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In the current era of noisy intermediate-scale quantum (NISQ) technology, quantum devices present new avenues for addressing complex, real-world challenges including potentially NP-hard optimization problems. This work aims to showcase how the inherent noise in NISQ devices can be leveraged to solve such real-world problems effectively. Utilizing a D-Wave quantum annealer and IonQ's gate-based NISQ computers, we generate and analyze solutions for managing train traffic under stochastic disturbances. Our case study focuses on the Baltimore Light RailLink, which embodies the characteristics of both tramway and railway networks. We explore the feasibility of using NISQ technology to model the stochastic nature of disruptions in these transportation systems. Our research marks the inaugural application of both quantum computing paradigms to tramway and railway rescheduling, highlighting the potential of quantum noise as a beneficial resource in complex optimization scenarios.

We demonstrate the unique capabilities of the Wigner function, particularly in its positive and negative parts, for exploring the phase diagram of the spin$-(\frac{1}{2\!}-\!\frac{1}{2})$ and spin$-(\frac{1}{2}\!-\!1)$ Ising-Heisenberg chains. We highlight the advantages and limitations of the phase space approach in comparison with the entanglement concurrence in detecting phase boundaries. We establish that the equal angle slice approximation in the phase space is an effective method for capturing the essential features of the phase diagram, but falls short in accurately assessing the negativity of the Wigner function for the homogeneous spin$-(\frac{1}{2}\!-\!\frac{1}{2})$ Ising-Heisenberg chain. In contrast, we find for the inhomogeneous spin$-(\frac{1}{2}\!-\!1)$ chain that an integral over the entire phase space is necessary to accurately capture the phase diagram of the system. This distinction underscores the sensitivity of phase space methods to the homogeneity of the quantum system under consideration., Comment: 15 pages, 6 figures

It is a well-understood fact that the transport of excitations throughout a lattice is intimately governed by the underlying structures. Hence, it is only natural to recognize that also the dispersion of information has to depend on the lattice geometry. In the present work, we demonstrate that two-dimensional lattices described by the Bose-Hubbard model exhibit information scrambling for systems as little as two hexagons. However, we also find that the OTOC shows the exponential decay characteristic for quantum chaos only for a judicious choice of local observables. More generally, the OTOC is better described by Gaussian-exponential convolutions, which alludes to the close similarity of information scrambling and decoherence theory., Comment: 9 pages, 10 figures

In this work, we analyze the local certification of unitary quantum channels, which is a natural extension of quantum hypothesis testing. A particular case of a quantum channel operating on two systems corresponding to product states at the input, is considered. The goal is to minimize the probability of the type II error, given a specified maximum probability of the type I error, considering assistance through entanglement with auxiliary systems. Our result indicates connection of the local certification problem with a product numerical range of unitary matrices. We show that the optimal local strategy does not need usage of auxiliary systems and requires only single round of one-way classical communication. Moreover, we compare local and global certification strategies and show that typically local strategies are optimal, yet in some extremal cases, where global strategies make no errors, local ones may fail miserably. Finally, some application for local certification of von Neumann measurements are discussed as well.

Motivated by recent efforts to develop quantum computing for practical, industrial-scale challenges, we demonstrate the effectiveness of state-of-the-art hybrid (not necessarily quantum) solvers in addressing the business-centric optimization problem of scheduling Automatic Guided Vehicles (AGVs). Some solvers can already leverage noisy intermediate-scale quantum (NISQ) devices. In our study, we utilize D-Wave hybrid solvers that implement classical heuristics with potential assistance from a quantum processing unit. This hybrid methodology performs comparably to existing classical solvers. However, due to the proprietary nature of the software, the precise contribution of quantum computation remains unclear. Our analysis focuses on a practical, business-oriented scenario: scheduling AGVs within a factory constrained by limited space, simulating a realistic production setting. Our approach maps a realistic AGVs problem onto one reminiscent of railway scheduling and demonstrates that the AGVs problem is better suited to quantum computing than its railway counterpart, the latter being denser in terms of the average number of constraints per variable. The main idea here is to highlight the potential usefulness of a hybrid approach for handling AGVs scheduling problems of practical sizes. We show that a scenario involving up to 21 AGVs, significant due to possible deadlocks, can be efficiently addressed by a hybrid solver in seconds.

We investigate the computational efficiency and thermodynamic cost of the D-Wave quantum annealer under reverse-annealing with and without pausing. Our experimental results demonstrate that the combination of reverse-annealing and pausing leads to improved computational efficiency while minimizing the thermodynamic cost compared to reverse-annealing alone. Moreover, we find that the magnetic field has a positive impact on the performance of the quantum annealer during reverse-annealing but becomes detrimental when pausing is involved. Our results provide strategies for optimizing the performance and energy consumption of quantum annealing systems employing reverse-annealing protocols., Comment: 9 pages, 7 figures

Current generation quantum annealers have already proven to be successful problem-solvers. Yet, quantum annealing is still very much in its infancy, with suboptimal applicability. For instance, to date it is still an open question which annealing protocol causes the fewest diabatic excitations for a given eigenspectrum, and even whether there is a universally optimal strategy. Therefore, in this paper, we report analytical and numerical studies of the diabatic excitations arising from nonlinear protocols applied to the transverse field Ising chain, the exactly solvable model that serves as a quantum annealing playground. Our analysis focuses on several driving schemes that inhibit or facilitate the dynamic phases discussed in a previous work. Rather remarkably, we find that the paradigmatic Kibble-Zurek behavior can be suppressed with ``pauses'' in the evolution, both for crossing and for stopping at the quantum critical point of the system.

Finding the ground state of the Ising spin-glass is an important and challenging problem (NP-hard, in fact) in condensed matter physics. However, its applications spread far beyond physic due to its deep relation to various combinatorial optimization problems, such as travelling salesman or protein folding. Sophisticated and promising new methods for solving Ising instances rely on quantum resources. In particular, quantum annealing is a quantum computation paradigm, that is especially well suited for Quadratic Unconstrained Binary Optimization (QUBO). Nevertheless, commercially available quantum annealers (i.e., D-Wave) are prone to various errors, and their ability to find low energetic states (corresponding to solutions of superior quality) is limited. This naturally calls for a post-processing procedure to correct errors (capable of lowering the energy found by the annealer). As a proof-of-concept, this work combines the recent ideas revolving around the DIRAC architecture with the Chimera topology and applies them in a real-world setting as an error-correcting scheme for quantum annealers. Our preliminary results show how to correct states output by quantum annealers using reinforcement learning. Such an approach exhibits excellent scalability, as it can be trained on small instances and deployed for large ones. However, its performance on the chimera graph is still inferior to a typical algorithm one could incorporate in this context, e.g., simulated annealing.

Currently, existing quantum annealers have proven themselves as viable technology for the first practical applications in the noisy-intermediate-scale-quantum era. However, to fully exploit their capabilities, a comprehensive characterization of their finite-time excitations is instrumental. To this end, we develop a phase diagram for driven Ising chains, from which the scaling behavior of the excess work can be read off as a function of process duration and system size. "Fast" processes are well described by the Kibble-Zurek mechanism; "slow" processes are governed by effective Landau-Zener dynamics; and "very slow" processes can be approximated with adiabatic perturbation theory.

We are in the Noisy Intermediate-Scale Quantum (NISQ) devices' era, in which quantum hardware has become available for application in real-world problems. However, demonstrating the usefulness of such NISQ devices are still rare. In this work, we consider a practical railway dispatching problem: delay and conflict management on single-track railway lines. We examine the issue of train dispatching consequences caused by the arrival of an already delayed train to the network segment being considered. This problem is computationally hard and needs to be solved almost in real-time. We introduce a quadratic unconstrained binary optimization (QUBO) model of this problem, compatible with the emerging quantum annealing technology. The model's instances can be executed on present-day quantum annealers. As a proof-of-concept, we solve selected real-life problems from the Polish railway network using D-Wave quantum annealers. As a reference, we also provide solutions calculated with classical methods, including those relevant to the community (linear integer programming) and a sophisticated algorithm based on tensor networks for solving Ising instances. Our preliminary results illustrate the degree of difficulty of real-life railway instances for the current quantum annealing technology. Moreover, our analysis shows that the new generation of quantum annealers (the advantage system) perform much worse on those instances than its predecessor., Comment: arXiv admin note: substantial text overlap with arXiv:2010.08227

In this work, we consider a practical railway dispatching problem: delay and conflict management on a single-track railway line. We examine the issue of train dispatching consequences caused by the arrival of an already delayed train to the segment being considered. This problem is computationally hard and often needs to be solved timely. Here, we introduce a quadratic unconstrained binary optimization (QUBO) model of the problem in question, compatible with the emerging quantum annealing technology. The model's instances can be executed on present-day quantum annealers. As a proof-of-concept, we solve selected real-life problems from the Polish railway network using D-Wave quantum annealers. As a reference, we also provide solutions calculated with classical methods, including those relevant to the community (linear integer programming) and a sophisticated algorithm based on tensor networks for solving QUBO problems., Comment: 36 pages, 17 figures, secondrevision based on reviewers' comments

Recent years have witnessed an unprecedented increase in experiments and hybrid simulations involving quantum computers. In particular, quantum annealers. Although quantum supremacy has not been established thus far, there exist a plethora of algorithms promising to outperform classical computers in the near-term future. Here, we propose a parallel in time approach to simulate dynamical systems designed to be executed already on present-day quantum annealers. In essence, purely classical methods for solving dynamics systems are serial. Therefore, their parallelization is substantially limited. In the presented approach, however, the time evolution is rephrased as a ground--state search of a classical Ising model. Such a problem is solved intrinsically in parallel by quantum computers. The main idea is exemplified by simulating the Rabi oscillations generated by a two-level quantum system (i.e. qubit) experimentally., Comment: 6 pages, 4 figures

We demonstrate how to compute the low energy spectrum for small ($N\le 50$), but otherwise arbitrary, spin-glass instances using modern Graphics Processing Units or similar heterogeneous architecture. Our algorithm performs an exhaustive (i.e., brute-force) search of all possible configurations to select $S\ll 2^N$ lowest ones together with their corresponding energies. We mainly focus on the Ising model defined on an arbitrary graph. An open-source implementation based on CUDA Fortran and a suitable Python wrapper are provided. As opposed to heuristic approaches, ours is exact and thus can serve as a references point to benchmark other algorithms and hardware, including quantum and digital annealers. Our implementation offers unprecedented speed and efficiency already visible on commodity hardware. At the same time, it can be easily launched on professional, high-end graphics cards virtually at no extra effort. As a practical application, we employ it to demonstrate that the recent Matrix Product State based algorithm-despite its one-dimensional nature-can still accurately approximate the low energy spectrum of fully connected graphs of size $N$ approaching $50$.

We are at the verge of a new era, which will be dominated by Noisy Intermediate-Scale Quantum Devices. Prototypical examples for these new technologies are present-day quantum annealers. In the present work, we investigate to what extent static disorder generated by an external source of noise does not have to be detrimental, but can actually assist quantum annealers in achieving better performance. In particular, we analyze the graph coloring problem that can be solved on a sparse topology (i.e. chimera graph) via suitable embedding. We show that specifically tailored disorder can enhance the fidelity of the annealing process and thus increase the overall performance of the annealer., Comment: 7 pages, 5 figures

For a given Hamiltonian $H$ on a multipartite quantum system, one is interested in finding the energy $E_0$ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value $\lambda_{\rm min}^{\otimes}$ of $H$ among all product states. For several concrete model Hamiltonians, we investigate the difference $\lambda_{\rm min}^{\otimes}-E_0$, called separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable among a given state is sufficiently distant from the average value, the state is almost surely entangled., Comment: 8 pages total, 3 figures

We devise a deterministic algorithm to efficiently sample high-quality solutions of certain spin-glass systems that encode hard optimization problems. We employ tensor networks to represent the Gibbs distribution of all possible configurations. Using approximate tensor-network contractions, we are able to efficiently map the low-energy spectrum of some quasi-two-dimensional Hamiltonians. We exploit the local nature of the problems to compute spin-glass droplets geometries, which provides a new form of compression of the low-energy spectrum. It naturally extends to sampling, which otherwise, for exact contraction, is $\#$P-complete. In particular, for one of the hardest known problem-classes devised on chimera graphs known as deceptive cluster loops and for up to $2048$ spins, we find on the order of $10^{10}$ degenerate ground states in a single run of our algorithm, computing better solutions than have been reported on some hard instances. Our gradient-free approach could provide new insight into the structure of disordered spin-glass complexes, with ramifications both for machine learning and noisy intermediate-scale quantum devices., Comment: Final version; 7+6 pages, 4+6 figures; added benchmarks on time-to-solution and fair sampling/counting versus PT and QA. Code and problem instances can be found at https://github.com/marekrams/tnac4o

We conduct experimental simulations of many body quantum systems using a \emph{hybrid} classical-quantum algorithm. In our setup, the wave function of the transverse field quantum Ising model is represented by a restricted Boltzmann machine. This neural network is then trained using variational Monte Carlo assisted by a D-Wave quantum sampler to find the ground state energy. Our results clearly demonstrate that already the first generation of quantum computers can be harnessed to tackle non-trivial problems concerning physics of many body quantum systems., Comment: 6 pages, 4 figures

Near term quantum hardware promises unprecedented computational advantage. Crucial in its development is the characterization and minimization of computational errors. We propose the use of the quantum fluctuation theorem to benchmark the performance of quantum annealers. This versatile tool provides simple means to determine whether the quantum dynamics are unital, unitary, and adiabatic, or whether the system is prone to thermal noise. Our proposal is experimentally tested on two generations of the D-Wave machine, which illustrates the sensitivity of the fluctuation theorem to the smallest aberrations from ideal annealing., Comment: 5 pages, 4 figure

The shift of interest from general purpose quantum computers to adiabatic quantum computing or quantum annealing calls for a broadly applicable and easy to implement test to assess how quantum or adiabatic is a specific hardware. Here we propose such a test based on an exactly solvable many body system -- the quantum Ising chain in transverse field -- and implement it on the D-Wave machine. An ideal adiabatic quench of the quantum Ising chain should lead to an ordered broken symmetry ground state with all spins aligned in the same direction. An actual quench can be imperfect due to decoherence, noise, flaws in the implemented Hamiltonian, or simply too fast to be adiabatic. Imperfections result in topological defects: Spins change orientation, kinks punctuating ordered sections of the chain. The number of such defects quantifies the extent by which the quantum computer misses the ground state, and is, therefore, imperfect., Comment: 8 pages, 7 figures, to appear in Scientific Reports, authors' list completed

The ground state of the one-dimensional Bose-Hubbard model at unit filling undergoes the Mott-superfluid quantum phase transition. It belongs to the Kosterlitz-Thouless universality class with an exponential divergence of the correlation length in place of the usual power law. We present numerical simulations of a linear quench both from the Mott insulator to superfluid and back. The results satisfy the scaling hypothesis that follows from the Kibble-Zurek mechanism (KZM). In the superfluid-to-Mott quenches there is no significant excitation in the superfluid phase despite its gaplessness. Since all critical superfluid ground states are qualitatively similar, the excitation begins to build up only after crossing the critical point when the ground state begins to change fundamentally. The last process falls into the KZM framework., Comment: To appear in PRB

We investigate $\mathcal{P}\mathcal{T}$-symmetric quantum systems ultra-weakly coupled to an environment. We find that such open systems evolve under $\mathcal{P}\mathcal{T}$-symmetric, purely dephasing and unital dynamics. The dynamical map describing the evolution is then determined explicitly using a quantum canonical transformation. Furthermore, we provide an explanation of why $\mathcal{P}\mathcal{T}$-symmetric dephasing type interactions lead to \emph{critical slowing down of decoherence}. This effect is further exemplified with an experimentally relevant system -- a $\mathcal{P}\mathcal{T}$-symmetric qubit easily realizable, \emph{e.g.}, in optical or microcavity experiments., Comment: 5 pages, 1 figure

A non-commuting measurement transfers, via the apparatus, information encoded in a system's state to the external "observer". Classical measurements determine properties of physical objects. In the quantum realm, the very same notion restricts the recording process to orthogonal states as only those are distinguishable by measurements. Therefore, even a possibility to describe physical reality by means of non-hermitian operators should \emph{volens nolens} be excluded as their eigenstates are not orthogonal. Here, we show that non-hermitian operators with real spectrum can be treated within the standard framework of quantum mechanics. Furthermore, we propose a quantum canonical transformation that maps hermitian systems onto non-hermitian ones. Similar to classical inertial forces this transformation is accompanied by an energetic cost pinning the system on the unitary path.

Thermodynamics is the phenomenological theory of heat and work. Here we analyze to what extent quantum thermodynamic relations are immune to the underlying mathematical formulation of quantum mechanics. As a main result, we show that the Jarzynski equality holds true for all non-hermitian quantum systems with real spectrum. This equality expresses the second law of thermodynamics for isothermal processes arbitrarily far from equilibrium. In the quasistatic limit however, the second law leads to the Carnot bound which is fulfilled even if some eigenenergies are complex provided they appear in conjugate pairs. Furthermore, we propose two setups to test our predictions. Namely with strongly interacting excitons and photons in a semiconductor microcavity and in the non-hermitian tight-binding model., Comment: 9 pages, 2 figures

When a system is driven across a quantum critical point at a constant rate its evolution must become non-adiabatic as the relaxation time $\tau$ diverges at the critical point. According to the Kibble-Zurek mechanism (KZM), the emerging post-transition excited state is characterized by a finite correlation length $\hat\xi$ set at the time $\hat t=\hat \tau$ when the critical slowing down makes it impossible for the system to relax to the equilibrium defined by changing parameters. This observation naturally suggests a dynamical scaling similar to renormalization familiar from the equilibrium critical phenomena. We provide evidence for such KZM-inspired spatiotemporal scaling by investigating an exact solution of the transverse field quantum Ising chain in the thermodynamic limit., Comment: 15 pages, 11 figures; 1 figure added, version accepted in Phys. Rev. B

The Carnot statement of the second law of thermodynamics poses an upper limit on the efficiency of all heat engines. Recently, it has been studied whether generic quantum features such as coherence and quantum entanglement could allow for quantum devices with efficiencies larger than the Carnot efficiency. The present study shows that this is not permitted by the laws of thermodynamics. In particular, we will show that rather the definition of heat has to be modified to account for the thermodynamic cost for maintaining coherence and entanglement. Our theoretical findings are numerically illustrated for an experimentally relevant example from optomechanics., Comment: 6 pages, 3 figure; published version

Obtaining a thermodynamically accurate phase diagram through numerical calculations is a computationally expensive problem that is crucially important to understanding the complex phenomena of solid state physics, such as superconductivity. In this work we show how this type of analysis can be significantly accelerated through the use of modern GPUs. We illustrate this with a concrete example of free energy calculation in multi-band iron-based superconductors, known to exhibit a superconducting state with oscillating order parameter. Our approach can also be used for classical BCS-type superconductors. With a customized algorithm and compiler tuning we are able to achieve a 19x speedup compared to the CPU (119x compared to a single CPU core), reducing calculation time from minutes to mere seconds, enabling the analysis of larger systems and the elimination of finite size effects., Comment: 20 pages, 6 figures