Classical chaotic systems are distinguished by their sensitive dependence on initial conditions. The absence of this property in quantum systems has lead to a number of proposals for perturbation-based characterizations of quantum chaos, including linear growth of entropy, exponential decay of fidelity, and hypersensitivity to perturbation. All of these accurately predict chaos in the classical limit, but it is not clear that they behave the same far from the classical realm. We investigate the dynamics of a family of quantizations of the baker's map, which range from a highly entangling unitary transformation to an essentially trivial shift map. Linear entropy growth and fidelity decay are exhibited by this entire family of maps, but hypersensitivity distinguishes between the simple dynamics of the trivial shift map and the more complicated dynamics of the other quantizations. This conclusion is supported by an analytical argument for short times and numerical evidence at later times.
Hypersensitivity to perturbation is a criterion for chaos based on the question of how much information about a perturbing environment is needed to keep the entropy of a Hamiltonian system from increasing. We demonstrate numerically that hypersensitivity to perturbation is present in the following quantum maps: the quantum kicked top, the quantum baker's map, the quantum lazy baker's map, and the quantum sawtooth and cat maps.
Hypersensitivity to perturbation is a criterion for chaos based on the question of how much information about a perturbing environment is needed to keep the entropy of a Hamiltonian system from increasing. In this paper we give a brief overview of our work on hypersensitivity to perturbation in classical and quantum systems.
By numerically simulating an implementation of the quantum baker's map on a 3-qubit NMR quantum computer based on the molecule trichloroethylene, we demonstrate the feasibility of quantum chaos experiments on present-day quantum computers. We give detailed descriptions of proposed experiments that investigate (a) the rate of entropy increase due to decoherence and (b) the phenomenon of hypersensitivity to perturbation.
We study the properties of eigenstates of an operating quantum computer which simulates the dynamical evolution in the regime of quantum chaos. Even if the quantum algorithm is polynomial in number of qubits $n_q$, it is shown that the ideal eigenstates become mixed and strongly modified by static imperfections above a certain threshold which drops exponentially with $n_q$. Above this threshold the quantum eigenstate entropy grows linearly with $n_q$ but the computation remains reliable during a time scale which is polynomial in the imperfection strength and in $n_q$.
We study the chaotic behaviour and the quantum-classical correspondence for the baker's map. Correspondence between quantum and classical expectation values is investigated and it is numerically shown that it is lost at the logarithmic timescale. The quantum chaos degree is computed and it is demonstrated that it describes the chaotic features of the model. The correspondence between classical and quantum chaos degrees is considered.
The modular multiplication operator, a central subroutine in Shor's factoring algorithm, is shown to be a coherent superposition of two quantum bakers maps when the multiplier is 2. The classical limit of the maps being completely chaotic, it is shown that there exist perturbations that push the modular multiplication operator into regimes of generic quantum chaos with spectral fluctuations that are those of random matrices. For the initial state of relevance to Shor's algorithm we study fidelity decay due to phase and bit-flip errors in a single qubit and show exponential decay with shoulders…
It is pointed out that an exactly solvable permutation operator, viewed as the quantization of cyclic shifts, is useful in constructing a basis in which to study the quantum baker's map, a paradigm system of quantum chaos. In the basis of this operator the eigenfunctions of the quantum baker's map are compressed by factors of around five or more. We show explicitly its connection to an operator that is closely related to the usual quantum baker's map. This permutation operator has interesting connections to the art of shuffling cards as well as to the quantum factoring algorithm of Shor via th…
We show that the quantum baker's map, a prototypical map invented for theoretical studies of quantum chaos, has a very simple realization in terms of quantum gates. Chaos in the quantum baker's map could be investigated experimentally on a quantum computer based on only 3 qubits.
This paper reports on the experimental implementation of the quantum baker's map via a three bit nuclear magnetic resonance (NMR) quantum information processor. The experiments tested the sensitivity of the quantum chaotic map to perturbations. In the first experiment, the map was iterated forward and then backwards to provide benchmarks for intrinsic errors and decoherence. In the second set of experiments, the least significant qubit was perturbed in between the iterations to test the sensitivity of the quantum chaotic map to applied perturbations. These experiments are used to investigate p…
A set of quantum states, dynamically related to the classical periodic orbits of a chaotic map, is used as a basis in which the description of the eigenstates of its quantum version is greatly simplified. This set can be improved with the inclusion of short time propagation along the stable and unstable manifolds of the periodic orbits resulting in a construction similar to the scar functions of Vergini (Jour. Phys. A \textbf{33}, (2000) 4709). The average participation ratio is used to quantify the quality of the basis.
The characteristic stretching and squeezing of chaotic motion is linearized within the finite number of phase space domains which subdivide a classical baker map. Tensor products of such maps are also chaotic, but a more interesting generalized baker map arises if the stacking orders for the factor maps are allowed to interact. These maps are readily quantized, in such a way that the stacking interaction is entirely attributed to primary qubits in each map, if each subsystem has power-of-two Hilbert space dimension. We here study the particular example of two baker maps that interact via a con…
For chaotic classical systems, the distribution of return times to a small region of phase space is universal. We propose a simple tool to investigate multiple returns in quantum systems. Numerical evidence for the baker map and kicked top points, also in the quantum case, at a universal distribution.
The out-of-time-ordered correlator (OTOC) is a measure of quantum chaos that is being vigorously investigated. Analytically accessible simple models that have long been studied in other contexts could provide insights into such measures. This paper investigates the OTOC in the quantum bakers map which is the quantum version of a simple and exactly solvable model of deterministic chaos that caricatures the action of kneading dough. Exact solutions based on the semiquantum approximation are derived that tracks very well the correlators till the Ehrenfest time. The growth occurs at the exponentia…
We rationalize the somewhat surprising efficacy of the Hadamard transform in simplifying the eigenstates of the quantum baker's map, a paradigmatic model of quantum chaos. This allows us to construct closely related, but new, transforms that do significantly better, thus nearly solving for many states of the quantum baker's map. These new transforms, which combine the standard Fourier and Hadamard transforms in an interesting manner, are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit-flip (parity) and possess bit-reversal (time-re…
We show that the coarse-grained quantum baker's map exhibits a linear entropy increase at an asymptotic rate given by the Kolmogorov-Sinai entropy of the classical chaotic baker's map. The starting point of our analysis is a symbolic representation of the map on a string of $N$ qubits, i.e., an $N$-bit register of a quantum computer. To coarse-grain the quantum evolution, we make use of the decoherent histories formalism. As a byproduct, we show that the condition of medium decoherence holds asymptotically for the coarse-grained quantum baker's map.
We consider continuous observation of the nonlinear dynamics of single atom trapped in an optical cavity by a standing wave with intensity modulation. The motion of the atom changes the phase of the field which is then monitored by homodyne detection of the output field. We show that the conditional Hilbert space dynamics of this system, subject to measurement induced perturbations, depends strongly on whether the corresponding classical dynamics is regular or chaotic. If the classical dynamics is chaotic the distribution of conditional Hilbert space vectors corresponding to different observat…
We investigate how classical predictability of the coarse-grained evolution of the quantum baker's map depends on the character of the coarse-graining. Our analysis extends earlier work by Brun and Hartle [Phys. Rev. D 60, 123503 (1999)] to the case of a chaotic map. To quantify predictability, we compare the rate of entropy increase for a family of coarse-grainings in the decoherent histories formalism. We find that the rate of entropy increase is dominated by the number of scales characterising the coarse-graining.
Quantum baker`s map is a model of chaotic system. We study quantum dynamics for the quantum baker's map. We use the Schack and Caves symbolic description of the quantum baker`s map. We find an exact expression for the expectation value of the time dependent position operator. A relation between quantum and classical trajectories is investigated. Breakdown of the quantum-classical correspondence at the logarithmic timescale is rigorously established.
We study relevant features of the spectrum of the quantum open baker map. The opening consists of a cut along the momentum $p$ direction of the 2-torus phase space, modelling an open chaotic cavity. We study briefly the classical forward trapped set and analyze the corresponding quantum nonunitary evolution operator. The distribution of eigenvalues depends strongly on the location of the escape region with respect to the central discontinuity of this map. This introduces new ingredients to the association among the classical escape and quantum decay rates. Finally, we could verify that the val…
We provide compelling evidence for the presence of quantum chaos in the unitary part of Shor's factoring algorithm. In particular we analyze the spectrum of this part after proper desymmetrization and show that the fluctuations of the eigenangles as well as the distribution of the eigenvector components follow the CUE ensemble of random matrices, of relevance to quantized chaotic systems that violate time-reversal symmetry. However, as the algorithm tracks the evolution of a single state, it is possible to employ other operators, in particular it is possible that the generic quantum chaos foun…
We conjecture that in chaotic quantum systems with escape the intensity statistics for resonance states universally follows an exponential distribution. This requires a scaling by the multifractal mean intensity which depends on the system and the decay rate of the resonance state. We numerically support the conjecture by studying the phase-space Husimi function and the position representation of resonance states of the chaotic standard map, the baker map, and a random matrix model, each with partial escape.
We examine the conjecture that entropy production in subsystems of a given system can be used as a dynamical criterion for quantum chaos in the latter. Numerical results are presented for finite dimensional spin systems as also for the quantum baker's map. Of especial importance is the power spectrum of the entropy production which gets progressively more and more broad-banded as the degree of correlation in the Hamiltonian matrix is made to decrease.
A measure describing the chaos of a dynamics was introduced by two complexities in information dynamics, and it is called the chaos degree. In particular, the entropic chaos degree has been used to characterized several dynamical maps such that logistis, Baker's, Tinckerbel's in classical or quantum systems. In this paper, we give a new treatment of quantum chaos by defining the entropic chaos degree for quantum transition dynamics, and we prove that every non-chaotic quantum dynamics, e.g., dissipative dynamics, has zero chaos degree. A quantum spin 1/2 system is studied by our chaos degree, …
We analyze certain eigenstates of the quantum baker's map and demonstrate, using the Walsh-Hadamard transform, the emergence of the ubiquitous Thue-Morse sequence, a simple sequence that is at the border between quasi-periodicity and chaos, and hence is a good paradigm for quantum chaotic states. We show a family of states that are also simply related to Thue-Morse sequence, and are strongly scarred by short periodic orbits and their homoclinic excursions. We give approximate expressions for these states and provide evidence that these and other generic states are multifractal.
We introduce a class of functions that limit to multifractal measures and which arise when one takes the Fourier transform of the Hadamard transform. This introduces generalizations of the Fourier transform of the well-studied and ubiquitous Thue-Morse sequence, and introduces also generalizations to other intriguing sequences. We show their relevance to quantum chaos, by displaying quantum eigenfunctions of the quantum bakers map that are approximated well by such measures, thereby extending our recent work where we pointed to the existence of ``Thue-Morse'' states.
We study the stability of classical structures in chaotic systems when a dissipative quantum evolution takes place. We consider a paradigmatic model, the quantum baker map in contact with a heat bath at finite temperature. We analyze the behavior of the purity, fidelity and Husimi distributions corresponding to initial states localized on short periodic orbits (scar functions) and map eigenstates. Scar functions, that have a fundamental role in the semiclassical description of chaotic systems, emerge as very robust against environmental perturbations. This is confirmed by the study of other st…
We have studied two complementary decoherence measures purity and fidelity for a generic diffusive noise in two different chaotic systems (the baker and the cat maps). For both quantities, we have found classical structures in quantum mechanics - the scar functions - that are specially stable when subjected to environmental perturbations. We show that these quantum states constructed on classical invariants are the most robust significant quantum distributions in generic dissipative maps.
The classical Bernoulli and baker maps are two simple models of deterministic chaos. On the level of ensembles, it has been shown that the time evolution operator for these maps admits generalized spectral representations in terms of decaying eigenfunctions. We introduce the quantum version of the Bernoulli map. We define it as a projection of the quantum baker map. We construct a quantum analogue of the generalized spectral representation, yielding quantum decaying states represented by density matrices. The quantum decaying states develop a quasi-fractal shape limited by the quantum uncertai…
We study the behavior of an open quantum system, with an $N$--dimensional space of states, whose density matrix evolves according to a non--unitary map defined in two steps: A unitary step, where the system evolves with an evolution operator obtained by quantizing a classically chaotic map (baker's and Harper's map are the two examples we consider). A non--unitary step where the evolution operator for the density matrix mimics the effect of diffusion in the semiclassical (large $N$) limit. The process of decoherence and the transition from quantum to classical behavior are analyzed in detail b…
We show that particle transport in a uniform, quantum multi-baker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semi-classical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time from diffusive to ballistic motion on the order of the inverse of Planck's constant. We can argue, that for a large class of 1D quantum random walks, sim…
An exact general formula for the matrix elements of the evolution operator in quantum theory is established. The formul ("ABC-formula") has the form <U(t)>=exp(At+B+C(t)). The constants A and B and the decreasing function C(t) are computed in perturbation theory. The ABC-formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory. The formula includes the higher order corrections to the well known Weisskopf-Wigner approximation and to the stochastic (van Hove) limit which are widely used in considerations of problems of radiation, decay, quantum de…
Using quantum maps we study the accuracy of semiclassical trace formulas. The role of chaos in improving the semiclassical accuracy, in some systems, is demonstrated quantitatively. However, our study of the standard map cautions that this may not be most general. While studying a sawtooth map we demonstrate the rather remarkable fact that at the level of the time one trace even in the presence of fixed points on singularities the trace formula may be exact, and in any case has no logarithmic divergences observed for the quantum bakers map. As a byproduct we introduce fantastic periodic curves…
We derive an expression for the mean square displacement of a particle whose motion is governed by a uniform, periodic, quantum multi-baker map. The expression is a function of both time, $t$, and Planck's constant, $\hbar$, and allows a study of both the long time, $t\to\infty$, and semi-classical, $\hbar\to 0$, limits taken in either order. We evaluate the expression using random matrix theory as well as numerically, and observe good agreement between both sets of results. The long time limit shows that particle transport is generically ballistic, for any fixed value of Planck's constant. Ho…
In this work we study several models of decoherence and how different quantum maps and algorithms react when perturbed by them. Following closely Ref. [1], generalizations of the three paradigmatic one single qubit quantum channels (these are the depolarizing channel, the phase damping channel and the amplitude damping channel) for the case of an arbitrarily-sized finite-dimensional Hilbert space are presented, as well as other types of noise in phase space. More specifically, Grover's search algorithm's response to decoherence is analyzed; together with those of a family of quantum versions o…
Spectral properties of evolution operators corresponding to random maps and quantized chaotic systems strongly interacting with an environment can be described by the ensemble of non-hermitian random matrices from the real Ginibre ensemble. We analyze evolution operators Psi=Psi_s...Psi_1 representing the composition of s random maps and demonstrate that their complex eigenvalues are asymptotically described by the law of Burda et al. obtained for a product of s independent random complex Ginibre matrices. Numerical data support the conjecture that the same results are applicable to characteri…
A non-Hermitian PT-symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. The presence of PT-symmetry can lead to stable mixed regular chaotic behaviour without sinks or sources for subcritical values of the gain-loss parameter. This is an example of what is known in classical dynamical systems as reversible dynamical systems. For large values of the kicking strength a strange attractor is observed that also persists if…
