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Quantum Annealing (QA) relies on mixing two Hamiltonian terms, a simple driver and a complex problem Hamiltonian, in a linear combination. The time-dependent schedule for this mixing is often taken to be linear in time: improving on this linear choice is known to be essential and has proven to be difficult. Here, we present different techniques for improving on the linear-schedule QA along two directions, conceptually distinct but leading to similar outcomes: 1) the first approach consists of constructing a Trotter-digitized QA (dQA) with schedules parameterized in terms of Fourier modes or Chebyshev polynomials, inspired by the Chopped Random Basis algorithm for optimal control in continuous time; 2) the second approach is technically a Quantum Approximate Optimization Algorithm (QAOA), whose solutions are found iteratively using linear interpolation or expansion in Fourier modes. Both approaches emphasize finding smooth optimal schedule parameters, ultimately leading to hybrid quantum-classical variational algorithms of the alternating Hamiltonian Ansatz type. We apply these techniques to MaxCut problems on weighted 3-regular graphs with N = 14 sites, focusing on hard instances that exhibit a small spectral gap, for which a standard linear-schedule QA performs poorly. We characterize the physics behind the optimal protocols for both the dQA and QAOA approaches, discovering shortcuts to adiabaticity-like dynamics. Furthermore, we study the transferability of such smooth solutions among hard instances of MaxCut at different circuit depths. Finally, we show that the smoothness pattern of these protocols obtained in a digital setting enables us to adapt them to continuous-time evolution, contrarily to generic non-smooth solutions. This procedure results in an optimized QA schedule that is implementable on analog devices.

Finding a local Hamiltonian H^ that has a given many-body wave function |ψ.

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.

The Kuramoto model serves as a paradigm for describing spontaneous synchronization in a system of classical interacting rotors. In this paper, we extend this model to the quantum domain by coupling quantum interacting rotors to external baths following the Caldeira-Leggett approach. Studying the mean-field model in the overdamped limit using Feynman-Vernon theory, we show how quantum mechanics modifies the phase diagram. Specifically, we demonstrate that quantum fluctuations hinder the emergence of synchronization, albeit not entirely suppressing it. We examine the phase transition into the synchronized phase at various temperatures, revealing that classical results are recovered at high temperatures while a quantum phase transition occurs at zero temperature. Additionally, we derive an analytical expression for the critical coupling, highlighting its dependence on the model parameters, and examine the differences between classical and quantum behavior.

We show that a two-body Jastrow wave function is able to capture the ground-state properties of the S=1 Heisenberg chain with nearest-neighbor superexchange J and single-ion anisotropy term D, in both the topological and large-D phases (with D/J≥0). Here, the optimized Jastrow pseudopotential assumes a very simple form in Fourier space, i.e., vq≈1/q2, which is able to give rise to a finite string-order parameter in the topological regime. The results are analyzed by using an exact mapping from the quantum expectation values over the variational state to the classical partition function of the one-dimensional Coulomb gas of particles with charge q=±1. Here, two phases are present at low temperatures: the first one is a diluted gas of dipoles (bound neutral pairs of particles), which are randomly oriented (describing the large-D phase); the other one is a dense liquid of dipoles, which are aligned thanks to the residual dipole-dipole interactions (describing the topological phase, with the finite string order being related to the dipole alignment). Our results provide an insightful interpretation of the ground-state nature of the spin-1 antiferromagnetic Heisenberg model.

Here, we focus on the problem of minimizing complex classical cost functions associated with prototypical discrete neural networks, specifically the paradigmatic Hopfield model and binary perceptron. We show that the adiabatic time evolution of QA can be efficiently represented as a suitable Tensor Network. This representation allows for simple classical simulations, well-beyond small sizes amenable to exact diagonalization techniques. We show that the optimized state, expressed as a Matrix Product State (MPS), can be recast into a Quantum Circuit, whose depth scales only linearly with the system size and quadratically with the MPS bond dimension. This may represent a valuable starting point allowing for further circuit optimization on near-term quantum devices.

A large ongoing research effort focuses on variational quantum algorithms (VQAs), representing leading candidates to achieve computational speed-ups on current quantum devices. The scalability of VQAs to a large number of qubits, beyond the simulation capabilities of classical computers, is still debated. Two major hurdles are the proliferation of low-quality variational local minima, and the exponential vanishing of gradients in the cost-function landscape, a phenomenon referred to as barren plateaus. In this work, we show that by employing iterative search schemes, one can effectively prepare the ground state of paradigmatic quantum many-body models, also circumventing the barren plateau phenomenon. This is accomplished by leveraging the transferability to larger system sizes of a class of iterative solutions, displaying an intrinsic smoothness of the variational parameters, a result that does not extend to other solutions found via random-start local optimization. Our scheme could be directly tested on near-term quantum devices, running a refinement optimization in a favorable local landscape with nonvanishing gradients.

We consider the two-dimensional quantum Ising model, in absence of disorder, subject to periodic imperfect global spin flips. We show by a combination of exact diagonalization and tensor-network methods that the system can sustain a spontaneously broken discrete time-translation symmetry. Employing careful scaling analysis, we show the feasibility of a two-dimensional discrete time-crystal (DTC) prethermal phase. Despite an unbounded energy pumped into the system, in the high-frequency limit, a well-defined effective Hamiltonian controls a finite-temperature intermediate regime, wherein local time averages are described by thermal averages. As a consequence, the long-lived stability of the DTC relies on the existence of a long-range ordered phase at finite temperature. Interestingly, even for large deviations from the perfect spin flip, we observe a nonperturbative change in the decay rate of the order parameter, which is related to the long-lived stability of the magnetic domains in 2D.

We study the dynamics of the quenched Anderson model at finite temperature using matrix product states (MPSs). Exploiting a chain mapping for the electron bath, we investigate the entanglement structure in the MPS for various orderings of the two chains, which emerge from the thermofield transformation employed to deal with nonzero temperature. We show that merging both chains can significantly lower the entanglement at finite temperatures as compared to an intuitive nearest-neighbor implementation of the Hamiltonian. Analyzing the population of the free bath modes - possible when simulating the full dynamics of impurity plus bath - we find clear signatures of the Kondo effect in the quench dynamics.

We propose an efficient algorithm to numerically solve Anderson impurity problems using matrix product states. By introducing a modified chain mapping we obtain significantly lower entanglement, as compared to all previous attempts, while keeping the short-range nature of the couplings. Employing a thermofield transformation, our approach naturally extends to finite temperatures, with applications to dynamical mean field theory, nonequilibrium dynamics, and quantum transport.

We show that the quantum approximate optimization algorithm (QAOA) can construct, with polynomially scaling resources, the ground state of the fully connected p-spin Ising ferromagnet, a problem that notoriously poses severe difficulties to a vanilla quantum annealing (QA) approach due to the exponentially small gaps encountered at first-order phase transition for p≥3. For a target ground state at arbitrary transverse field, we find that an appropriate QAOA parameter initialization is necessary to achieve good performance of the algorithm when the number of variational parameters 2P is much smaller than the system size N because of the large number of suboptimal local minima. Instead, when P exceeds a critical value PN∗N, the structure of the parameter space simplifies, as all minima become degenerate. This allows achieving the ground state with perfect fidelity with a number of parameters scaling extensively with N and with resources scaling polynomially with N.

We propose a reinforcement learning (RL) scheme for feedback quantum control within the quantum approximate optimization algorithm (QAOA). We reformulate the QAOA variational minimization as a learning task, where an RL agent chooses the control parameters for the unitaries, given partial information on the system. Such an RL scheme finds a policy converging to the optimal adiabatic solution of the quantum Ising chain that can also be successfully transferred between systems with different sizes, even in the presence of disorder. This allows for immediate experimental verification of our proposal on more complicated models: The RL agent is trained on a small control system, simulated on classical hardware, and then tested on a larger physical sample.

We investigate the effect of dissipation from a thermal environment on topological pumping in the periodically-driven Rice-Mele model. We report that dissipation can improve the robustness of pumping quantisation in a regime of finite driving frequencies. Specifically, in this regime, lowerature dissipative dynamics can lead to a pumped charge that is much closer to the Thouless quantised value, compared to a coherent evolution. We understand this effect in the Floquet framework: dissipation increases the population of a Floquet band which shows a topological winding, where pumping is essentially quantised. This finding is a step towards understanding a potentially very useful resource to exploit in experiments, where dissipation effects are unavoidable. We consider small couplings with the environment and we use a Bloch-Redfield quantum master equation approach for our numerics: comparing these results with an exact MPS numerical treatment we find that the quantum master equation works very well also at low temperature, a quite remarkable fact.

We revisit here the Kibble-Zurek mechanism for superfluid bosons slowly driven across the transition toward the Mott-insulating phase. By means of a combination of the time-dependent variational principle and a tree-tensor network, we characterize the current flowing during annealing in a ring-shaped one-dimensional Bose-Hubbard model with artificial classical gauge field on up to 32 lattice sites. We find that the superfluid current shows, after an initial decrease, persistent oscillations which survive even when the system is well inside the Mott insulating phase. We demonstrate that the amplitude of such oscillations is connected to the residual energy, characterizing the creation of defects while crossing the quantum critical point, while their frequency matches the spectral gap in the Mott insulating phase. Our predictions can be verified in future atomtronics experiments with neutral atoms in ring-shaped traps. We believe that the proposed setup provides an interesting but simple platform to study the nonequilibrium quantum dynamics of persistent currents experimentally.

We investigate the effects of disorder on a periodically driven one-dimensional model displaying quantized topological transport. We show that, while instantaneous eigenstates are necessarily Anderson localized, the periodic driving plays a fundamental role in delocalizing Floquet states over the whole system, henceforth allowing for a steady-state nearly quantized current. Remarkably, this is linked to a localization-delocalization transition in the Floquet states at strong disorder, which occurs for periodic driving corresponding to a nontrivial loop in the parameter space. As a consequence, the Floquet spectrum becomes continuous in the delocalized phase, in contrast with a pure-point instantaneous spectrum.

We present a study of the digitized quantum annealing protocol proposed by R. Barends et al. [Nature (London) 534, 222 (2016)NATUAS0028-083610.1038/nature17658]. Our analysis, performed on the benchmark case of a transverse Ising chain problem, shows that the algorithm has a well-defined optimal working point for the annealing time τPopt, scaling as τPopt∼P, where P is the number of digital Trotter steps, beyond which the residual energy error shoots up toward the value characteristic of the maximally disordered state. We present an analytical analysis for the translationally invariant transverse Ising chain case, but our numerical evidence suggests that this scenario is more general, surviving, for instance, the presence of disorder.

We show that a hierarchy of topological phases in one dimension - a topological Devil's staircase - can emerge at fractional filling fractions in interacting systems, whose single-particle band structure describes a topological or a crystalline topological insulator. Focusing on a specific example in the BDI class, we present a field-theoretical argument based on bosonization that indicates how the system, as a function of the filling fraction, hosts a series of density waves. Subsequently, based on a numerical investigation of the low-lying energy spectrum, Wilczek-Zee phases, and entanglement spectra, we show that they are symmetry protected topological phases. In sharp contrast to the non-interacting limit, these topological density waves do not follow the bulk-edge correspondence, as their edge modes are gapped. We then discuss how these results are immediately applicable to models in the AIII class, and to crystalline topological insulators protected by inversion symmetry. Our findings are immediately relevant to cold atom experiments with alkaline-earth atoms in optical lattices, where the band structure properties we exploit have been recently realized.

We discuss and qualify the connection between two separate phenomena in the physics of nanoscale friction, general in nature and relevant to experiments. The first is thermally assisted lubricity (TAL), i.e., the low-velocity regime where a nanosized dry slider exhibits a viscouslike friction despite corrugations that would otherwise imply hard stick-slip friction. The second is the occurrence of negative dissipated work (NDW) events in sampling the work probability distribution. The abundance, or scarcity due to insufficient sampling, of these NDW events implies experimental fulfillment or violation of the celebrated Jarzynski equality (JE) of nonequilibrium statistical mechanics. We show, both analytically and in simulations of the one-dimensional point slider Prandtl-Tomlinson model, that a general crossover can be individuated as the total frictional work per cycle crosses kBT. Below such crossover, the TAL regime holds, the dissipation is essentially linear, and the numerical validation for the JE is feasible (i.e., does not require an exponentially large sampling size). Above it, the dissipation profile departs from linearity and gains its hard stick-slip features, and the mandatory sampling for the JE becomes exponentially large. In addition, we derive a parameter-free formula expressing the linear velocity coefficient of viscous friction, correcting previous empirically parameterized expressions. With due caution, the connection between friction and work tails can be extended beyond a single degree of freedom to more complex sliders, thus inviting realistic crosscheck experiments. Of importance for experimental nanofriction will be the search for NDW tails in the sliding behavior of trapped cold ions, and alternatively checking for TAL in the sliding pattern of dragged colloid monolayers as well as in forced protein unwinding.

Simulated quantum annealing (SQA) is a classical computational strategy that emulates a quantum annealing (QA) dynamics through a path-integral Monte Carlo whose parameters are changed during the simulation. Here we apply SQA to the one-dimensional transverse field Ising chain, where previous works have shown that, in the presence of disorder, a coherent QA provides a quadratic speedup with respect to classical simulated annealing, with a density of Kibble-Zurek defects decaying as ρKZQA∼(log10τ)-2 as opposed to ρKZSA∼(log10τ)-1, τ being the total annealing time, while for the ordered case both give the same power law ρKZQA≈ρKZSA∼τ-1/2. We show that the dynamics of SQA, while correctly capturing the Kibble-Zurek scaling τ-1/2 for the ordered case, is unable to reproduce the QA dynamics in the disordered case at intermediate τ. We analyze and discuss several issues related to the choice of the Monte Carlo moves (local or global in space), the time-continuum limit needed to eliminate the Trotter-discretization error, and the long autocorrelation times shown by a local-in-space Monte Carlo dynamics for large disordered samples.

We investigate the use of variational wave functions that mimic stochastic recurrent neural networks, specifically, unrestricted Boltzmann machines, as guiding functions in projective quantum Monte Carlo (PQMC) simulations of quantum spin models. As a preliminary step, we investigate the accuracy of such unrestricted neural network states as variational Ansätze for the ground state of the ferromagnetic quantum Ising chain. We find that by optimizing just three variational parameters, independently on the system size, accurate ground-state energies are obtained, comparable to those previously obtained using restricted Boltzmann machines with few variational parameters per spin. Chiefly, we show that if one uses optimized unrestricted neural network states as guiding functions for importance sampling, the efficiency of the PQMC algorithms is greatly enhanced, drastically reducing the most relevant systematic bias, namely, the one due to the finite random-walker population. The scaling of the computational cost with the system size changes from the exponential scaling characteristic of PQMC simulations performed without importance sampling, to a polynomial scaling, apparently even at the ferromagnetic quantum critical point. The important role of the protocol chosen to sample hidden-spin configurations, in particular at the critical point, is analyzed. We discuss the implications of these findings for what concerns the problem of simulating adiabatic quantum optimization using stochastic algorithms on classical computers.