arXiv subjects

Davide Vodola

Publications and source records attributed to Davide Vodola.

24 records · Page 2Linked to original sources

arXiv2019

Analytical percolation theory for topological color codes under qubit loss

Quantum information theory has shown strong connections with classical statistical physics. For example, quantum error correcting codes like the surface and the color code present a tolerance to qubit loss that is related to the classical percolation threshold of the lattices where the codes are defined. Here we explore such connection to study analytically the tolerance of the color code when the protocol introduced in [Phys. Rev. Lett. $\textbf{121}$, 060501 (2018)] to correct qubit losses is applied. This protocol is based on the removal of the lost qubit from the code, a neighboring qubit, and the lattice edges where these two qubits reside. We first obtain analytically the average fraction of edges $ r(p) $ that the protocol erases from the lattice to correct a fraction $ p $ of qubit losses. Then, the threshold $ p_c $ below which the logical information is protected corresponds to the value of $ p $ at which $ r(p) $ equals the bond-percolation threshold of the lattice. Moreover, we prove that the logical information is protected if and only if the set of lost qubits does not include the entire support of any logical operator. The results presented here open a route to an analytical understanding of the effects of qubit losses in topological quantum error codes.

quant-ph↗

arXiv2018

Algebraic Localization from Power-Law Interactions in Disordered Quantum Wires

We analyze the effects of disorder on the correlation functions of one-dimensional quantum models of fermions and spins with long-range interactions that decay with distance $\ell$ as a power-law $1/\ell^α$. Using a combination of analytical and numerical results, we demonstrate that power-law interactions imply a long-distance algebraic decay of correlations within disordered-localized phases, for all exponents $α$. The exponent of algebraic decay depends only on $α$, and not, e.g., on the strength of disorder. We find a similar algebraic localization for wave-functions. These results are in contrast to expectations from short-range models and are of direct relevance for a variety of quantum mechanical systems in atomic, molecular and solid-state physics.

cond-mat.dis-nn↗

arXiv2016

Effective Theory and Breakdown of Conformal Symmetry in a Long-Range Quantum Chain

We deal with the problem of studying the effective theories and the symmetries of long-range models around critical points. We focus in particular on the Kitaev chain with long-range pairings decaying with distance as power-law with exponent $α$. This is a quadratic solvable model, yet displaying non-trivial quantum phase transitions. By renormalization group approach we derive first the effective theory close to the critical line at positive chemical potential. This is the sum of two terms: a Dirac action $S_D$, found in the short-range Ising universality class, and a CS breaking term $S_{AN}$. While $S_D$ originates from low-energy excitations, $S_{AN}$ derives from the higher energy modes where singularities develop, due to the long-range nature of the model. At criticality $S_{AN}$ flows to zero if $α> 2$, while if $α< 2$ it dominates and determines the breakdown of the CS. Out of criticality $S_{AN}$ breaks, in the considered approximation, the effective Lorentz invariance (ELI) for every finite $α$. As $α$ increases, such ELI breakdown becomes less and less pronounced and in the short-range limit $α\to \infty$ the ELI is restored. To test the validity of the effective theory, we compared the two-fermion static correlations and the von Neumann entropy obtained from them with the ones calculated on the lattice, finding agreement. From the effective theory one can also see that new phases emerge for $α< 1$. Finally we show that at every finite $α$ the critical exponents are the same as for the short-range ($α\to \infty$) model. At variance, for the critical line with negative chemical potential, only $S_{AN}$ ($S_D$) is present for $1 < α< 2$ (for $α> 2$). Close to this line, where the minimum of the spectrum coincides with the momentum where singularities develop, the critical exponents change exactly where CS is broken.

cond-mat.str-el↗

arXiv2015

Long-range Ising and Kitaev Models: Phases, Correlations and Edge Modes

We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the 1D Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance $r$ as $1/r^α$, as well as a related class of fermionic Hamiltonians that generalise the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents $α$, based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for $α\lesssim1$, while connected correlation functions can decay with a hybrid exponential and power-law behaviour, with a power that is $α$-dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every $α$. Along the critical lines, for all models breaking of conformal symmetry is argued at low enough $α$. For the fermionic models we show that the edge modes, massless for $α\gtrsim 1$, can acquire a mass for $α< 1$. The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.

cond-mat.str-el↗

arXiv2014

Kitaev chains with long-range pairing

We propose and analyze a generalization of the Kitaev chain for fermions with long-range $p$-wave pairing, which decays with distance as a power-law with exponent $α$. Using the integrability of the model, we demonstrate the existence of two types of gapped regimes, where correlation functions decay exponentially at short range and algebraically at long range ($α> 1$) or purely algebraically ($α< 1$). Most interestingly, along the critical lines, long-range pairing is found to break conformal symmetry for sufficiently small $α$. This is accompanied by a violation of the area law for the entanglement entropy in large parts of the phase diagram in the presence of a gap, and can be detected via the dynamics of entanglement following a quench. Some of these features may be relevant for current experiments with cold atomic ions.

cond-mat.str-el↗

arXiv2014

Dynamics of entanglement entropy and entanglement spectrum crossing a quantum phase transition

We study the time evolution of entanglement entropy and entanglement spectrum in a finite-size system which crosses a quantum phase transition at different speeds. We focus on the Ising model with a time-dependent magnetic field, which is linearly tuned on a time scale $τ$ . The time evolution of the entanglement entropy displays different regimes depending on the value of $τ$, showing also oscillations which depend on the instantaneous energy spectrum. The entanglement spectrum is characterized by a rich dynamics where multiple crossings take place with a gap-dependent frequency. Moreover, we investigate the Kibble-Zurek scaling of entanglement entropy and Schmidt gap.

cond-mat.stat-mech↗