Network of Coupled Stochastic Oscillators and One-Way Quantum Computations

1. 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

2. Classical and quantum computations Programmable computers perform computational tasks by algorithmic means. Universal Turing Machineis 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.Itgave rise to the concept of a Universal quantum computer. Quantum computation algorithmsare based on evolution of somequantumsystemandexploitataion of quantum physics laws for computationperformance. The quantum analogue of classical bit is qubit, a two-level quantum system in the sate of quantum superposition of and In classical computations the basic piece of the yes – no information is the bit Aftermeasurement Quantum computing can put muchmore computational powerthan classical computing.

3. Logical quantum networks Gubit -a two-level quantum-mechanical system- issingle processing unit oflogicalquantum network One-qubit gates specify unitary transformations of single qubit states Two-qubit gates specify qubit interactions Diagram of logical quantum network Quantum computationspermit to realizea 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)

4. 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 CQCisa cluster of entangled qubits;informationprocessing and information readout are realized viaspecially organized sequence ofоne-qubit measurements; the choice of measurement sequencedefines quantum computation algorithm itself. Cluster entanglement is gradually destroyedunder the measurements (controlled changing of cluster entanglement degree); soacluster can be used only oncefor computation performance. Information processingin CQCschemes is really carried out atclassical 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 realizationofGrover’s algorithmof quantum searchthrough unstructured data.

5. A beam of polarized light as a qubit. Quantum and classical levels of description Quantum level Stokes parameters - densitymatrix 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 polarizationof light beam) form the Bloch sphere (known as Poincare spherein optics).

6. 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 areown oscillator frequencies. are radii of limit cycles, Consider linearly coupled pair of chaotically modulated oscillators with and are stationary random functions with zero means. where 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

7. 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 statewith (ensemble of linearlypolarizedphotons) unpolarized photons). Polarization entangled state.

8. 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-statevector forj-thoscillator - collection of internal parameters of j-th oscillator - matrices, characterizing oscillatory coupling • four-component functions, specifying external actions on j-th network oscillator

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

10. Calculation of cluster entanglement degree changing In viewof opticalinterpretation of N-qubit cluster entanglement degree decreaseaccompanied the process of CQC computations can be exactly calculated as light polarization degree increaseafter 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.

11. Conclusions Qubitadmits 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 lightpolarization degree. Qubit model isdesigned 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-qubitcluster. 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 transformedlight polarization degree increase in terms of Stokes parameters.