Quantum Non-Markovianity: Characterization, Quantification and Detection
A Rivas, SF Huelga, MB Plenio
We present a comprehensive and up to date review on the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.
Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers
O Viyuela, A Rivas and M A Martin-Delgado
We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in two-dimensional fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number \(n_U\). With it, we study thermal topological phases in several two-dimensional models of topological insulators and superconductors, computing phase diagrams where the temperature \(T\) is on an equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between two nontrivial phases in a model with high Chern numbers. At small temperatures we recover the standard topological phases as the Uhlmann number approaches to the Chern number.
Quantum Google algorithm
GD Paparo, M Müller, F Comellas, MA Martin-Delgado
We review the main findings on the ranking capabilities of the recently proposed Quantum PageRank algorithm (G.D. Paparo et al., Sci. Rep. 2, 444 (2012) and G.D. Paparo et al., Sci. Rep. 3, 2773 (2013)) applied to large complex networks. The algorithm has been shown to identify unambiguously the underlying topology of the network and to be capable of clearly highlighting the structure of secondary hubs of networks. Furthermore, it can resolve the degeneracy in importance of the low-lying part of the list of rankings. Examples of applications include real-world instances from the WWW, which typically display a scale-free network structure and models of hierarchical networks. The quantum algorithm has been shown to display an increased stability with respect to a variation of the damping parameter, present in the Google algorithm, and a more clearly pronounced power-law behaviour in the distribution of importance among the nodes, as compared to the classical algorithm.
Quantum computations on a topologically encoded qubit
D Nigg, M Müller, E A Martinez, P Schindler, M Hennrich, T Monz, M A Martin-Delgado, R Blatt
The construction of a quantum computer remains a fundamental scientific and technological challenge because of the influence of unavoidable noise. Quantum states and operations can be protected from errors through the use of protocols for quantum computing with faulty components. We present a quantum error-correcting code in which one qubit is encoded in entangled states distributed over seven trapped-ion qubits. The code can detect one bit flip error, one phase flip error, or a combined error of both, regardless on which of the qubits they occur. We applied sequences of gate operations on the encoded qubit to explore its computational capabilities. This seven-qubit code represents a fully functional instance of a topologically encoded qubit, or color code, and opens a route toward fault-tolerant quantum computing.
Efficient algorithm to compute the Berry conductivity
A Dauphin, M Müller, MA Martin-Delgado
We propose and construct a numerical algorithm to calculate the Berry conductivity in topological band insulators. The method is applicable to cold atom systems as well as solid state setups, both for the insulating case where the Fermi energy lies in the gap between two bulk bands as well as in the metallic regime. This method interpolates smoothly between both regimes. The algorithm is gauge-invariant by construction, efficient, and yields the Berry conductivity with known and controllable statistical error bars. We apply the algorithm to several paradigmatic models in the field of topological insulators, including Haldaneʼs model on the honeycomb lattice, the multi-band Hofstadter model, and the BHZ model, which describes the 2D spin Hall effect observed in CdTe/HgTe/CdTe quantum well heterostructures.
Quantum speedup for active learning agents
GD Paparo, V Dunjko, A Makmal, MA Martin-Delgado, HJ Briegel
Can quantum mechanics help us in building intelligent robots and agents? One of the defining characteristics of intelligent behavior is the capacity to learn from experience. However, a major bottleneck for agents to learn in any real-life situation is the size and complexity of the corresponding task environment. Owing to, e.g., a large space of possible strategies, learning is typically slow. Even for a moderate task environment, it may simply take too long to rationally respond to a given situation. If the environment is impatient, allowing only a certain time for a response, an agent may then be unable to cope with the situation and to learn at all. Here we show that quantum physics can help and provide a significant speed-up for active learning as a genuine problem of artificial intelligence. We introduce a large class of quantum learning agents for which we show a quadratic boost in their active learning efficiency over their classical analogues. This result will be particularly relevant for applications involving complex task environments.
Quantum Google algorithm
GD Paparo, M Müller, F Comellas, MA Martin-Delgado
We review the main findings on the ranking capabilities of the recently proposed Quantum PageRank algorithm (G.D. Paparo et al., Sci. Rep. 2, 444 (2012) and G.D. Paparo et al., Sci. Rep. 3, 2773 (2013)) applied to large complex networks. The algorithm has been shown to identify unambiguously the underlying topology of the network and to be capable of clearly highlighting the structure of secondary hubs of networks. Furthermore, it can resolve the degeneracy in importance of the low-lying part of the list of rankings. Examples of applications include real-world instances from the WWW, which typically display a scale-free network structure and models of hierarchical networks. The quantum algorithm has been shown to display an increased stability with respect to a variation of the damping parameter, present in the Google algorithm, and a more clearly pronounced power-law behaviour in the distribution of importance among the nodes, as compared to the classical algorithm.
Uhlmann phase as a topological measure for one-dimensional fermion systems
O Viyuela, A Rivas, MA Martin-Delgado
We introduce the Uhlmann geometric phase as a tool to characterize symmetry-protected topological phases in one-dimensional fermion systems, such as topological insulators and superconductors. Since this phase is formulated for general mixed quantum states, it provides a way to extend topological properties to finite temperature situations. We illustrate these ideas with some paradigmatic models and find that there exists a critical temperature Tc at which the Uhlmann phase goes discontinuously and abruptly to zero. This stands as a borderline between two different topological phases as a function of the temperature. Furthermore, at small temperatures we recover the usual notion of topological phase in fermion systems.
Quantum-information engines with many-body states attaining optimal extractable work with quantum control
JM Diaz de la Cruz, MA Martin-Delgado
We introduce quantum information engines that extract work from quantum states and a single thermal reservoir. They may operate under three general conditions: i/ Unitarily Steered evolution (US); ii/ Irreversible Thermalization (IT) and iii/ Isothermal Relaxation (IR), and hence are called USITIR machines. They include novel engines without traditional feedback control mechanisms, as well as versions which also include them. Explicit constructions of USITIR engines are presented for one- and two-qubit states and their maximum extractable work is computed, which is optimal. Optimality is achieved when the notions of controllable thermalizability and density matrix controllability are fullfilled. Then, many-body extensions of USITIR engines are also analyzed and conditions for optimal work extraction are identified. When they are not met, we measure their lack of optimality by means of newly defined uncontrollable entropies, that are explicitly computed for some selected examples. This includes cases of distinguishable and indistinguishable particles.
Reducing space-time to binary information
S Weinfurtner, G De las Cuevas, MA Martin-Delgado, HJ Briegel
We present a new description of discrete space-time in 1+1 dimensions in terms of a set of elementary geometrical units that represent its independent classical degrees of freedom. This is achieved by means of a binary encoding that is ergodic in the class of space-time manifolds respecting coordinate invariance of general relativity. Space-time fluctuations can be represented in a classical lattice gas model whose Boltzmann weights are constructed with the discretized form of the Einstein-Hilbert action. Within this framework, it is possible to compute basic quantities such as the Ricci curvature tensor and the Einstein equations, and to evaluate the path integral of discrete gravity. The description as a lattice gas model also provides a novel way of quantization and, at the same time, to quantum simulation of fluctuating space-time.