25 research outputs found
Quasi Markovian behavior in mixing maps
We consider the time dependent probability distribution of a coarse grained
observable Y whose evolution is governed by a discrete time map. If the map is
mixing, the time dependent one-step transition probabilities converge in the
long time limit to yield an ergodic stochastic matrix. The stationary
distribution of this matrix is identical to the asymptotic distribution of Y
under the exact dynamics. The nth time iterate of the baker map is explicitly
computed and used to compare the time evolution of the occupation probabilities
with those of the approximating Markov chain. The convergence is found to be at
least exponentially fast for all rectangular partitions with Lebesgue measure.
In particular, uniform rectangles form a Markov partition for which we find
exact agreement.Comment: 16 pages, 1 figure, uses elsart.sty, to be published in Physica D
Special Issue on Predictability: Quantifying Uncertainty in Models of Complex
Phenomen
Parallel Quantum Computing Emulation
Quantum computers provide a fundamentally new computing paradigm that
promises to revolutionize our ability to solve broad classes of problems.
Surprisingly, the basic mathematical structures of gate-based quantum
computing, such as unitary operations on a finite-dimensional Hilbert space,
are not unique to quantum systems but may be found in certain classical systems
as well.
Previously, it has been shown that one can represent an arbitrary multi-qubit
quantum state in terms of classical analog signals using nested quadrature
amplitude modulated signals. Furthermore, using digitally controlled analog
electronics one may manipulate these signals to perform quantum gate operations
and thereby execute quantum algorithms. The computational capacity of a single
signal is, however, limited by the required bandwidth, which scales
exponentially with the number of qubits when represented using frequency-based
encoding.
To overcome this limitation, we introduce a method to extend this approach to
multiple parallel signals. Doing so allows a larger quantum state to be
emulated with the same gate time required for processing frequency-encoded
signals. In the proposed representation, each doubling of the number of signals
corresponds to an additional qubit in the spatial domain. Single quit gate
operations are similarly extended so as to operate on qubits represented using
either frequency-based or spatial encoding schemes. Furthermore, we describe a
method to perform gate operations between pairs of qubits represented using
frequency or spatial encoding or between frequency-based and spatially encoded
qubits. Finally, we describe how this approach may be extended to represent
qubits in the time domain as well.Comment: 9 pages, 4 figures, 2018 IEEE International Conference on Rebooting
Computing (ICRC