Quantum Monte Carlo simulations of fidelity at magnetic quantum phase transitions

Modular Many-Body Quantum Sensors

Quantum many-body systems undergoing phase transitions have been proposed as probes enabling beyond-classical enhancement of sensing precision. However, this enhancement is usually limited to a very narrow region around the critical point. Here, we systematically develop a modular approach for introducing multiple phase transitions in a many-body system. This naturally allows us to enlarge the region of quantum-enhanced precision by encompassing the newly created phase boundaries. Our approach is general and can be applied to both symmetry-breaking and topological quantum sensors. In symmetry-breaking sensors, we show that the newly created critical points inherit the original universality class and a simple total magnetization measurement already suffices to locate them. In topological sensors, our modular construction creates multiple bands which leads to a rich phase diagram. In both cases, Heisenberg scaling for Hamiltonian parameter estimation is achieved at all the phase boundaries. This can be exploited to create a global sensor which significantly outperforms a uniform probe.

Critical Quantum Metrology Assisted by Real-Time Feedback Control

We investigate critical quantum metrology, that is, the estimation of parameters in many-body systems close to a quantum critical point, through the lens of Bayesian inference theory. We first derive a no-go result stating that any nonadaptive strategy will fail to exploit quantum critical enhancement (i.e., precision beyond the shot-noise limit) for a sufficiently large number of particles N whenever our prior knowledge is limited. We then consider different adaptive strategies that can overcome this no-go result and illustrate their performance in the estimation of (i) a magnetic field using a probe of 1D spin Ising chain and (ii) the coupling strength in a Bose-Hubbard square lattice. Our results show that adaptive strategies with real-time feedback control can achieve sub-shot-noise scaling even with few measurements and substantial prior uncertainty.

Quantum criticality of a Z_{3}-symmetric spin chain with long-range interactions

Based on large-scale density matrix renormalization group techniques, we investigate the critical behaviors of quantum three-state Potts chains with long-range interactions. Using fidelity susceptibility as an indicator, we obtain a complete phase diagram of the system. The results show that as the long-range interaction power α increases, the critical points f_{c}^{*} shift towards lower values. In addition, the critical threshold α_{c}(≈1.43) of the long-range interaction power is obtained for the first time by a nonperturbative numerical method. This indicates that the critical behavior of the system can be naturally divided into two distinct universality classes, namely the long-range (α<α_{c}) and short-range (α>α_{c}) universality classes, qualitatively consistent with the classical ϕ^{3} effective field theory. This work provides a useful reference for further research on phase transitions in quantum spin chains with long-range interaction.

Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity

In quantum and quantum-inspired machine learning, a key step is to embed the data in the quantum space known as Hilbert space. Studying quantum kernel function, which defines the distances among the samples in the Hilbert space, belongs to the fundamental topics in this direction. In this work, we propose a tunable quantum-inspired kernel function (QIKF) named rescaled logarithmic fidelity (RLF) and a non-parametric algorithm for the semi-supervised learning in the quantum space. The rescaling takes advantage of the non-linearity of the kernel to tune the mutual distances of samples in the Hilbert space, and meanwhile avoids the exponentially-small fidelities between quantum many-qubit states. Being non-parametric excludes the possible effects from the variational parameters, and evidently demonstrates the properties of the kernel itself. Our results on the hand-written digits (MNIST dataset) and movie reviews (IMDb dataset) support the validity of our method, by comparing with the standard fidelity as the QIKF as well as several well-known non-parametric algorithms (naive Bayes classifiers, k-nearest neighbors, and spectral clustering). High accuracy is demonstrated, particularly for the unsupervised case with no labeled samples and the few-shot cases with small numbers of labeled samples. With the visualizations by t-stochastic neighbor embedding, our results imply that the machine learning in the Hilbert space complies with the principles of maximal coding rate reduction, where the low-dimensional data exhibit within-class compressibility, between-class discrimination, and overall diversity. The proposed QIKF and semi-supervised algorithm can be further combined with the parametric models such as tensor networks, quantum circuits, and quantum neural networks.

Driving Enhanced Quantum Sensing in Partially Accessible Many-Body Systems

The ground-state criticality of many-body systems is a resource for quantum-enhanced sensing, namely, the Heisenberg precision limit, provided that one has access to the whole system. We show that, for partial accessibility, the sensing capabilities of a block of spins in the ground state reduces to the sub-Heisenberg limit. To compensate for this, we drive the Hamiltonian periodically and use a local steady state for quantum sensing. Remarkably, the steady-state sensing shows a significant enhancement in precision compared to the ground state and even achieves super-Heisenberg scaling for low frequencies. The origin of this precision enhancement is related to the closing of the Floquet quasienergy gap. It is in close correspondence with the vanishing of the energy gap at criticality for ground-state sensing with global accessibility. The proposal is general to all the integrable models and can be implemented on existing quantum devices.

Dynamic Framework for Criticality-Enhanced Quantum Sensing

Quantum criticality, as a fascinating quantum phenomenon, may provide significant advantages for quantum sensing. Here we propose a dynamic framework for quantum sensing with a family of Hamiltonians that undergo quantum phase transitions (QPTs). By giving the formalism of the quantum Fisher information (QFI) for quantum sensing based on critical quantum dynamics, we demonstrate its divergent feature when approaching the critical point. We illustrate the basic principle and the details of experimental implementation using quantum Rabi model. The framework is applicable to a variety of examples and does not rely on the stringent requirement for particular state preparation or adiabatic evolution. It is expected to provide a route towards the implementation of criticality-enhanced quantum sensing.

Ground-state properties of the one-dimensional transverse Ising model in a longitudinal magnetic field

The critical properties of the one-dimensional spin-1/2 transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility was used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J_{2}, we calculated the fidelity susceptibility as a function of the transverse field B_{x} and the longitudinal field B_{z}. The phase diagram in the (B_{x},B_{z})-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For B_{x}<1, we observed an antiferromagnetic phase for small values of B_{z} and a paramagnetic phase for large values of B_{z}. For B_{x}>1 and low B_{z}, we found a disordered phase that undergoes a second-order phase transition to a paramagnetic phase for large values of B_{z}.

Quantum fidelity approach to the ground-state properties of the one-dimensional axial next-nearest-neighbor Ising model in a transverse field

In this work we analyze the ground-state properties of the s=1/2 one-dimensional axial next-nearest-neighbor Ising model in a transverse field using the quantum fidelity approach. We numerically determined the fidelity susceptibility as a function of the transverse field B_{x} and the strength of the next-nearest-neighbor interaction J_{2}, for systems of up to 24 spins. We also examine the ground-state vector with respect to the spatial ordering of the spins. The ground-state phase diagram shows ferromagnetic, floating, and 〈2,2〉 phases, and we predict an infinite number of modulated phases in the thermodynamic limit (L→∞). Paramagnetism only occurs for larger magnetic fields. The transition lines separating the modulated phases seem to be of second order, whereas the line between the floating and the 〈2,2〉 phases is possibly of first order.

Applications of fidelity measures to complex quantum systems

We revisit fidelity as a measure for the stability and the complexity of the quantum motion of single-and many-body systems. Within the context of cold atoms, we present an overview of applications of two fidelities, which we call static and dynamical fidelity, respectively. The static fidelity applies to quantum problems which can be diagonalized since it is defined via the eigenfunctions. In particular, we show that the static fidelity is a highly effective practical detector of avoided crossings characterizing the complexity of the systems and their evolutions. The dynamical fidelity is defined via the time-dependent wave functions. Focusing on the quantum kicked rotor system, we highlight a few practical applications of fidelity measurements in order to better understand the large variety of dynamical regimes of this paradigm of a low-dimensional system with mixed regular-chaotic phase space.

One- and two-dimensional quantum models: Quenches and the scaling of irreversible entropy

Using the scaling relation of the ground state quantum fidelity, we propose the most generic scaling relations of the irreversible work (the residual energy) of a closed quantum system at absolute zero temperature when one of the parameters of its Hamiltonian is suddenly changed. We consider two extreme limits: the heat susceptibility limit and the thermodynamic limit. It is argued that the irreversible entropy generated for a thermal quench at low enough temperatures when the system is initially in a Gibbs state is likely to show a similar scaling behavior. To illustrate this proposition, we consider zero-temperature and thermal quenches in one-dimensional (1D) and 2D Dirac Hamiltonians where the exact estimation of the irreversible work and the irreversible entropy is possible. Exploiting these exact results, we then establish the following. (i) The irreversible work at zero temperature shows an appropriate scaling in the thermodynamic limit. (ii) The scaling of the irreversible work in the 1D Dirac model at zero temperature shows logarithmic corrections to the scaling, which is a signature of a marginal situation. (iii) Remarkably, the logarithmic corrections do indeed appear in the scaling of the entropy generated if the temperature is low enough while they disappear for high temperatures. For the 2D model, no such logarithmic correction is found to appear.

Generalized fidelity susceptibility at phase transitions

In the present work, we investigate the intrinsic relation between quantum fidelity susceptibility (QFS) and the dynamical structure factor. We give a concise proof of the QFS beyond the perturbation theory. With the QFS in the Lehmann representation, we point out that the QFS is actually the negative-two-power moment of dynamical structure factor and illuminate the inherent relation between physical quantities in the linear response theory. Moreover, we discuss the generalized fidelity susceptibility (GFS) of any quantum relevant operator, that may not be coupled to the driving parameter, and present similar scaling behaviors. Finally, we demonstrate that the QFS cannot capture the fourth-order quantum phase transition in a spin-1/2 anisotropic XY chain in the transverse alternating field, while a lower-order GFS can seize the criticalities.

A quantum fidelity study of the anisotropic next-nearest-neighbour triangular lattice Heisenberg model

Ground- and excited-state quantum fidelities in combination with generalized quantum fidelity susceptibilites, obtained from exact diagonalizations, are used to explore the phase diagram of the anisotropic next-nearest-neighbour triangular Heisenberg model. Specifically, the J'-J2 plane of this model, which connects the J1-J2 chain and the anisotropic triangular lattice Heisenberg model, is explored using these quantities. Through the use of a quantum fidelity associated with the first excited-state, in addition to the conventional ground-state fidelity, the BKT-type transition and Majumdar-Ghosh point of the J1-J2 chain (J'=0) are found to extend into the J'-J2 plane and connect with points on the J2=0 axis thereby forming bounded regions in the phase diagram. These bounded regions are then explored through the generalized quantum fidelity susceptibilities χρ, χ₁₂₀°, χD and χCAF which are associated with the spin stiffness, 120° spiral order parameter, dimer order parameter and collinear antiferromagnetic order parameter respectively. These quantities are believed to be extremely sensitive to the underlying phase and are thus well suited for finite-size studies. Analysis of the fidelity susceptibilities suggests that the J', J2≪J phase of the anisotropic triangular model is either a collinear antiferromagnet or possibly a gapless disordered phase that is directly connected to the Luttinger phase of the J1-J2 chain. Furthermore, the outer region is dominated by incommensurate spiral physics as well as dimer order.

Three-site interacting spin chain in a staggered field: fidelity versus Loschmidt echo

We study the the ground state fidelity and the ground state Loschmidt echo of a three-site interacting XX chain in presence of a staggered field which exhibits special types of quantum phase transitions due to change in the topology of the Fermi surface, apart from quantum phase transitions from gapped to gapless phases. We find that, on one hand, the fidelity is able to detect only the boundaries separating the gapped from the gapless phase; it is completely insensitive to the phase transition from the two Fermi points region to the four Fermi points region lying within this gapless phase. On the other hand, the Loschmidt echo shows a dip only at a special point in the entire phase diagram and hence fails to detect any quantum phase transition associated with the present model. We provide appropriate arguments in support of this anomalous behavior.

Criticalities of the transverse- and longitudinal-field fidelity susceptibilities for the d=2 quantum Ising model

The inner product between the ground-state eigenvectors with proximate interaction parameters, namely, the fidelity, plays a significant role in the quantum dynamics. In this paper, the critical behaviors of the transverse- and longitudinal-field fidelity susceptibilities for the d=2 quantum (transverse-field) Ising model are investigated by means of the numerical-diagonalization method; the former susceptibility has been investigated rather extensively. The critical exponents for these fidelity susceptibilities are estimated as α(F)((t))=0.752(24) and α(F)((h))=1.81(13), respectively. These indices are independent, and suffice for obtaining conventional critical indices such as ν=0.624(12) and γ=1.19(13).

Quantum phases of dimerized and frustrated Heisenberg spin chains with s = 1/2, 1 and 3/2: an entanglement entropy and fidelity study

We study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with frustration (next-nearest-neighbor interaction J2) and dimerization (δ). In particular, we analyze the behaviors of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2-δ plane. All the calculations in this work are based on numerical exact diagonalizations of finite systems.

Assisted finite-rate adiabatic passage across a quantum critical point: exact solution for the quantum Ising model

The dynamics of a quantum phase transition is inextricably woven with the formation of excitations, as a result of critical slowing down in the neighborhood of the critical point. We design a transitionless quantum driving through a quantum critical point, allowing one to access the ground state of the broken-symmetry phase by a finite-rate quench of the control parameter. The method is illustrated in the one-dimensional quantum Ising model in a transverse field. Driving through the critical point is assisted by an auxiliary Hamiltonian, for which the interplay between the range of the interaction and the modes where excitations are suppressed is elucidated.

Lower and upper bounds on the fidelity susceptibility

We derive upper and lower bounds on the fidelity susceptibility in terms of macroscopic thermodynamical quantities, such as the susceptibilities and thermal average values. The quality of the bounds is checked against the exact expressions for a single spin in an external magnetic field. Their usefulness is illustrated by two examples of many-particle models which are exactly solved in the thermodynamic limit: the Dicke superradiance model and the single-impurity Kondo model. It is shown that, when divergent behavior is considered, the fidelity susceptibility and the thermodynamic susceptibility are equivalent for a large class of models exhibiting critical behavior.

Finite-temperature fidelity susceptibility for one-dimensional quantum systems

We calculate the fidelity susceptibility χf for the Luttinger model and show that there is a universal contribution linear in temperature T (or inverse length 1/L). Furthermore, we develop an algorithm--based on a lattice path integral approach--to calculate the fidelity F(T) in the thermodynamic limit for one-dimensional quantum systems. We check the Luttinger model predictions by calculating χf(T) analytically for free spinless fermions and numerically for the XXZ chain. Finally, we study χf at the two phase transitions in this model.