computation, randomness and chaos
Quantum mechanics has had an
enormous technological and societal impact. To grasp this point, it is
sufficient to cite the invention of the
transistor, perhaps the most remarkable among the countless other
applications of quantum mechanics. It is also easy to see the enormous
impact of computers on everyday life. The importance of computers is
such that it is appropriate to say that we are now living in the
information age. This information revolution became possible thanks to
the invention of the transistor, that is, thanks to the synergy between
computer science and quantum physics.
Today this synergy offers completely new opportunities and promises
exciting advances in both fundamental science and technological
application. We are referring here to the fact that quantum mechanics can be used to
process and transmit information 
Miniaturization provides us
with an intuitive way of understanding why, in the near future, quantum
laws will become important for computation. The electronics industry
for computers grows hand-in-hand with the decrease in size of
integrated circuits. This miniaturization is necessary to increase
computational power, that is, the number of floating-point operations
per second (flops) a computer can perform. The progress in the
miniaturization process may be quantified empirically in Moore's law:
the number of transistors on a single integrated-circuit chip doubles
every 18-24 months. Extrapolating Moore's law, one would estimate that
around the year 2020 we shall reach the atomic size for storing a
single bit of
information. At that point, quantum effects will become unavoidably
Therefore, quantum physics sets fundamental limitations on the size of
the circuit components.
The first question under debate is whether it would be more convenient
to push the silicon-based transistor to its physical limits or instead
develop alternative devices, such as quantum dots, single-electron
transistors or molecular switches. A common feature of all these
devices is that they are at the nanometre length scale, and therefore
quantum effects play a crucial role.
So far, we have talked about quantum switches that could substitute
silicon-based transistors and possibly be connected together to execute
classical algorithms based on Boolean logic. In this perspective,
quantum effects are simply unavoidable corrections that must be taken
account owing to the nanometre size of the switches. A quantum computer
represents a radically different challenge: the aim is to build a machine based on quantum logic,
that is, a machine that can process the information and perform logic
operations in agreement with the laws of quantum mechanics.
Development of quantum algorithms for
A relevant class of quantum algorithms is the simulation of
We have proposed  a quantum algorithm which uses the number of
qubits in an optimal way and efficiently simulates a physical model
with rich and complex dynamics described by the quantum sawtooth map.
We have demonstrated that complex dynamics could be simulated already
with less than 10 qubits, while 40 qubits would allow one to make
computations inaccessible to present-day supercomputers.
Husimi functions for the
sawtooth map in action-angle variables, for n=6 (top left), n=9 (top
right), n=16 qubits (bottom left) and classical Poincare
section (bottom right).
computation of dynamical localization
Dynamical localization is one of the most interesting phenomena
that characterize the quantum behavior of classically chaotic systems:
quantum interference effects suppress classical diffusion leading to
exponentially localized wave functions.
We have shown  that quantum computers can simulate efficiently the
quantum localization of classical chaos. The speed up with respect to
classical computation is
quadratic. The localization effect was studied experimentally by the
group of David Cory at MIT by emulating the dynamics if the
quantum sawtooth map in the perturbative regime on a three-qubit
nuclear magnetic resonance quantum information processor (see M.K.
Henry, J. Emerson, R. Martinez, and D.G. Cory, Phys. Rev. A 74, 062317
Dynamical localization in the
sawtooth map model simulated using six qubits
of imperfections on the stability of quantum computation
We have studied the effect of
static imperfections in the quantum computer hardware on
the stability of quantum computation.
We have found that a reliable quantum computation is possible up
to a time scale which is polynomial in the number of qubits.
The errors generated by these imperfections are more dangerous
than the errors of random noise in gate operations .
Husimi functions for the sawtooth map in action-angle variables,
in the presence of static imperfections, for n=6 (top left), n=9 (top
right), n=16 qubits (bottom left) and classical Poincare
section (bottom right) in the presence of round-off errors.
Dynamics of entanglement in quantum
computers with imperfections
Objective: understand the time
scales for the stability of entanglement (a key resource for quantum
computation and information) under decoherence and imperfection effects.
We have studied  the evolution of the entanglement of formation
between two nearest neighbor qubits in a lattice, which initially are
maximally entangled or separable.
We have characterized three regimes:
a) Perturbative regime: the entanglement is stable against
b) Crossever regime: imperfections degrade the concurrence of an
initially entangled pair but can also drive a significant entanglement
c) Ergodic regime: a pair of qubits becomes entangled with the
rest of the lattice and the concurrence of the pair drops to zero
The stability of the entanglement of formation in an operating quantum
computer has been investigated in .
We have discussed behavior of entanglement across a transition to chaos
Concurrence saturation values for
different number of qubits, starting from a Bell state (left) or a
separable state (right)
Entanglement, Bell's inequalities, randomness and the
random states carry a lot of entanglement and entanglement
has no analogue in classical mechanics,
one can conclude that random states are highly non-classical. On the
other hand, for chaotic map the classical limit can be recovered when
the dimension of the Hilbert space diverges, and (ergodic) random
states in a way mimic classical microcanonical density. How can we
reconcile this apparent contradiction? We have considered  the detection of entanglement for
random states by means of witness
operators. While the entanglement content of random pure states
is almost maximal, we have shown that, due to the complexity of such
states, the detection of their entanglement is difficult. Moreover, the
entanglement detection probability drops exponentially when considering
mixtures of random states. Our results can be used to explain the
emergence of classicality in coarse
grained quantum chaotic dynamics. We also explored the violation of Bell's inequalities in the limit of high-dimensional systems , naturally arising when exploring the quantum-to-classical transition.
Schematic drawing of entanglement witnesses, see the review paper
Robust and efficient generator of multipartite entanglement
Quantum chaotic maps can
efficiently generate pseudo-random states carrying almost maximal
multipartite entanglement, as characterized by the probability
distribution of bipartite entanglement between all possible
bipartitions of the system. We have shown  that such multipartite
entanglement is robust, in the sense that, when realistic noise is
considered, distillable entanglement of bipartitions remains almost
maximal up to a noise strength that drops only polynomially with the
number of qubits.
Stability border for distillable
entanglement of bipartitions
 G. Benenti, G. Casati and G. Strini,
Principles of Quantum Computation and
Information (World Scientific,
 G. Benenti, G. Casati, S. Montangero and D.L. Shepelyansky, Efficient quantum computing of complex
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 G. Benenti, G. Casati, S. Montangero and D.L. Shepelyansky, Dynamical localization simulated on a
few-qubit quantum computer, Phys. Rev. A 67, 052312 (2003).
 S. Montangero, G. Benenti and R. Fazio, Dynamics of entanglement in quantum
computers with imperfections, Phys. Rev. Lett. 91, 187901 (2003).
 D. Rossini, G. Benenti and G. Casati, Entanglement echoes in quantum computation,
Phys. Rev. A 69, 052317 (2004).
 C. Mejia-Monasterio, G. Benenti, G.G. Carlo and G. Casati, Entanglement across a transition to
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