Quantum 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 [1]

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 approximately
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 dominant.
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 to
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 into
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 dynamical systems

A relevant class of quantum algorithms is the simulation of dynamical systems.
 We have proposed [2] 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).



Quantum 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 [3] 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 (2006)).
More recently, we have shown [4] that dynamical localization of the quantum sawtooth map can be simulated on actual, small-scale quantum processors. Our results demonstrate that quantum computing of dynamical localization may become a convenient tool for evaluating advances in quantum hardware performances.





Dynamical localization in the sawtooth map model simulated using six qubits



 
Dynamical localization simulated in real (IBMQ) quantum processors


Effects 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 [2].





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 [5] 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 imperfections

b) Crossever regime: imperfections degrade the concurrence of an initially entangled pair but can also drive a significant entanglement generation

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 [6].

We have discussed behavior of entanglement across a transition to chaos in [7].




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 classical limit
Since 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 [8] 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 [9], naturally arising when exploring the quantum-to-classical transition.



Schematic drawing of entanglement witnesses, see the review paper [10]

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 [11] 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


References

[1] G. Benenti, G. Casati, D. Rossini and G. Strini, Principles of Quantum Computation and Information (A comprehensive textbook) (World Scientific, Singapore, 2019).
[2]  G. Benenti, G. Casati, S. Montangero and D.L. Shepelyansky, Efficient quantum computing of complex dynamics, Phys. Rev. Lett. 87, 227901 (2001).
[3]  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).
[4] A. Pizzamiglio, S. Y. Chang, M. Bondani, S. Montangero, D. Gerace and G. Benenti, Dynamical localization simulated on actual quantum hardware, Entropy 23, 654 (2021).
[5] S. Montangero, G. Benenti and R. Fazio, Dynamics of entanglement in quantum computers with imperfections, Phys. Rev. Lett. 91, 187901 (2003).
[6] D. Rossini, G. Benenti and G. Casati, Entanglement echoes in quantum computation,  Phys. Rev. A 69, 052317 (2004).
[7] C. Mejia-Monasterio, G. Benenti, G.G. Carlo and G. Casati, Entanglement across a transition to quantum chaos, Phys. Rev. A 71, 062324 (2005).
[8]  M. Znidaric, T. Prosen, G. Benenti and G. Casati, Detecting entanglement of random states with an entanglement witness, J. Phys. A. 40, 13787 (2007).
[9]  W. Weiss, G. Benenti, G. Casati, I. Guarneri, T. Calarco, M. Paternostro and S. Montangero, Violation of Bell inequalities in larger Hilbert spaces: robustness and challenges, New J. Phys. 18, 013021 (2016).
[10]  G. Benenti, Entanglement, randomness and chaos, Riv. Nuovo Cimento 32, 105 (2009)
[11]  D. Rossini and G. Benenti, Robust and efficient generator of almost maximal multipartite entanglement, Phys. Rev. Lett. 100, 060501 (2008).