19 research outputs found
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 as
, 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 , 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 , while connected correlation functions can
decay with a hybrid exponential and power-law behaviour, with a power that is
-dependent. Interestingly, for the fermionic models we provide an exact
analytical derivation for the decay of the correlation functions at every
. Along the critical lines, for all models breaking of conformal
symmetry is argued at low enough . For the fermionic models we show
that the edge modes, massless for , can acquire a mass for
. 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.Comment: 15 pages, 8 figure
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 . The time evolution of the
entanglement entropy displays different regimes depending on the value of
, 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.Comment: Accepted for publication in Phys. Rev.
Kitaev chains with long-range pairing
We propose and analyze a generalization of the Kitaev chain for fermions with
long-range -wave pairing, which decays with distance as a power-law with
exponent . 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 () or
purely algebraically (). Most interestingly, along the critical
lines, long-range pairing is found to break conformal symmetry for sufficiently
small . 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.Comment: 5+3 pages, 4+2 figure
Correlations and Quantum Dynamics of 1D Fermionic Models: New Results for the Kitaev Chain with Long-Range Pairing
In the first part of the thesis, we propose an exactly-solvable one-dimensional model for fermions with long-range p-wave pairing decaying with distance as a power law. We studied the phase diagram by analyzing the critical lines, the decay of correlation functions and the scaling of the von Neumann entropy with the system size. We found two gapped regimes, where correlation functions decay (i) exponentially at short range and algebraically at long range, (ii) purely algebraically. In the latter the entanglement entropy is found to diverge logarithmically. Most interestingly, along the critical lines, long-range pairing breaks also the conformal symmetry. This can be detected via the dynamics of entanglement following a quench. In the second part of the thesis we studied the evolution in time of the entanglement entropy for the Ising model in a transverse field varying linearly in time with different velocities. We found different regimes: an adiabatic one (small velocities) when the system evolves according the instantaneous ground state; a sudden quench (large velocities) when the system is essentially frozen to its initial state; and an intermediate one, where the entropy starts growing linearly but then displays oscillations (also as a function of the velocity). Finally, we discussed the Kibble-Zurek mechanism for the transition between the paramagnetic and the ordered phase.In the first part of the thesis, we propose an exactly-solvable one-dimensional model for fermions with long-range p-wave pairing decaying with distance as a power law. We studied the phase diagram by analyzing the critical lines, the decay of correlation functions and the scaling of the von Neumann entropy with the system size. We found two gapped regimes, where correlation functions decay (i) exponentially at short range and algebraically at long range, (ii) purely algebraically. In the latter the entanglement entropy is found to diverge logarithmically. Most interestingly, along the critical lines, long-range pairing breaks also the conformal symmetry. This can be detected via the dynamics of entanglement following a quench. In the second part of the thesis we studied the evolution in time of the entanglement entropy for the Ising model in a transverse field varying linearly in time with different velocities. We found different regimes: an adiabatic one (small velocities) when the system evolves according the instantaneous ground state; a sudden quench (large velocities) when the system is essentially frozen to its initial state; and an intermediate one, where the entropy starts growing linearly but then displays oscillations (also as a function of the velocity). Finally, we discussed the Kibble-Zurek mechanism for the transition between the paramagnetic and the ordered phase
Bayesian Optimization for QAOA
The Quantum Approximate Optimization Algorithm (QAOA) adopts a hybrid
quantum-classical approach to find approximate solutions to variational
optimization problems. In fact, it relies on a classical subroutine to optimize
the parameters of a quantum circuit. In this work we present a Bayesian
optimization procedure to fulfil this optimization task, and we investigate its
performance in comparison with other global optimizers. We show that our
approach allows for a significant reduction in the number of calls to the
quantum circuit, which is typically the most expensive part of the QAOA. We
demonstrate that our method works well also in the regime of slow circuit
repetition rates, and that few measurements of the quantum ansatz would already
suffice to achieve a good estimate of the energy. In addition, we study the
performance of our method in the presence of noise at gate level, and we find
that for low circuit depths it is robust against noise. Our results suggest
that the method proposed here is a promising framework to leverage the hybrid
nature of QAOA on the noisy intermediate-scale quantum devices
Characterizing quantum instruments: from non-demolition measurements to quantum error correction
In quantum information processing quantum operations are often processed
alongside measurements which result in classical data. Due to the information
gain of classical measurement outputs non-unitary dynamical processes can take
place on the system, for which common quantum channel descriptions fail to
describe the time evolution. Quantum measurements are correctly treated by
means of so-called quantum instruments capturing both classical outputs and
post-measurement quantum states. Here we present a general recipe to
characterize quantum instruments alongside its experimental implementation and
analysis. Thereby, the full dynamics of a quantum instrument can be captured,
exhibiting details of the quantum dynamics that would be overlooked with common
tomography techniques. For illustration, we apply our characterization
technique to a quantum instrument used for the detection of qubit loss and
leakage, which was recently implemented as a building block in a quantum error
correction (QEC) experiment (Nature 585, 207-210 (2020)). Our analysis reveals
unexpected and in-depth information about the failure modes of the
implementation of the quantum instrument. We then numerically study the
implications of these experimental failure modes on QEC performance, when the
instrument is employed as a building block in QEC protocols on a logical qubit.
Our results highlight the importance of careful characterization and modelling
of failure modes in quantum instruments, as compared to simplistic
hardware-agnostic phenomenological noise models, which fail to predict the
undesired behavior of faulty quantum instruments. The presented methods and
results are directly applicable to generic quantum instruments.Comment: 28 pages, 21 figure
Algebraic localization from power-law couplings 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 x as a power law 1/x^a. 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 a. The exponent of algebraic decay depends only on a, 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
Twins Percolation for Qubit Losses in Topological Color Codes
We establish and explore a new connection between quantum information theory
and classical statistical mechanics by studying the problem of qubit losses in
2D topological color codes. We introduce a protocol to cope with qubit losses,
which is based on the identification and removal of a twin qubit from the code,
and which guarantees the recovery of a valid three-colorable and trivalent
reconstructed color code lattice. Moreover, we show that determining the
corresponding qubit loss error threshold is equivalent to a new generalized
classical percolation process. We numerically compute the associated qubit loss
thresholds for two families of 2D color code and find that these are close to
satisfying the fundamental limit close to the
50% as imposed by the no-cloning theorem. Our findings reveal a new connection
between topological color codes and percolation theory, show high robustness of
color codes against qubit loss, and are relevant for implementations of quantum
error correction in various physical platforms.Comment: 6+7 pages, 3+7 figure
Corrélations et dynamique quantique de modÚles de fermions 1D : nouveaux résultats sur la chaßne de Kitaev avec pairing à longue portée
In the first part of the thesis, we propose an exactly-solvable one-dimensional model for fermions with long-range p-wave pairing decaying with distance â as a power law 1/âα. We studied the phase diagram by analyzing the critical lines, the decay of correlation functions and the scaling of the von Neumann entropy with the system size. We found two gapped regimes, where correlation functions decay (i) exponentially at short range and algebraically at long range (α > 1), (ii) purely algebraically (α 1), (ii) ouÌ elles tombent aÌ puissance seulement (α < 1). Dans la seconde reÌgion lâentropie dâintrication dâun sous-systeÌme diverge logarithmiquement. Remarquablement, sur les lignes critiques, le pairing aÌ long rayon brise la symeÌtrie conforme du modeÌle pour des α suffisamment petits. On a prouveÌ ça en calculant aussi lâeÌvolution temporelle de lâentropie dâintrication apreÌs un quench. Dans la seconde partie de la theÌse nous avons analyseÌ la dynamique de lâentropie dâintrication du modeÌle dâIsing avec un champ magneÌtique qui deÌpend lineÌairement du temps avec de diffeÌrentes vitesses. Nous avons un reÌgime adiabatique (de basses vitesses) lorsque le systeÌme eÌvolue selon son eÌtat fondamental instantaneÌ; un sudden quench (de hautes vitesses) lorsque le systeÌme est congeleÌ dans son eÌtat initial; un reÌgime intermeÌdiaire ouÌ lâentropie croiÌt lineÌairement et, ensuite, elle montre des oscillations du moment que le systeÌme se trouve dans une superposition des eÌtats exciteÌs de lâHamiltonienne instantaneÌe. Nous avons discuteÌ aussi du meÌcanisme de Kibble-Zurek pour la transition entre la phase paramagneÌtique et antiferromagneÌtique