Quantum Ising Chain Research Articles

Fixing detailed balance in ancilla-based dissipative state engineering

Dissipative state engineering is a general term for a protocol which prepares the ground state of a complex many-body Hamiltonian using engineered dissipation or engineered environments. Recently, it was shown that a version of this protocol, where the engineered environment consists of one or more dissipative qubit ancillas tuned to be resonant with the low-energy transitions of a many-body system, resulted in the combined system evolving to reasonable approximation to the ground state. This potentially broadens the applicability of the method beyond nonfrustrated systems, to which it was previously restricted. Here we argue that this approach has an intrinsic limitation because the ancillas, seen as an effective bath by the system in the weak-coupling limit, do not give the detailed balance expected for a true zero-temperature environment. Our argument is based on the study of a similar approach employing linear coupling to bosonic ancillas. We explore overcoming this limitation using a recently developed open quantum systems technique called pseudomodes. With a simple example model of a one-dimensional quantum Ising chain, we show that detailed balance can be fixed, and a more accurate estimation of the ground state obtained, at the cost of two additional unphysical dissipative modes and the extrapolation error of implementing those modes in physical systems. Published by the American Physical Society 2024

Controlling Energy Storage Crossing Quantum Phase Transitions in an Integrable Spin Quantum Battery.

We investigate the performance of a one-dimensional dimerized XY chain as a spin quantum battery. Such integrable model shows a rich quantum phase diagram that emerges through a mapping of the spins onto auxiliary fermionic degrees of freedom. We consider a charging protocol relying on the double quench of an internal parameter, namely the strength of the dimerization, and address the energy stored in the systems. We observe three distinct regimes, depending on the timescale characterizing the duration of the charging: a short-time regime related to the dynamics of the single dimers, a long-time regime related to the recurrence time of the system at finite size, and a thermodynamic limit time regime. In the latter, the energy stored is almost unaffected by the charging time and the precise values of the charging parameters, provided the quench crosses a quantum phase transition. Finally, we analytically prove that the three-timescale behavior and the strong dependence of the energy stored on the quantum phase diagram also hold in the quantum Ising chain in a transverse field. Our results can play a relevant role in the design of stable solid-state quantum batteries.

Noninvertible Symmetries Act Locally by Quantum Operations.

Noninvertible symmetries of quantum field theories and many-body systems generalize the concept of symmetries by allowing noninvertible operations in addition to more ordinary invertible ones described by groups. The aim of this Letter is to point out that these noninvertible symmetries act on local operators by quantum operations, i.e., completely positive maps between density matrices, which form a natural class of operations containing both unitary evolutions and measurements and play an important role in quantum information theory. This observation will be illustrated by the Kramers-Wannier duality of the one-dimensional quantum Ising chain, which is a prototypical example of noninvertible symmetry operations.

Measurement-induced entanglement transition in chaotic quantum Ising chain

Measurement-induced entanglement transition in chaotic quantum Ising chain

Frustrating Quantum Batteries

We propose to use a quantum spin chain as a device to store and release energy coherently and we investigate the interplay between its internal correlations and outside decoherence. We employ the quantum Ising chain in a transverse field and our charging protocol consists of a sudden global quantum quench in the external field to take the system out of equilibrium. Interactions with the environment and decoherence phenomena can dissipate part of the work that the chain can supply after being charged, measured by the ergotropy. We find that overall, the system shows remarkably better performance, in terms of resilience, charging time, and energy storage, when topological frustration is introduced by setting antiferromagnetic interactions with an odd number of sites and periodic boundary conditions. Moreover, we show that in a simple discharging protocol to an external spin, only the frustrated chain can transfer work and not just heat. Published by the American Physical Society 2024

The quantum Ising chain for beginners

We present here various techniques to work with clean and disordered quantum Ising chains, for the benefit of students and non-experts. Starting from the Jordan-Wigner transformation, which maps spin-1/2 systems into fermionic ones, we review some of the basic approaches to deal with the superconducting correlations that naturally emerge in this context. In particular, we analyze the form of the ground state and excitations of the model, relating them to the symmetry-breaking physics, and illustrate aspects connected to calculating dynamical quantities, thermal averages, correlation functions, and entanglement entropy. A few problems provide simple applications of the techniques.

Towards Adiabatic Quantum Computing Using Compressed Quantum Circuits

We describe tensor network algorithms to optimize quantum circuits for adiabatic quantum computing. To suppress diabatic transitions, we include counterdiabatic driving in the optimization and utilize variational matrix product operators to represent adiabatic gauge potentials. Traditionally, Trotter product formulas are used to turn adiabatic time evolution into quantum circuits and the addition of counterdiabatic driving increases the circuit depth per time step. Instead, we classically optimize a parameterized quantum circuit of fixed depth to simultaneously capture adiabatic evolution together with counterdiabatic driving over many time steps. The methods are applied to the ground-state preparation of quantum Ising chains with transverse and longitudinal fields. We show that the classically optimized circuits can significantly outperform Trotter product formulas. Additionally, we discuss how the approach can be used for combinatorial optimization. Published by the American Physical Society 2024

Dynamical quantum phase transitions following a noisy quench

We study how time-dependent energy fluctuations impact the dynamical quantum phase transitions (DQPTs) following a noisy ramped quench of the transverse magnetic field in a quantum Ising chain. By numerically solving the stochastic Schrödinger equation of the mode-decoupled fermionic Hamiltonian of the problem, we identify two generic scenarios: Depending on the amplitude of the noise and the rate of the ramp, the expected periodic sequence of noiseless DQPTs may either be uniformly shifted in time or else replaced by a disarray of closely spaced DQPTs. Guided by an exact noise master equation, we trace the phenomenon to the interplay between noise-induced excitations which accumulate during the quench and the near-adiabatic dynamics of the massive modes of the system. Our analysis generalizes to any one-dimensional fermionic two-band model subject to a noisy quench. Published by the American Physical Society 2024

Parent Hamiltonian Reconstruction via Inverse Quantum Annealing.

Finding a local Hamiltonian H[over ^] that has a given many-body wave function |ψ⟩ as its ground state, i.e., a parent Hamiltonian, is a challenge of fundamental importance in quantum technologies. Here we introduce a numerical method, inspired by quantum annealing, that efficiently performs this task through an artificial inverse dynamics: a slow deformation of the states |ψ(λ(t))⟩, starting from a simple state |ψ_{0}⟩ with a known H[over ^]_{0}, generates an adiabatic evolution of the corresponding Hamiltonian. We name this approach inverse quantum annealing. The method, implemented through a projection onto a set of local operators, only requires the knowledge of local expectation values, and, for long annealing times, leads to an approximate parent Hamiltonian whose degree of locality depends on the correlations built up by the states |ψ(λ)⟩. We illustrate the method on two paradigmatic models: the Kitaev fermionic chain and a quantum Ising chain in longitudinal and transverse fields.

Phase diagram near the quantum critical point in Schwinger model at θ = π: analogy with quantum Ising chain

Abstract The Schwinger model, 1D quantum electrodynamics, has CP symmetry at θ = π due to the topological nature of the θ term. At zero temperature, it is known that as the fermion mass increases, the system undergoes a second-order phase transition to the CP broken phase, which belongs to the same universality class as the quantum Ising chain. In this paper, we obtain the phase diagram near the quantum critical point (QCP) in the temperature and fermion mass plane using first-principle Monte Carlo simulations, while avoiding the sign problem by using the lattice formulation of the bosonized Schwinger model. Specifically, we perform a detailed investigation of the correlation function of the electric field near the QCP and find that its asymptotic behavior can be described by the universal scaling function of the quantum Ising chain. This finding indicates the existence of three regions near the QCP, each characterized by a specific asymptotic form of the correlation length, and demonstrates that the CP symmetry is restored at any nonzero temperature, entirely analogous to the quantum Ising chain. The range of the scaling behavior is also examined and found to be particularly wide.

Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

When a time propagator eδtA for duration δt consists of two noncommuting parts A=X+Y, Trotterization approximately decomposes the propagator into a product of exponentials of X and Y. Various Trotterization formulas have been utilized in quantum and classical computers, but much less is known for the Trotterization with the time-dependent generator A(t). Here, for A(t) given by the sum of two operators X and Y with time-dependent coefficients A(t)=x(t)X+y(t)Y, we develop a systematic approach to derive high-order Trotterization formulas with minimum possible exponentials. In particular, we obtain fourth-order and sixth-order Trotterization formulas involving seven and fifteen exponentials, respectively, which are no more than those for time-independent generators. We also construct another fourth-order formula consisting of nine exponentials having a smaller error coefficient. Finally, we numerically benchmark the fourth-order formulas in a Hamiltonian simulation for a quantum Ising chain, showing that the 9-exponential formula accompanies smaller errors per local quantum gate than the well-known Suzuki formula.

Multipartite entanglement in the measurement-induced phase transition of the quantum Ising chain

Multipartite entanglement in the measurement-induced phase transition of the quantum Ising chain

Monte Carlo matrix-product-state approach to the false vacuum decay in the monitored quantum Ising chain

In this work we characterize the false vacuum decay in the ferromagnetic quantum Ising chain with a weak longitudinal field subject to continuous monitoring of the local magnetization. Initializing the system in a metastable state, the false vacuum, we study the competition between coherent dynamics, which tends to create resonant bubbles of the true vacuum, and measurements which induce heating and reduce the amount of quantum correlations. To this end we exploit a numerical approach based on the combination of matrix product states with stochastic quantum trajectories which allows for the simulation of the trajectory-resolved non-equilibrium dynamics of interacting many-body systems in the presence of continuous measurements. We show how the presence of measurements affects the false vacuum decay: At short times the departure from the local minimum is accelerated while at long times the system thermalizes to an infinite-temperature incoherent mixture. For large measurement rates the system enters a quantum Zeno regime. The false vacuum decay and the thermalization physics are characterized in terms of the magnetization, connected correlation function, and the trajectory-resolved entanglement entropy.

Full counting statistics as probe of measurement-induced transitions in the quantum Ising chain

Non-equilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce different non-equilibrium regimes and an abrupt change in the scaling-law of the bipartite entanglement entropy. However, our understanding of how these regimes appear{, how they affect the statistics of local quantities and}, finally whether they survive in the thermodynamic limit, is much less established. Here we investigate measurement-induced phase transitions in the Quantum Ising chain coupled to a monitoring environment. In particular we show that local projective measurements induce a quantitative modification of the out-of-equilibrium probability distribution function of the local magnetization. Starting from a GHZ state, the relaxation of the paramagnetic and the ferromagnetic order is analysed. In particular we describe how the probability distributions associated to them show different behaviour depending on the measurement rate.

Kibble-Zurek scaling in the quantum Ising chain with a time-periodic perturbation

Kibble-Zurek scaling in the quantum Ising chain with a time-periodic perturbation

Exceptional points in the Baxter-Fendley free parafermion model

Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. Free parafermions are a generalisation of this idea to \mathbb{Z}_NℤN-symmetric clock models. In 1989 Baxter discovered a non-Hermitian but \mathcal{PT}𝒫𝒯-symmetric model directly generalising the Ising chain, which was later described by Fendley as a free parafermion spectrum. By extending the model’s magnetic field parameter to the complex plane, it is shown that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general \mathcal{PT}𝒫𝒯-symmetric at these exceptional points, their proximity can have a profound impact on the model on the \mathcal{PT}𝒫𝒯-symmetric real line. Furthermore, in certain cases an exceptional point may appear on the real line (with negative field).

Self-correcting quantum many-body control using reinforcement learning with tensor networks

Quantum many-body control is a central milestone en route to harnessing quantum technologies. However, the exponential growth of the Hilbert space dimension with the number of qubits makes it challenging to classically simulate quantum many-body systems and, consequently, to devise reliable and robust optimal control protocols. Here we present a framework for efficiently controlling quantum many-body systems based on reinforcement learning (RL). We tackle the quantum-control problem by leveraging matrix product states (1) for representing the many-body state and (2) as part of the trainable machine learning architecture for our RL agent. The framework is applied to prepare ground states of the quantum Ising chain, including states in the critical region. It allows us to control systems far larger than neural-network-only architectures permit, while retaining the advantages of deep learning algorithms, such as generalizability and trainable robustness to noise. In particular, we demonstrate that RL agents are capable of finding universal controls, of learning how to optimally steer previously unseen many-body states and of adapting control protocols on the fly when the quantum dynamics is subject to stochastic perturbations. Furthermore, we map our RL framework to a hybrid quantum–classical algorithm that can be performed on noisy intermediate-scale quantum devices and test it under the presence of experimentally relevant sources of noise.

Nonlocality and entanglement in measured critical quantum Ising chains

We study the effects of measurements, performed with a finite density in space, on the ground state of the one-dimensional transverse-field Ising model at criticality. Local degrees of freedom in critical states exhibit long-range entanglement, and as a result, local measurements can have highly nonlocal effects. Our analytical investigation of correlations and entanglement in the ensemble of measured states is based on properties of the Ising conformal field theory (CFT), where measurements appear as $(1+0)$-dimensional defects in the $(1+1)$-dimensional Euclidean spacetime. So that we can verify our predictions using large-scale free-fermion numerics, we restrict ourselves to parity-symmetric measurements. To describe their averaged effects analytically we use a replica approach, and we show that the defect arising in the replica theory is an irrelevant perturbation to the Ising CFT. Strikingly, the asymptotic scalings of averaged correlations and entanglement entropy are therefore unchanged relative to the ground state. In contrast, the defect generated by postselecting on the most likely measurement outcomes is exactly marginal. We then find that the exponent governing postmeasurement order parameter correlations, as well as the ``effective central charge'' governing the scaling of entanglement entropy, vary continuously with the density of measurements in space. Our work establishes connections between the effects of measurements on many-body quantum states and of physical defects on low-energy equilibrium properties.

Ising analogs of quantum spin chains with multispin interactions

A new family of free fermionic quantum spin chains with multispin interactions was recently introduced. Here we show that it is possible to build standard quantum Ising chains---but with inhomogeneous couplings---which have the same spectra as the novel spin chains with multispin interactions. The Ising models are obtained by associating an antisymmetric tridiagonal matrix to the polynomials that characterize the quasienergies of the system via a modified Euclidean algorithm. For the simplest nontrivial case, corresponding to the Fendley model, the phase diagram of the inhomogeneous Ising model is investigated numerically. It is characterized by gapped phases separated by critical lines with order-disorder transitions depending on the parity of the total number of energy density operators in the Hamiltonian.

Two-dimensional coherent spectrum of interacting spinons from matrix product states

We compute numerically the second- and third-order nonlinear magnetic susceptibilities of an Ising ladder model in the context of two-dimensional coherent spectroscopy by using the infinite time-evolving block decimation method. The Ising ladder model couples a quantum Ising chain to a bath of polarized spins, thereby effecting the decay of spinon excitations. We show that its third-order susceptibility contains a robust spinon echo signal in the weak-coupling regime, which appears in the two-dimensional coherent spectrum as a highly anisotropic peak on the frequency plane. The spinon echo peak reveals the dynamical properties of the spinons. In particular, the spectral peak corresponding to the high-energy spinons, which couple to the bath, is suppressed with increasing coupling, whereas those corresponding to low-energy spinons do not show any significant changes.