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 previous predicted properties of quantum chaotic dynamics.
This short paper presents an efficient implementation of Baker-Campbell-Hausdorff formula for calculating the logarithm of product of two possibly non-commutative Lie group elements using only Lie algebra terms.
We present two complementary ways in which Saraceno's symmetric version of the quantum baker's map can be written as a shift map on a string of quantum bits. One of these representations leads naturally to a family of quantizations of the baker's map.
We derive a simple closed form for the matrix elements of the quantum baker's map that shows that the map is an approximate shift in a symbolic representation based on discrete phase space. We use this result to give a formal proof that the quantum baker's map approaches a classical Bernoulli shift in the limit of a small effective Plank's constant.
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.
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 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…
The quantum baker map possesses two symmetries: a canonical "spatial" symmetry, and a time-reversal symmetry. We show that, even when these features are taken into account, the asymptotic entangling power of the baker's map does not always agree with the predictions of random matrix theory. We have verified that the dimension of the Hilbert space is the crucial parameter which determines whether the entangling properties of the baker are universal or not. For power-of-two dimensions, i.e., qubit systems, an anomalous entangling power is observed; otherwise the behavior of the baker is consiste…
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 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.
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 present here a canonical quantization for the baker's map. The method we use is quite different from that used in Balazs and Voros (ref. \QCITE{cite}{}{BV}) and Saraceno (ref. \QCITE{cite}{}{S}). We first construct a natural ``baker covering map'' on the plane $\QTO{mathbb}{\mathbb{R}}^{2}$. We then use as the quantum algebra of observables the subalgebra of operators on $L^{2}(\QTO{mathbb}{\mathbb{R}}) $ generated by $\left\{\exp (2πi\hat{x}) ,\exp (2πi\hat{p}) \right\} $ . We construct a unitary propagator such that as $\hbar \to 0$ the classical dynamics is returned. For Planck's constan…
We present here a complete description of the quantization of the baker's map. The method we use is quite different from that used in Balazs and Voros [BV] and Saraceno [S]. We use as the quantum algebra of observables the operators generated by {exp(2 Pi ix),exp (2 Pi ip)} and construct a unitary propagator such that as Planck's constant tends to zero,the classical dynamics is returned. For Planck's constant satisfying the integrality condition 1/N with N even, and for periodic boundary conditions for the wave functions on the torus, we show that the dynamics can be reduced to the dynamics on…
We show that the class of quantum baker's maps defined by Schack and Caves have the proper classical limit provided the number of momentum bits approaches infinity. This is done by deriving a semi-classical approximation to the coherent-state propagator.
Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method. We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map (the latter happens to be unitary but the former is not). In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers. In the case of the baker map it is shown that nonunitarity is restricted to a one-qubit operator (close to a Hadamard gate for some p…
We present a broad family of quantum baker maps that generalize the proposal of Schack and Caves to any even Hilbert space with arbitrary boundary conditions. We identify a structure, common to all maps consisting of a simple kernel perturbed by diffraction effects. This "essential" baker's map has a different semiclassical limit and can be diagonalized analytically for Hilbert spaces spanned by qubits. In all cases this kernel provides an accurate approximation to the spectral properties - eigenvalues and eigenfunctions - of all the different quantizations.
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 investigate entanglement production in a class of quantum baker's maps. The dynamics of these maps is constructed using strings of qubits, providing a natural tensor-product structure for application of various entanglement measures. We find that, in general, the quantum baker's maps are good at generating entanglement, producing multipartite entanglement amongst the qubits close to that expected in random states. We investigate the evolution of several entanglement measures: the subsystem linear entropy, the concurrence to characterize entanglement between pairs of qubits, and two proposal…
We propose a generalization of the model of classical baker map on the torus, in which the images of two parts of the phase space do overlap. This transformation is irreversible and cannot be quantized by means of a unitary Floquet operator. A corresponding quantum system is constructed as a completely positive map acting in the space of density matrices. We investigate spectral properties of this super-operator and their link with the increase of the entropy of initially pure states.
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…
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…
We study the quantum mechanics of a generalized version of the baker's map. We show that the Ruelle resonances (which govern the approach to ergodicity of classical distributions on phase space) also appear in the quantum correlation functions of observables at different times, and hence control the statistical variance of matrix elements of observables (in the basis of eigenstates of the quantum time evolution operator). We illustrate this with numerical results.
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…
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.
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 unita…
We define a class of dynamical systems on the sphere analogous to the baker map on the torus. The classical maps are characterized by dynamical entropy equal to ln 2. We construct and investigate a family of the corresponding quantum maps. In the simplest case of the model the system does not possess a time reversal symmetry and the quantum map is represented by real, orthogonal matrices of even dimension. The semiclassical ensemble of quantum maps, obtained by averaging over a range of matrix sizes, displays statistical properties characteristic of circular unitary ensemble. Time evolution of…
Lie-Algebraic Bounds on Quantum Control Time via the Baker-Campbell-Hausdorff Formula
NICT（情报通信研究机构）；Faculty of Informatics, Shizuoka University（静冈大学信息学部）；Faculty of Engineering, University of Toyama（富山大学工学部）；Department of Chemistry and Materials Science, Osaka Metropolitan University（大阪公立大学化学材料科学系）
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 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…
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 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.
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.
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 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.
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 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.
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, …
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 study the gate-based implementation of the binary reflected Gray code (BRGC) and binary code of the unitary time evolution operator due to the Laplacian discretized on a lattice with periodic boundary conditions. We find that the resulting Trotter error is independent of system size for a fixed lattice spacing through the Baker-Campbell-Hausdorff formula. We then present our algorithm for building the BRGC quantum circuit. For an adiabatic evolution time $t$ with this circuit, and spectral norm error $ε$, we find the circuit cost (number of gates) and depth required are $\mc{O}(t^2 n A D /ε…
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.
