement sequence defines quantum computation algorithm itself. Cluster entanglement is gradually destroyed under the measurements (controlled changing of cluster entanglement degree); so a cluster can be used only once for computation performance. Information processing in CQC schemes is really carried out at classical physics level. Quantum-mechanical principles are used only for “preparation” of entangled qubit cluster. Advantage: CQC computation schemes permit to overcome quantum coherence destruction, caused by measurements (the main difficulty of logical quantum networks). CQC schemes are ideally suitable for realization of Grover’s algorithm of quantum search through unstructured data.

A beam of polarized light as a qubit A beam of polarized light as a qubit. Quantum and classical levels of description Quantum level Stokes parameters - density matrix of pure state. Classical level - electrical field of electro-magnetic wave Stokes parameters in optics parameters of polarization ellipse Pure quantum-mechanical states (states of full polarization of light beam) form the Bloch sphere (known as Poincare sphere in optics).

Оsсillatory model of qubit . Oscillatory qubit model is designed as a pair of chaotically modulated limit cycle oscillators. Let the dynamical equations for unperturbed pair of uncoupled oscillators be : where are radii of limit cycles, are own oscillator frequencies. Consider linearly coupled pair of chaotically modulated oscillators with where and are stationary random functions with zero means. Then the system of ODE, governing the dynamics of the coupled pair can be written as The system (1) of ODE, governing qubit model, can be rewritten in variables

Qubit states (polarization states of light) . Pure state (ensemble of elliptically polarized photons Pure state (ensemble of circularly polarized photons Pure state (ensemble of Mixed state with (ensemble of linearly polarized photons) unpolarized photons). Polarization entangled state.

Dynamical equations for entangled qubit cluster Network of coupled stochastic oscillators simulates a cluster of entangled qubits. The qubit cluster in maximally entangled state corresponds to a beam of quasi-monochromatic unpolarized light, composed of N independent sub-beams. The dynamical system for oscillatory network can be written as - 4D-state vector for j-th oscillator - collection of internal parameters of j-th oscillator - matrices, characterizing oscillatory coupling four-component functions, specifying external actions on j-th network oscillator

Example of one-qubit gate ( polarized light beam transmission through linear polarizer ) Methods of classical ellipsometry The Jones matrix of ideal absorptive linear polarizer

Calculation of cluster entanglement degree changing In view of optical interpretation of N-qubit cluster entanglement degree decrease accompanied the process of CQC computations can be exactly calculated as light polarization degree increase after light transfer through the system of optical devices, corresponding to the set of one-qubit gates, specified by CQC scheme. The Stokes parameters provide proper tools to accurate polarization degree calculation after light transfer through given sequence of optical devices.

Conclusions Qubit admits optical interpretation - as a polarization state of a beam of quasi-monochromatic light (that can be adequately described at classical level). A cluster of N entangled qubits can be interpreted as a beam of unpolarized light, composed of N independent sub-beams. One-qubit gates can be modeled as actions of typical optical devices, modifying of light polarization. Cluster entanglement degree, decreased in the process of CQC computations, is directry related to light polarization degree. Qubit model is designed as a stochastic oscillator. It simulates electric field behavior of light beam and imitates correctly both pure and mixed qubit states. A network of coupled stochastic oscillators is designed as a model of N-qubit cluster. The network DS can provide a detailed analysis of cluster state behavior. Network state is changed in a discrete manner via external actions on single network oscillators. Cluster entanglement degree decrease can be accurately calculated as transformed light polarization degree increase in terms of Stokes parameters.