arXiv subjects

Paolo Zanardi

Publications and source records attributed to Paolo Zanardi.

At least 55 records · Page 3Linked to original sources

arXiv2016

Adiabaticity in open quantum systems

We provide a rigorous generalization of the quantum adiabatic theorem for open systems described by a Markovian master equation with time-dependent Liouvillian $\mathcal{L}(t)$. We focus on the finite system case relevant for adiabatic quantum computing and quantum annealing. Adiabaticity is defined in terms of closeness to the instantaneous steady state. While the general result is conceptually similar to the closed system case, there are important differences. Namely, a system initialized in the zero-eigenvalue eigenspace of $\mathcal{L}(t)$ will remain in this eigenspace with a deviation that is inversely proportional to the total evolution time $T$. In the case of a finite number of level crossings the scaling becomes $T^{-η}$ with an exponent $η$ that we relate to the rate of the gap closing. For master equations that describe relaxation to thermal equilibrium, we show that the evolution time $T$ should be long compared to the corresponding minimum inverse gap squared of $\mathcal{L}(t)$. Our results are illustrated with several examples.

quant-ph↗

arXiv2015

Dynamical Response Theory for Driven-Dissipative Quantum Systems

We discuss dynamical response theory of driven-dissipative quantum systems described by Markovian Master Equations generating semi-groups of maps. In this setting thermal equilibrium states are replaced by non-equilibrium steady states and dissipative perturbations are considered besides the Hamiltonian ones. We derive explicit expressions for the linear dynamical response functions for generalized dephasing channels and for Davies thermalizing generators. We introduce the notion of maximal harmonic response and compute it exactly for a single qubit channel. Finally, we analyze linear response near dynamical phase transitions in quasi-free open quantum systems. It is found that the effect of the dynamical phase transition shows up in a peak at the edge of the spectrum in the imaginary part of the dynamical response function.

quant-ph↗

arXiv2015

Quantum algorithms for topological and geometric analysis of big data

Extracting useful information from large data sets can be a daunting task. Topological methods for analyzing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying such topological features -- connected components, holes, or voids -- and for determining how such features persist as the data is viewed at different scales. This paper provides quantum algorithms for calculating Betti numbers in persistent homology, and for finding eigenvectors and eigenvalues of the combinatorial Laplacian. The algorithms provide an exponential speedup over classical algorithms for topological data analysis.

quant-ph↗

arXiv2015

Quantum Adiabatic Markovian Master Equations

We develop from first principles Markovian master equations suited for studying the time evolution of a system evolving adiabatically while coupled weakly to a thermal bath. We derive two sets of equations in the adiabatic limit, one using the rotating wave (secular) approximation that results in a master equation in Lindblad form, the other without the rotating wave approximation but not in Lindblad form. The two equations make markedly different predictions depending on whether or not the Lamb shift is included. Our analysis keeps track of the various time- and energy-scales associated with the various approximations we make, and thus allows for a systematic inclusion of higher order corrections, in particular beyond the adiabatic limit. We use our formalism to study the evolution of an Ising spin chain in a transverse field and coupled to a thermal bosonic bath, for which we identify four distinct evolution phases. While we do not expect this to be a generic feature, in one of these phases dissipation acts to increase the fidelity of the system state relative to the adiabatic ground state.

quant-ph↗

arXiv2015

Dissipative Universal Lindbladian Simulation

It is by now well understood that quantum dissipative processes can be harnessed and turned into a resource for quantum-information processing tasks. In this paper we demonstrate yet another way in which this is true by providing a dissipation-assisted protocol for the simulation of general Markovian dynamics. More precisely, we show how a suitable coherent coupling of a quantum system to a set of Markovian dissipating qubits allows one to enact an effective Liouvillian generator of any Lindbladian form. This effective dynamical generator arises from high-order virtual-dissipative processes and governs the system dynamics exactly in the limit of infinitely fast dissipation. Applications to the simulation of collective decoherence are discussed as an illustration.

quant-ph↗

arXiv2015

Theory of temporal fluctuations in isolated quantum systems

When an isolated quantum system is driven out of equilibrium, expectation values of general observables start oscillating in time. This article reviews the general theory of such temporal fluctuations. We first survey some results on the strength of such temporal fluctuations. For example temporal fluctuations are exponentially small in the system's volume for generic systems whereas they fall-off algebraically in integrable systems. We then concentrate on the the so-called quench scenario where the system is driven out-of-equilibrium under the application of a sudden perturbation. For sufficiently small perturbations, temporal fluctuations of physical observables can be characterized in full generality and can be used as an effective tool to probe quantum criticality of the underlying model. In the off-critical region the distribution becomes Gaussian. Close to criticality the distribution becomes a universal function uniquely characterized by a single critical exponent, that we compute explicitly. This contrasts standard equilibrium quantum fluctuations for which the critical distribution depends on a numerable set of critical coefficients and is known only for limited examples. The possibility of using temporal fluctuations to determine pseudo-critical boundaries in optical lattice experiments is further reviewed.

cond-mat.stat-mech↗

arXiv2015

Local convertibility of the ground state of the perturbed Toric code

We present analytical and numerical studies of the behaviour of the $α$-Renyi entropies in the Toric code in presence of several types of perturbations aimed at studying the simulability of these perturbations to the parent Hamiltonian using local operations and classical communications (LOCC) - a property called local-convertibility. In particular, the derivatives, with respect to the perturbation parameter, present different signs for different values of $α$ within the topological phase. From the information-theoretic point of view, this means that such ground states cannot be continuously deformed within the topological phase by means of catalyst assisted local operations and classical communications (LOCC). Such LOCC differential convertibility is on the other hand always possible in the trivial disordered phase. The non-LOCC convertibility is remarkable because it can be computed on a system whose size is independent of correlation length. This method can therefore constitute an experimentally feasible witness of topological order.

quant-ph↗

arXiv2014

Geometry, robustness, and emerging unitarity in dissipation-projected dynamics

Quantum information can be encoded in the set of steady-states (SSS) of a driven-dissipative system. Non steady-states are separated by a large dissipative gap that adiabatically decouples them way while the dynamics inside the SSS is governed by an effective, dissipation-projected, Hamiltonian. The latter results from a highly non-trivial interplay between a weak driving with the fast relaxation process that continuously projects the system back to the SSS. This amounts to a novel type of environment-induced quantum Zeno effect. We prove that the dissipation-projected dynamics is of geometric nature and that it is robust against different types of hamiltonian and dissipative perturbations. Remarkably, in some cases an effective unitary dynamics can emerge out of purely dissipative interactions.

quant-ph↗

arXiv2014

Coherent quantum dynamics in steady-state manifolds of strongly dissipative systems

It has been recently realized that dissipative processes can be harnessed and exploited to the end of coherent quantum control and information processing. In this spirit we consider strongly dissipative quantum systems admitting a non-trivial manifold of steady states. We show how one can enact adiabatic coherent unitary manipulations e.g., quantum logical gates, inside this {steady-state} manifold by adding a weak, time-rescaled, Hamiltonian term into the system's Liouvillian. The effective long-time dynamics is governed by a projected Hamiltonian which results from the interplay between the weak unitary control and the fast relaxation process. The leakage outside the {steady-state} manifold entailed by the Hamiltonian term is suppressed by an environment-induced symmetrization of the dynamics. We present applications to quantum-computation in decoherence-free subspaces and noiseless subsystems and numerical analysis of non-adiabatic errors.

quant-ph↗

arXiv2014

Local random quantum circuits: ensemble CP maps and swap algebras

We define different classes of local random quantum circuits (L-RQC) and show that: a) statistical properties of L-RQC are encoded into an associated family of completely positive maps and b) average purity dynamics can be described by the action of these maps on operator algebras of permutations (swap algebras). An exactly solvable one-dimensional case is analyzed to illustrate the power of the swap algebra formalism. More in general, we prove short time area-law bounds on average purity for uncorrelated L-RQC and infinite time results for both the uncorrelated and correlated cases.

quant-ph↗

arXiv2013

Fluctuation theorems for quantum processes

We present fluctuation theorems and moment generating function equalities for generalized thermodynamic observables and quantum dynamics described by completely positive trace preserving (CPTP) maps, with and without feedback control. Our results include the quantum Jarzynski equality and Crooks fluctuation theorem, and clarify the special role played by the thermodynamic work and thermal equilibrium states in previous studies. We show that for a specific class of generalized measurements, which include projective measurements, unitality replaces microreversibility as the condition for the physicality of the reverse process in our fluctuation theorems. We present an experimental application of our theory to the problem of extracting the system-bath coupling magnitude, which we do for a system of pairs of coupled superconducting flux qubits undergoing quantum annealing.

quant-ph↗

arXiv2013

Quantum information-geometry of dissipative quantum phase transitions

A general framework for analyzing the recently discovered phase transitions in the steady state of dissipation-driven open quantum systems is still missing. In order to fill this gap we extend the so-called fidelity approach to quantum phase transitions to open systems whose steady state is a Gaussian Fermionic state. We endow the manifold of correlations matrices of steady-states with a metric tensor g measuring the distinguishability distance between solutions corresponding to different set of control parameters. The phase diagram can be then mapped out in terms of the scaling-behavior of g and connections with the Liouvillean gap and the model correlation functions unveiled. We argue that the fidelity approach, thanks to its differential-geometric and information-theoretic nature, provides novel insights on dissipative quantum critical phenomena as well as a general and powerful strategy to explore them.

quant-ph↗

arXiv2013

On the size of linear superpositions in molecular nanomagnets

Molecular nanomagnets are zero-dimensional spin systems, that exhibit quantum mechanical behavior at low temperatures. Exploiting quantum-information theoretic measures, we quantify here the size of linear superpositions that can be generated within the ground multiplet of high- and low-spin nanomagnets. Amongst the former class of systems, we mainly focus on Mn12 and Fe8. General criteria for further increasing such sizes are also outlined, and illustrated in the case of spin rings. The actual character (micro or macroscopic) of linear superpositions in low-spin systems is inherently ill-defined. Here, this issue is addressed with specific reference to the Cr7Ni and V15 molecules, characterized by an S=1/2 ground state. In both cases, the measures we obtain are larger than those of a single s=1/2 spin, but not proportionate to the number and value of the constituent spins.

quant-ph↗

arXiv2013

Universal time-fluctuations in near-critical out-of-equilibrium quantum dynamics

Out of equilibrium quantum systems, on top of quantum fluctuations, display complex temporal patterns. Such time fluctuations are generically exponentially small in the system volume and can be therefore safely ignored in most of the cases. However, if one consider small quench experiments, time fluctuations can be greatly enhanced. We show that time fluctuations may become stronger than other forms of equilibrium quantum fluctuations if the quench is performed close to a critical point. For sufficiently relevant operators the full distribution function of dynamically evolving observable expectation values, becomes a universal function uniquely characterized by the critical exponents and the boundary conditions. At regular points of the phase diagram and for non sufficiently relevant operators the distribution becomes Gaussian. Our predictions are confirmed by an explicit calculation on the quantum Ising model.

quant-ph↗

arXiv2013

Local response of topological order to an external perturbation

We study the behavior of the Rényi entropies for the toric code subject to a variety of different perturbations, by means of 2D density matrix renormalization group and analytical methods. We find that Rényi entropies of different index α display derivatives with opposite sign, as opposed to typical symmetry breaking states, and can be detected on a very small subsystem regardless of the correlation length. This phenomenon is due to the presence in the phase of a point with flat entanglement spectrum, zero correlation length, and area law for the entanglement entropy. We argue that this kind of splitting is common to all the phases with a certain group theoretic structure, including quantum double models, cluster states, and other quantum spin liquids. The fact that the size of the subsystem does not need to scale with the correlation length makes it possible for this effect to be accessed experimentally.

quant-ph↗

arXiv2013

Probability density of quantum expectation values

We consider the quantum expectation value \mathcal{A}=\<ψ|A|ψ\> of an observable A over the state |ψ\> . We derive the exact probability distribution of \mathcal{A} seen as a random variable when |ψ\> varies over the set of all pure states equipped with the Haar-induced measure. The probability density is obtained with elementary means by computing its characteristic function, both for non-degenerate and degenerate observables. To illustrate our results we compare the exact predictions for few concrete examples with the concentration bounds obtained using Levy's lemma. Finally we comment on the relevance of the central limit theorem and draw some results on an alternative statistical mechanics based on the uniform measure on the energy shell.

quant-ph↗

arXiv2013

Entanglement susceptibility: Area laws and beyond

Generic quantum states in the Hilbert space of a many body system are nearly maximally entangled whereas low energy physical states are not; the so-called area laws for quantum entanglement are widespread. In this paper we introduce the novel concept of entanglement susceptibility by expanding the 2-Renyi entropy in the boundary couplings. We show how this concept leads to the emergence of area laws for bi-partite quantum entanglement in systems ruled by local gapped Hamiltonians. Entanglement susceptibility also captures quantitatively which violations one should expect when the system becomes gapless. We also discuss an exact series expansion of the 2-Renyi entanglement entropy in terms of connected correlation functions of a boundary term. This is obtained by identifying Renyi entropy with ground state fidelity in a doubled and twisted theory.

quant-ph↗

arXiv2013

Gaussian Equilibration

A finite quantum system evolving unitarily equilibrates in a probabilistic fashion. In the general many-body setting the time-fluctuations of an observable \mathcal{A} are typically exponentially small in the system size. We consider here quasi-free Fermi systems where the Hamiltonian and observables are quadratic in the Fermi operators. We first prove a novel bound on the temporal fluctuations Δ\mathcal{A}^{2} and then map the equilibration dynamics to a generalized classical XY model in the infinite temperature limit. Using this insight we conjecture that, in most cases, a central limit theorem can be formulated leading to what we call Gaussian equilibration: observables display a Gaussian distribution with relative error Δ\mathcal{A}/\bar{\mathcal{A}}=O(L^{-1/2}) where L is the dimension of the single particle space. The conjecture, corroborated by numerical evidence, is proven analytically under mild assumptions for the magnetization in the quantum XY model and for a class of observables in a tight-binding model. We also show that the variance is discontinuous at the transition between a quasi-free model and a non-integrable one.

cond-mat.stat-mech↗