1. 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

2. 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.

3. 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)

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 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.

5. 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).

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
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

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 state with
(ensemble of
linearly polarized photons)
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-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

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 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.

11. 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.