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We developed a general framework for hybrid quantum-classical computing of molecular and periodic embedding approaches based on an orbital space separation of the fragment and environment degrees of freedom. We demonstrate its potential by presenting a specific implementation of periodic range-separated DFT coupled to a quantum circuit ansatz, whereby the variational quantum eigensolver and the quantum equation-of-motion algorithm are used to obtain the low-lying spectrum of the embedded fragment Hamiltonian. Application of this scheme to study localized electronic states in materials is showcased through the accurate prediction of the optical properties of the neutral oxygen vacancy in magnesium oxide (MgO). Despite some discrepancies in the position of the main absorption band, the method demonstrates competitive performance compared to state-of-the-art ab initio approaches, particularly evidenced by the excellent agreement with the experimental photoluminescence emission peak.

Quantum computers (QC) could harbor the potential to significantly advance materials simulations, particularly at the atomistic scale involving strongly correlated fermionic systems where an accurate description of quantum many-body effects scales unfavorably with size. While a full-scale treatment of condensed matter systems with currently available noisy quantum computers remains elusive, quantum embedding schemes like dynamical mean-field theory (DMFT) allow the mapping of an effective, reduced subspace Hamiltonian to available devices to improve the accuracy of ab initio calculations such as density functional theory (DFT). Here, we report on the development of a hybrid quantum-classical DFT+DMFT simulation framework which relies on a quantum impurity solver based on the Lehmann representation of the impurity Green's function. Hardware experiments with up to 14 qubits on the IBM Quantum system are conducted, using advanced error mitigation methods and a novel calibration scheme for an improved zero-noise extrapolation to effectively reduce adverse effects from inherent noise on current quantum devices. We showcase the utility of our quantum DFT+DMFT workflow by assessing the correlation effects on the electronic structure of a real material, Ca2CuO2Cl2, and by carefully benchmarking our quantum results with respect to exact reference solutions and experimental spectroscopy measurements., Comment: 25 pages, 6 figures, supplementary information

The real-time simulation of large many-body quantum systems is a formidable task, that may only be achievable with a genuine quantum computational platform. Currently, quantum hardware with a number of qubits sufficient to make classical emulation challenging is available. This condition is necessary for the pursuit of a so-called quantum advantage, but it also makes verifying the results very difficult. In this manuscript, we flip the perspective and utilize known theoretical results about many-body quantum critical dynamics to benchmark quantum hardware and various error mitigation techniques on up to 133 qubits. In particular, we benchmark against known universal scaling laws in the Hamiltonian simulation of a time-dependent transverse field Ising Hamiltonian. Incorporating only basic error mitigation and suppression methods, our study shows reliable control up to a two-qubit gate depth of 28, featuring a maximum of 1396 two-qubit gates, before noise becomes prevalent. These results are transferable to applications such as Hamiltonian simulation, variational algorithms, optimization, or quantum machine learning. We demonstrate this on the example of digitized quantum annealing for optimization and identify an optimal working point in terms of both circuit depth and time step on a 133-site optimization problem., Comment: 11 pages, 4 pages appendix, 10 figures

Hybrid Tensor Networks (hTN) offer a promising solution for encoding variational quantum states beyond the capabilities of efficient classical methods or noisy quantum computers alone. However, their practical usefulness and many operational aspects of hTN-based algorithms, like the optimization of hTNs, the generalization of standard contraction rules to an hybrid setting, and the design of application-oriented architectures have not been thoroughly investigated yet. In this work, we introduce a novel algorithm to perform ground state optimizations with hybrid Tree Tensor Networks (hTTNs), discussing its advantages and roadblocks, and identifying a set of promising applications. We benchmark our approach on two paradigmatic models, namely the Ising model at the critical point and the Toric code Hamiltonian. In both cases, we successfully demonstrate that hTTNs can improve upon classical equivalents with equal bond dimension in the classical part., Comment: 19 pages, 16 figures

We introduce a novel computational framework for excited-states molecular quantum dynamics simulations driven by quantum computing-based electronic-structure calculations. This framework leverages the fewest-switches surface-hopping method for simulating the nuclear dynamics, and calculates the required excited-state transition properties with different flavors of the quantum subspace expansion and quantum equation-of-motion algorithms. We apply our method to simulate the collision reaction between a hydrogen atom and a hydrogen molecule. For this system, we critically compare the accuracy and efficiency of different quantum subspace expansion and equation-of-motion algorithms and show that only methods that can capture both weak and strong electron correlation effects can properly describe the non-adiabatic effects that tune the reactive event.

Optimal Transport (OT) has fueled machine learning (ML) across many domains. When paired data measurements $(\boldsymbol{\mu}, \boldsymbol{\nu})$ are coupled to covariates, a challenging conditional distribution learning setting arises. Existing approaches for learning a $\textit{global}$ transport map parameterized through a potentially unseen context utilize Neural OT and largely rely on Brenier's theorem. Here, we propose a first-of-its-kind quantum computing formulation for amortized optimization of contextualized transportation plans. We exploit a direct link between doubly stochastic matrices and unitary operators thus unravelling a natural connection between OT and quantum computation. We verify our method (QontOT) on synthetic and real data by predicting variations in cell type distributions conditioned on drug dosage. Importantly we conduct a 24-qubit hardware experiment on a task challenging for classical computers and report a performance that cannot be matched with our classical neural OT approach. In sum, this is a first step toward learning to predict contextualized transportation plans through quantum computing., Comment: ICML 2024

Randomized measurement protocols such as classical shadows represent powerful resources for quantum technologies, with applications ranging from quantum state characterization and process tomography to machine learning and error mitigation. Recently, the notion of measurement dual frames, in which classical shadows are generalized to dual operators of POVM effects, resurfaced in the literature. This brought attention to additional degrees of freedom in the post-processing stage of randomized measurements that are often neglected by established techniques. In this work, we leverage dual frames to construct improved observable estimators from informationally complete measurement samples. We introduce novel classes of parametrized frame superoperators and optimization-free dual frames based on empirical frequencies, which offer advantages over their canonical counterparts while retaining computational efficiency. Remarkably, this comes at almost no quantum or classical cost, thus rendering dual frame optimization a valuable addition to the randomized measurement toolbox., Comment: 9+3 pages, 4 figures

We explore the first-order phase transition in the lattice Schwinger model in the presence of a topological $\theta$-term by means of the variational quantum eigensolver (VQE). Using two different fermion discretizations, Wilson and staggered fermions, we develop parametric ansatz circuits suitable for both discretizations, and compare their performance by simulating classically an ideal VQE optimization in the absence of noise. The states obtained by the classical simulation are then prepared on the IBM's superconducting quantum hardware. Applying state-of-the art error-mitigation methods, we show that the electric field density and particle number, observables which reveal the phase structure of the model, can be reliably obtained from the quantum hardware. To investigate the minimum system sizes required for a continuum extrapolation, we study the continuum limit using matrix product states, and compare our results to continuum mass perturbation theory. We demonstrate that taking the additive mass renormalization into account is vital for enhancing the precision that can be obtained with smaller system sizes. Furthermore, for the observables we investigate we observe universality, and both fermion discretizations produce the same continuum limit., Comment: 21 pages, 10 figures, 1 table

We propose a method to prepare Gaussian wave packets with momentum on top of the interacting ground state of a fermionic Hamiltonian. Using Givens rotation, we show how to efficiently obtain expectation values of observables throughout the evolution of the wave packets on digital quantum computers. We demonstrate our technique by applying it to the staggered lattice formulation of the Thirring model and studying the scattering of two wave packets. Monitoring the the particle density and the entropy produced during the scattering process, we characterize the phenomenon and provide a first step towards studying more complicated collision processes on digital quantum computers. In addition, we perform a small-scale demonstration on IBM's quantum hardware, showing that our method is suitable for current and near-term quantum devices.

Machine learning techniques are essential tools to compute efficient, yet accurate, force fields for atomistic simulations. This approach has recently been extended to incorporate quantum computational methods, making use of variational quantum learning models to predict potential energy surfaces and atomic forces from ab initio training data. However, the trainability and scalability of such models are still limited, due to both theoretical and practical barriers. Inspired by recent developments in geometric classical and quantum machine learning, here we design quantum neural networks that explicitly incorporate, as a data-inspired prior, an extensive set of physically relevant symmetries. We find that our invariant quantum learning models outperform their more generic counterparts on individual molecules of growing complexity. Furthermore, we study a water dimer as a minimal example of a system with multiple components, showcasing the versatility of our proposed approach and opening the way towards larger simulations. Our results suggest that molecular force fields generation can significantly profit from leveraging the framework of geometric quantum machine learning, and that chemical systems represent, in fact, an interesting and rich playground for the development and application of advanced quantum machine learning tools., Comment: 12 pages, 8 figures

Variational quantum algorithms represent a powerful approach for solving optimization problems on noisy quantum computers, with a broad spectrum of potential applications ranging from chemistry to machine learning. However, their performances in practical implementations crucially depend on the effectiveness of quantum circuit training, which can be severely limited by phenomena such as barren plateaus. While, in general, dissipation is detrimental for quantum algorithms, and noise itself can actually induce barren plateaus, here we describe how the inclusion of properly engineered Markovian losses after each unitary quantum circuit layer can restore the trainability of quantum models. We identify the required form of the dissipation processes and establish that their optimization is efficient. We benchmark our proposal in both a synthetic and a practical quantum chemistry example, demonstrating its effectiveness and potential impact across different domains., Comment: Comments are welcome

Nuclear quantum effects such as zero-point energy and hydrogen tunnelling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets whereas realistic simulations beyond the Born--Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H$_2$ and D$_2$ molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by a several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles (NEO-UCCSD) while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method -- an important step toward accurate simulations involving non-classical nuclei and non--Born--Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects and we demonstrate that inclusion of correlation energy systematically improves the prediction of $\Delta$ZPE.

Quantum computers have emerged as a promising platform to simulate the strong electron correlation that is crucial to catalysis and photochemistry. However, owing to the choice of a trial wave function employed in the popular hybrid quantum-classical variational quantum eigensolver (VQE) algorithm, the accurate simulation is restricted to certain classes of correlated phenomena. Herein, we combine the spin-flip (SF) formalism with the unitary coupled cluster with singles and doubles (UCCSD) method via the quantum equation-of-motion (qEOM) approach to allow for an efficient simulation of a large family of strongly correlated problems. In particular, we show that the developed qEOM-SF-UCCSD/VQE method outperforms its UCCSD/VQE counterpart for simulation of the cis-trans isomerization of ethylene and the automerization of cyclobutadiene. The predicted qEOM-SF-UCCSD/VQE barrier heights for these two problems are in a good agreement with the experimentally determined values. The methodological developments presented herein will further stimulate investigation of this approach for the simulation of other types of correlated/entangled phenomena on a quantum computer.

Cosmological simulations describing the evolution of density perturbations of a self-gravitating collisionless Dark Matter (DM) fluid in an expanding background, provide a powerful tool to follow the formation of cosmic structures over wide dynamic ranges. The most widely adopted approach, based on the N-body discretization of the collisionless Vlasov-Poisson (VP) equations, is hampered by an unfavorable scaling when simulating the wide range of scales needed to cover at the same time the formation of single galaxies and of the largest cosmic structures. The dynamics described by the VP equations is limited by the rapid increase of the number of resolution elements which is required to simulate an ever growing range of scales. Recent studies showed an interesting mapping of the 6-dimensional+1 (6D+1) VP problem into a more amenable 3D+1 non-linear Schr\"odinger-Poisson (SP) problem for simulating the evolution of DM perturbations. This opens up the possibility of improving the scaling of time propagation simulations using quantum computing. In this paper, we introduce a quantum algorithm for simulating the (SP) equation by adapting a variational real-time evolution approach to a self-consistent, non-linear, problem. To achieve this, we designed a novel set of quantum circuits that establish connections between the solution of the original Poisson equation and the solution of the corresponding time-dependent Schr\"odinger equation. We also analyzed how nonlinearity impacts the variance of observables. Furthermore, we explored how the spatial resolution behaves as the SP dynamics approaches the classical limit and discovered an empirical logarithmic relationship between the required number of qubits and the scale of the SP equation. This entire approach holds the potential to serve as an efficient alternative for solving the Vlasov-Poisson (VP) equation by means of classical algorithms., Comment: main text 13 pages, appendix 3 pages, 8 numbered figures, 2 tables. Submitted to Physical Review Research

In recent years, there has been tremendous progress in the development of quantum computing hardware, algorithms and services leading to the expectation that in the near future quantum computers will be capable of performing simulations for natural science applications, operations research, and machine learning at scales mostly inaccessible to classical computers. Whereas the impact of quantum computing has already started to be recognized in fields such as cryptanalysis, natural science simulations, and optimization among others, very little is known about the full potential of quantum computing simulations and machine learning in the realm of healthcare and life science (HCLS). Herein, we discuss the transformational changes we expect from the use of quantum computation for HCLS research, more specifically in the field of cell-centric therapeutics. Moreover, we identify and elaborate open problems in cell engineering, tissue modeling, perturbation modeling, and bio-topology while discussing candidate quantum algorithms for research on these topics and their potential advantages over classical computational approaches., Comment: 6 figures

Quantum computers offer an intriguing path for a paradigmatic change of computing in the natural sciences and beyond, with the potential for achieving a so-called quantum advantage, namely a significant (in some cases exponential) speed-up of numerical simulations. The rapid development of hardware devices with various realizations of qubits enables the execution of small scale but representative applications on quantum computers. In particular, the high-energy physics community plays a pivotal role in accessing the power of quantum computing, since the field is a driving source for challenging computational problems. This concerns, on the theoretical side, the exploration of models which are very hard or even impossible to address with classical techniques and, on the experimental side, the enormous data challenge of newly emerging experiments, such as the upgrade of the Large Hadron Collider. In this roadmap paper, led by CERN, DESY and IBM, we provide the status of high-energy physics quantum computations and give examples for theoretical and experimental target benchmark applications, which can be addressed in the near future. Having the IBM 100 x 100 challenge in mind, where possible, we also provide resource estimates for the examples given using error mitigated quantum computing.

Quantum machine learning with parametrised quantum circuits has attracted significant attention over the past years as an early application for the era of noisy quantum processors. However, the possibility of achieving concrete advantages over classical counterparts in practical learning tasks is yet to be demonstrated. A promising avenue to explore potential advantages is the learning of data generated by quantum mechanical systems and presented in an inherently quantum mechanical form. In this article, we explore the applicability of quantum-data learning to practical problems in high-energy physics, aiming to identify domain specific use-cases where quantum models can be employed. We consider quantum states governed by one-dimensional lattice gauge theories and a phenomenological quantum field theory in particle physics, generated by digital quantum simulations or variational methods to approximate target states. We make use of an ansatz based on quantum convolutional neural networks and numerically show that it is capable of recognizing quantum phases of ground states in the Schwinger model, (de)confinement phases from time-evolved states in the $\mathbb{Z}_2$ gauge theory, and that it can extract fermion flavor/coupling constants in a quantum simulation of parton shower. The observation of non-trivial learning properties demonstrated in these benchmarks will motivate further exploration of the quantum-data learning architecture in high-energy physics., Comment: 16 pages, 14 figures

The combined quantum electron-nuclear dynamics is often associated with the Born-Huang expansion of the molecular wave function and the appearance of nonadiabatic effects as a perturbation. On the other hand, native multicomponent representations of electrons and nuclei also exist, which do not rely on any a priori approximation. However, their implementation is hampered by prohibitive scaling costs and therefore quantum computers offer a unique opportunity for extending their use to larger systems. Here, we propose a quantum algorithm for the simulation of the time-evolution of molecular systems in the second quantization framework, which is applied to the simulation of the proton transfer dynamics in malonaldehyde. After partitioning the dynamics into slow and fast components, we show how the entanglement between the electronic and nuclear degrees of freedom can persist over long times if electrons are not adiabatically following the nuclear displacement. The proposed quantum algorithm may become a valid candidate for the study of electron-nuclear quantum phenomena when sufficiently powerful quantum computers become available., Comment: 14 pages, 10 figures, 1 table

The standard model (SM) of particle physics represents a theoretical paradigm for the description of the fundamental forces of nature. Despite its broad applicability, the SM does not enable the description of all physically possible events. The detection of events that cannot be described by the SM, which are typically referred to as anomalous, and the related potential discovery of exotic physical phenomena is a non-trivial task. The challenge becomes even greater with next-generation colliders that will produce even more events with additional levels of complexity. The additional data complexity motivates the search for unsupervised anomaly detection methods that do not require prior knowledge about the underlying models. In this work, we develop such a technique. More explicitly, we employ a quantum generative adversarial network to identify anomalous events. The method learns the background distribution from SM data and, then, determines whether a given event is characteristic for the learned background distribution. The proposed quantum-powered anomaly detection strategy is tested on proof-of-principle examples using numerical simulations and IBM Quantum processors. We find that the quantum generative techniques using ten times fewer training data samples can yield comparable accuracy to the classical counterpart for the detection of the Graviton and Higgs particles. Additionally, we empirically compute the capacity of the quantum model and observe an improved expressivity compared to its classical counterpart., Comment: For updated final version, see proceedings of the 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), vol 01, pages 331-341

State-of-the-art noisy digital quantum computers can only execute short-depth quantum circuits. Variational algorithms are a promising route to unlock the potential of noisy quantum computers since the depth of the corresponding circuits can be kept well below hardware-imposed limits. Typically, the variational parameters correspond to virtual $R_Z$ gate angles, implemented by phase changes of calibrated pulses. By encoding the variational parameters directly as hardware pulse amplitudes and durations we succeed in further shortening the pulse schedule and overall circuit duration. This decreases the impact of qubit decoherence and gate noise. As a demonstration, we apply our pulse-based variational algorithm to the calculation of the ground state of different hydrogen-based molecules (H$_2$, H$_3$ and H$_4$) using IBM cross-resonance-based hardware. We observe a reduction in schedule duration of up to $5\times$ compared to CNOT-based Ans\"atze, while also reducing the measured energy. In particular, we observe a sizable improvement of the minimal energy configuration of H$_3$ compared to a CNOT-based variational form. Finally, we discuss possible future developments including error mitigation schemes and schedule optimizations, which will enable further improvements of our approach paving the way towards the simulation of larger systems on noisy quantum devices.

Quantum computing is emerging as a new computational paradigm with the potential to transform several research fields, including quantum chemistry. However, current hardware limitations (including limited coherence times, gate infidelities, and limited connectivity) hamper the straightforward implementation of most quantum algorithms and call for more noise-resilient solutions. In quantum chemistry, the limited number of available qubits and gate operations is particularly restrictive since, for each molecular orbital, one needs, in general, two qubits. In this study, we propose an explicitly correlated Ansatz based on the transcorrelated (TC) approach, which transfers -- without any approximation -- correlation from the wavefunction directly into the Hamiltonian, thus reducing the number of resources needed to achieve accurate results with noisy, near-term quantum devices. In particular, we show that the exact transcorrelated approach not only allows for more shallow circuits but also improves the convergence towards the so-called basis set limit, providing energies within chemical accuracy to experiment with smaller basis sets and, therefore, fewer qubits. We demonstrate our method by computing bond lengths, dissociation energies, and vibrational frequencies close to experimental results for the hydrogen dimer and lithium hydride using just 4 and 6 qubits, respectively. Conventional methods require at least ten times more qubits for the same accuracy.

Interactions between electrons and phonons play a crucial role in quantum materials. Yet, there is no universal method that would simultaneously accurately account for strong electron-phonon interactions and electronic correlations. By combining methods of the variational quantum eigensolver and the variational non-Gaussian solver, we develop a hybrid quantum-classical algorithm suitable for this type of correlated systems. This hybrid method tackles systems with arbitrarily strong electron-phonon coupling without increasing the number of required qubits and quantum gates, as compared to purely electronic models. We benchmark the new method by applying it to the paradigmatic Hubbard-Holstein model at half filling, and show that it correctly captures the competition between charge density wave and antiferromagnetic phases, quantitatively consistent with exact diagonalization., Comment: 8 pages, 4 figures; 4 pages Supplemental Material

Nuclear quantum phenomena beyond the Born-Oppenheimer approximation are known to play an important role in a growing number of chemical and biological processes. While there exists no unique consensus on a rigorous and efficient implementation of coupled electron-nuclear quantum dynamics, it is recognised that these problems scale exponentially with system size on classical processors and therefore may benefit from quantum computing implementations. Here, we introduce a methodology for the efficient quantum treatment of the electron-nuclear problem on near-term quantum computers, based upon the Nuclear-Electronic Orbital (NEO) approach. We generalize the electronic two-qubit tapering scheme to include nuclei by exploiting symmetries inherent in the NEO framework; thereby reducing the hamiltonian dimension, number of qubits, gates, and measurements needed for calculations. We also develop parameter transfer and initialisation techniques, which improve convergence behavior relative to conventional initialisation. These techniques are applied to H$_2$ and malonaldehyde for which results agree with Nuclear-Electronic Orbital Full Configuration Interaction and Nuclear-Electronic Orbital Complete Active Space Configuration Interaction benchmarks for ground state energy to within $10^{-6}$ Ha and entanglement entropy to within $10^{-4}$. These implementations therefore significantly reduce resource requirements for full quantum simulations of molecules on near-term quantum devices while maintaining high accuracy., Comment: 26 pages, 7 figures, 10 tables

Quantum computing has emerged as a promising platform for simulating strongly correlated systems in chemistry, for which the standard quantum chemistry methods are either qualitatively inaccurate or too expensive. However, due to the hardware limitations of the available noisy near-term quantum devices, their application is currently limited only to small chemical systems. One way for extending the range of applicability can be achieved within the quantum embedding approach. Herein, we employ the projection-based embedding method for combining the variational quantum eigensolver (VQE) algorithm, although not limited to, with density functional theory (DFT). The developed VQE-in-DFT method is then implemented efficiently on a real quantum device and employed for simulating the triple bond breaking process in butyronitrile. The results presented herein show that the developed method is a promising approach for simulating systems with a strongly correlated fragment on a quantum computer. The developments as well as the accompanying implementation will benefit many different chemical areas including the computer aided drug design as well as the study of metalloenzymes with a strongly correlated fragment.

Much hope for finding new physics phenomena at microscopic scale relies on the observations obtained from High Energy Physics experiments, like the ones performed at the Large Hadron Collider (LHC). However, current experiments do not indicate clear signs of new physics that could guide the development of additional Beyond Standard Model (BSM) theories. Identifying signatures of new physics out of the enormous amount of data produced at the LHC falls into the class of anomaly detection and constitutes one of the greatest computational challenges. In this article, we propose a novel strategy to perform anomaly detection in a supervised learning setting, based on the artificial creation of anomalies through a random process. For the resulting supervised learning problem, we successfully apply classical and quantum Support Vector Classifiers (CSVC and QSVC respectively) to identify the artificial anomalies among the SM events. Even more promising, we find that employing an SVC trained to identify the artificial anomalies, it is possible to identify realistic BSM events with high accuracy. In parallel, we also explore the potential of quantum algorithms for improving the classification accuracy and provide plausible conditions for the best exploitation of this novel computational paradigm., Comment: 15 pages, 10 figures

We propose a new strategy for anomaly detection at the LHC based on unsupervised quantum machine learning algorithms. To accommodate the constraints on the problem size dictated by the limitations of current quantum hardware we develop a classical convolutional autoencoder. The designed quantum anomaly detection models, namely an unsupervised kernel machine and two clustering algorithms, are trained to find new-physics events in the latent representation of LHC data produced by the autoencoder. The performance of the quantum algorithms is benchmarked against classical counterparts on different new-physics scenarios and its dependence on the dimensionality of the latent space and the size of the training dataset is studied. For kernel-based anomaly detection, we identify a regime where the quantum model significantly outperforms its classical counterpart. An instance of the kernel machine is implemented on a quantum computer to verify its suitability for available hardware. We demonstrate that the observed consistent performance advantage is related to the inherent quantum properties of the circuit used., Comment: added dataset zenodo link, public github link for the code, fixed typos, and added some references

Gate-based quantum computers typically encode and process information in two-dimensional units called qubits. Using $d$-dimensional qudits instead may offer intrinsic advantages, including more efficient circuit synthesis, problem-tailored encodings and embedded error correction. In this work, we design a superconducting qudit-based quantum processor wherein the logical space of transmon qubits is extended to higher-excited levels. We propose a universal gate set featuring a two-qudit cross-resonance entangling gate, for which we predict fidelities beyond $99\%$ in the $d=4$ case of ququarts with realistic experimental parameters. Furthermore, we present a decomposition routine that compiles general qudit unitaries into these elementary gates, requiring fewer entangling gates than qubit alternatives. As proof-of-concept applications, we numerically demonstrate the synthesis of ${\rm SU}(16)$ gates for noisy quantum hardware and an embedded error correction sequence that encodes a qubit memory in a transmon ququart to protect against pure dephasing noise. We conclude that universal qudit control -- a valuable extension to the operational toolbox of superconducting quantum information processing -- is within reach of current transmon-based architectures and has applications to near-term and long-term hardware., Comment: 10+6 pages, 9+3 figures, revised version published in PRX Quantum

Random walks (or Markov chains) are models extensively used in theoretical computer science. Several tools, including analysis of quantities such as hitting and mixing times, are helpful for devising randomized algorithms. A notable example is Sch\"oning's algorithm for the satisfiability (SAT) problem. In this work, we use the density-matrix formalism to define a quantum Markov chain model which directly generalizes classical walks, and we show that a common tools such as hitting times can be computed with a similar formula as the one found in the classical theory, which we then apply to known quantum settings such as Grover's algorithm., Comment: 18 pages, 5 figures

The Shor-Laflamme distribution (SLD) of a quantum state is a collection of local unitary invariants that quantify $k$-body correlations. We show that the SLD of graph states can be derived by solving a graph-theoretical problem. In this way, the mean and variance of the SLD are obtained as simple functions of efficiently computable graph properties. Furthermore, this formulation enables us to derive closed expressions of SLDs for some graph state families. For cluster states, we observe that the SLD is very similar to a binomial distribution, and we argue that this property is typical for graph states in general. Finally, we derive an SLD-based entanglement criterion from the purity criterion and apply it to derive meaningful noise thresholds for entanglement. Our new entanglement criterion is easy to use and also applies to the case of higher-dimensional qudits. In the bigger picture, our results foster the understanding both of quantum error-correcting codes, where a closely related notion of Shor-Laflamme distributions plays an important role, and of the geometry of quantum states, where Shor-Laflamme distributions are known as sector length distributions., Comment: 22+18 pages, 8+6 figures, minor changes: title and motivation have been updated, appendix F has been removed to shorten the paper

Controlling selectivity of reactions is a quest in chemistry. Here, we demonstrate reversible and selective bond formation and dissociation promoted by tip-induced reduction-oxidation reactions on a surface. Molecular rearrangements leading to different constitutional isomers are selected by the polarity and magnitude of applied voltage pulses from the tip of a combined scanning tunneling and atomic force microscope. Characterization of voltage dependence of the reactions and determination of reaction rates demonstrate selectivity in constitutional isomerization reactions and provide insight into the underlying mechanisms. With support of density functional theory calculations, we find that the energy landscape of the isomers in different charge states is important to rationalize the selectivity. Tip-induced selective single-molecule reactions increase our understanding of redox chemistry and could lead to novel molecular machines.

The variational approach is a cornerstone of computational physics, considering both conventional and quantum computing computational platforms. The variational quantum eigensolver (VQE) algorithm aims to prepare the ground state of a Hamiltonian exploiting parametrized quantum circuits that may offer an advantage compared to classical trial states used, for instance, in quantum Monte Carlo or tensor network calculations. While traditionally, the main focus has been on developing better trial circuits, we show that the algorithm's success crucially depends on other parameters such as the learning rate, the number $N_s$ of measurements to estimate the gradient components, and the Hamiltonian gap $\Delta$. We first observe the existence of a finite $N_s$ value below which the optimization is impossible, and the energy variance resembles the behavior of the specific heat in second-order phase transitions. Secondly, when $N_s$ is above such threshold level, and learning is possible, we develop a phenomenological model that relates the fidelity of the state preparation with the optimization hyperparameters as well as $\Delta$. More specifically, we observe that the computational resources scale as $1/\Delta^2$, and we propose a symmetry-enhanced simulation protocol that should be used if the gap closes. We test our understanding on several instances of two-dimensional frustrated quantum magnets, which are believed to be the most promising candidates for near-term quantum advantage through variational quantum simulations., Comment: 12 pages, 13 figures

Machine Learning (ML) for Ligand Based Virtual Screening (LB-VS) is an important in-silico tool for discovering new drugs in a faster and cost-effective manner, especially for emerging diseases such as COVID-19. In this paper, we propose a general-purpose framework combining a classical Support Vector Classifier (SVC) algorithm with quantum kernel estimation for LB-VS on real-world databases, and we argue in favor of its prospective quantum advantage. Indeed, we heuristically prove that our quantum integrated workflow can, at least in some relevant instances, provide a tangible advantage compared to state-of-art classical algorithms operating on the same datasets, showing strong dependence on target and features selection method. Finally, we test our algorithm on IBM Quantum processors using ADRB2 and COVID-19 datasets, showing that hardware simulations provide results in line with the predicted performances and can surpass classical equivalents., Comment: 16 pages, 7 figures

We propose an improved quantum algorithm to calculate the Green's function through real-time propagation, and use it to compute the retarded Green's function for the 2-, 3- and 4-site Hubbard models. This novel protocol significantly reduces the number of controlled operations when compared to those previously suggested in literature. Such reduction is quite remarkable when considering the 2-site Hubbard model, for which we show that it is possible to obtain the exact time propagation of the $|N\pm 1\rangle$ states by exponentiating one single Pauli component of the Hamiltonian, allowing us to perform the calculations on an actual superconducting quantum processor.

Informationally complete (IC) positive operator-valued measures (POVMs) are generalized quantum measurements that offer advantages over the standard computational basis readout of qubits. For instance, IC-POVMs enable efficient extraction of operator expectation values, a crucial step in many quantum algorithms. POVM measurements are typically implemented by coupling one additional ancilla qubit to each logical qubit, thus imposing high demands on the device size and connectivity. Here, we show how to implement a general class of IC-POVMs without ancilla qubits. We exploit the higher-dimensional Hilbert space of a qudit in which qubits are often encoded. POVMs can then be realized by coupling each qubit to two of the available qudit states, followed by a projective measurement. We develop the required control pulse sequences and numerically establish their feasibility for superconducting transmon qubits through pulse-level simulations. Finally, we present an experimental demonstration of a qudit-space POVM measurement on IBM Quantum hardware. This paves the way to making POVM measurements broadly available to quantum computing applications., Comment: 9+10 pages, 5+5 figures, 1+1 tables

Solving electronic structure problems represents a promising field of application for quantum computers. Currently, much effort has been spent in devising and optimizing quantum algorithms for quantum chemistry problems featuring up to hundreds of electrons. While quantum algorithms can in principle outperform their classical equivalents, the polynomially scaling runtime, with the number of constituents, can still prevent quantum simulations of large scale systems. We propose a strategy to extend the scope of quantum computational methods to large scale simulations using a machine learning potential, trained on quantum simulation data. The challenge of applying machine learning potentials in today's quantum setting arises from the several sources of noise affecting the quantum computations of electronic energies and forces. We investigate the trainability of a machine learning potential selecting various sources of noise: statistical, optimization and hardware noise.Finally, we construct the first machine learning potential from data computed on actual IBM Quantum processors for a hydrogen molecule. This already would allow us to perform arbitrarily long and stable molecular dynamics simulations, outperforming all current quantum approaches to molecular dynamics and structure optimization., Comment: 12 pages, 8 figures

Accurate molecular force fields are of paramount importance for the efficient implementation of molecular dynamics techniques at large scales. In the last decade, machine learning methods have demonstrated impressive performances in predicting accurate values for energy and forces when trained on finite size ensembles generated with ab initio techniques. At the same time, quantum computers have recently started to offer new viable computational paradigms to tackle such problems. On the one hand, quantum algorithms may notably be used to extend the reach of electronic structure calculations. On the other hand, quantum machine learning is also emerging as an alternative and promising path to quantum advantage. Here we follow this second route and establish a direct connection between classical and quantum solutions for learning neural network potentials. To this end, we design a quantum neural network architecture and apply it successfully to different molecules of growing complexity. The quantum models exhibit larger effective dimension with respect to classical counterparts and can reach competitive performances, thus pointing towards potential quantum advantages in natural science applications via quantum machine learning., Comment: 12 pages, 7 figures

A central building block of many quantum algorithms is the diagonalization of Pauli operators. Although it is always possible to construct a quantum circuit that simultaneously diagonalizes a given set of commuting Pauli operators, only resource-efficient circuits are reliably executable on near-term quantum computers. Generic diagonalization circuits can lead to an unaffordable Swap-gate overhead on quantum devices with limited hardware connectivity. A common alternative is excluding two-qubit gates, however, this comes at the cost of restricting the class of diagonalizable sets of Pauli operators to tensor product bases (TPBs). In this letter, we introduce a theoretical framework for constructing hardware-tailored (HT) diagonalization circuits. We apply our framework to group the Pauli operators occurring in the decomposition of a given Hamiltonian into jointly-HT-diagonalizable sets. We investigate several classes of popular Hamiltonians and observe that our approach requires a smaller number of measurements than conventional TPB approaches. Finally, we experimentally demonstrate the practical applicability of our technique, which showcases the great potential of our circuits for near-term quantum computing., Comment: 4+34 pages, 3+12 figures, 1+11 tables. Main changes with respect to the previous version: SM Sec. II removed, note added to SM Sec. III, new SM Secs. VIII and IX about scalability and practicality, respectively

The simulation of quantum dynamics calls for quantum algorithms working in first quantized grid encodings. Here, we propose a variational quantum algorithm for performing quantum dynamics in first quantization. In addition to the usual reduction in circuit depth conferred by variational approaches, this algorithm also enjoys several advantages compared to previously proposed ones. For instance, variational approaches suffer from the need for a large number of measurements. However, the grid encoding of first quantized Hamiltonians only requires measuring in position and momentum bases, irrespective of the system size. Their combination with variational approaches is therefore particularly attractive. Moreover, heuristic variational forms can be employed to overcome the limitation of the hard decomposition of Trotterized first quantized Hamiltonians into quantum gates. We apply this quantum algorithm to the dynamics of several systems in one and two dimensions. Our simulations exhibit the previously observed numerical instabilities of variational time propagation approaches. We show how they can be significantly attenuated through subspace diagonalization at a cost of an additional $\mathcal{O}(MN^2)$ 2-qubit gates where $M$ is the number of dimensions and $N^M$ is the total number of grid points.

Transcorrelated methods provide an efficient way of partially transferring the description of electronic correlations from the ground state wavefunction directly into the underlying Hamiltonian. In particular, Dobrautz et al. [Phys. Rev. B, 99(7), 075119, (2019)] have demonstrated that the use of momentum-space representation, combined with a non-unitary similarity transformation, results in a Hubbard Hamiltonian that possesses a significantly more compact ground state wavefunction, dominated by a single Slater determinant. This compactness/single-reference character greatly facilitates electronic structure calculations. As a consequence, however, the Hamiltonian becomes non-Hermitian, posing problems for quantum algorithms based on the variational principle. We overcome these limitations with the ansatz-based quantum imaginary time evolution algorithm and apply the transcorrelated method in the context of digital quantum computing. We demonstrate that this approach enables up to 4 orders of magnitude more accurate and compact solutions in various instances of the Hubbard model at intermediate interaction strength ($U/t=4$), enabling the use of shallower quantum circuits for wavefunction ansatzes. In addition, we propose a more efficient implementation of the quantum imaginary time evolution algorithm in quantum circuits that is tailored to non-Hermitian problems. To validate our approach, we perform hardware experiments on the ibmq_lima quantum computer. Our work paves the way for the use of exact transcorrelated methods for the simulations of ab initio systems on quantum computers., Comment: 21 pages, 6 figures

Many-body Green's functions encode all the properties and excitations of interacting electrons. While these are challenging to be evaluated accurately on a classical computer, recent efforts have been directed towards finding quantum algorithms that may provide a quantum advantage for this task, exploiting architectures that will become available in the near future. In this work we introduce a novel near-term quantum algorithm for computing one-particle Green's functions via their Lehmann representation. The method is based on a generalization of the quantum equation of motion algorithm that gives access to the charged excitations of the system. We demonstrate the validity of the present proposal by computing the Green's function of a two-site Fermi-Hubbard model on a IBM quantum processor.

Variational quantum algorithms (VQAs) are increasingly being applied in simulations of strongly-bound (covalently bonded) systems using full molecular orbital basis representations. The application of quantum computers to the weakly-bound intermolecular and non-covalently bonded regime however has remained largely unexplored. In this work, we develop a coarse-grained representation of the electronic response that is ideally suited for determining the ground state of weakly interacting molecules using a VQA. We require qubit numbers that grow linearly with the number of molecules and derive scaling behaviour for the number of circuits and measurements required, which compare favourably to traditional variational quantum eigensolver methods. We demonstrate our method on IBM superconducting quantum processors and show its capability to resolve the dispersion energy as a function of separation for a pair of non-polar molecules - thereby establishing a means by which quantum computers can model Van der Waals interactions directly from zero-point quantum fluctuations. Within this coarse-grained approximation, we conclude that current-generation quantum hardware is capable of probing energies in this weakly bound but nevertheless chemically ubiquitous and biologically important regime. Finally, we perform experiments on simulated and real quantum computers for systems of three, four and five oscillators as well as oscillators with anharmonic onsite binding potentials; the consequences of the latter are unexamined in large systems using classical computational methods but can be incorporated here with low computational overhead., Comment: 28 pages, 12 figures. Published version. Additional experimental results

Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter-approximation of the time-evolution operator. This approach typically relies on deep circuits and is therefore hampered by the substantial limitations of available noisy and near-term quantum hardware. On the other hand, variational quantum algorithms have become an indispensable alternative, enabling small-scale simulations on present-day hardware. However, despite the recent development of variational quantum algorithms for quantum dynamics, a detailed assessment of their efficiency and scalability is yet to be presented. To fill this gap, we applied a variational quantum algorithm based on McLachlan's principle to simulate the dynamics of a spin-boson model subject to varying levels of realistic hardware noise as well as in different physical regimes, and discuss the algorithm's accuracy and scaling behavior as a function of system size. We observe a good performance of the variational approach used in combination with a general, physically motivated wavefunction ansatz, and compare it to the conventional first-order Trotter-evolution. Finally, based on this, we make scaling predictions for the simulation of a classically intractable system. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage for the solution of time-dependent problems., Comment: 8 pages, 4 figures

Near term quantum computers suffer from the presence of different noise sources. In order to mitigate for this effect and acquire results with significantly better accuracy, there is the urge of designing efficient error correction or error mitigation schemes. The cost of such techniques is usually high in terms of resource requirements, either in hardware or at the algorithmic level. In this work, we follow a pragmatic approach and we use repetition codes as scalable schemes with the potential to provide more accurate solutions to problems of interest in quantum chemistry and physics. We investigate different repetition code layouts and we propose a circular repetition scheme with connectivity requirements that are native on IBM Quantum hardware. We showcase our approach in multiple IBM Quantum devices and validate our results using a simplified theoretical noise model. We highlight the effect of using the proposed scheme in an electronic structure VQE calculation and in the simulation of time evolution for a quantum Ising model.

Quantum computation represents an emerging framework to solve lattice gauge theories (LGT) with arbitrary gauge groups, a general and long-standing problem in computational physics. While quantum computers may encode LGT using only polynomially increasing resources, a major openissue concerns the violation of gauge-invariance during the dynamics and the search for groundstates. Here, we propose a new class of parametrized quantum circuits that can represent states belonging only to the physical sector of the total Hilbert space. This class of circuits is compact yet flexible enough to be used as a variational ansatz to study ground state properties, as well as representing states originating from a real-time dynamics. Concerning the first application, the structure of the wavefunction ansatz guarantees the preservation of physical constraints such as the Gauss law along the entire optimization process, enabling reliable variational calculations. As for the second application, this class of quantum circuits can be used in combination with timedependent variational quantum algorithms, thus drastically reducing the resource requirements to access dynamical properties., Comment: 16 pages, 13 figures, published version

We develop a workflow to use current quantum computing hardware for solving quantum many-body problems, using the example of the fermionic Hubbard model. Concretely, we study a four-site Hubbard ring that exhibits a transition from a product state to an intrinsically interacting ground state as hopping amplitudes are changed. We locate this transition and solve for the ground state energy with high quantitative accuracy using a variational quantum algorithm executed on an IBM quantum computer. Our results are enabled by a variational ansatz that takes full advantage of the maximal set of commuting $\mathbb{Z}_2$ symmetries of the problem and a Lanczos-inspired error mitigation algorithm. They are a benchmark on the way to exploiting near term quantum simulators for quantum many-body problems.

Quantum machine learning could possibly become a valuable alternative to classical machine learning for applications in High Energy Physics by offering computational speed-ups. In this study, we employ a support vector machine with a quantum kernel estimator (QSVM-Kernel method) to a recent LHC flagship physics analysis: $t\bar{t}H$ (Higgs boson production in association with a top quark pair). In our quantum simulation study using up to 20 qubits and up to 50000 events, the QSVM-Kernel method performs as well as its classical counterparts in three different platforms from Google Tensorflow Quantum, IBM Quantum and Amazon Braket. Additionally, using 15 qubits and 100 events, the application of the QSVM-Kernel method on the IBM superconducting quantum hardware approaches the performance of a noiseless quantum simulator. Our study confirms that the QSVM-Kernel method can use the large dimensionality of the quantum Hilbert space to replace the classical feature space in realistic physics datasets.

Many prominent quantum computing algorithms with applications in fields such as chemistry and materials science require a large number of measurements, which represents an important roadblock for future real-world use cases. We introduce a novel approach to tackle this problem through an adaptive measurement scheme. We present an algorithm that optimizes informationally complete positive operator-valued measurements (POVMs) on the fly in order to minimize the statistical fluctuations in the estimation of relevant cost functions. We show its advantage by improving the efficiency of the variational quantum eigensolver in calculating ground-state energies of molecular Hamiltonians with extensive numerical simulations. Our results indicate that the proposed method is competitive with state-of-the-art measurement-reduction approaches in terms of efficiency. In addition, the informational completeness of the approach offers a crucial advantage, as the measurement data can be reused to infer other quantities of interest. We demonstrate the feasibility of this prospect by reusing ground-state energy-estimation data to perform high-fidelity reduced state tomography., Comment: Close to published version

We show that molecular spin qudits provide an ideal platform to simulate the quantum dynamics of photon fields strongly interacting with matter. The basic unit of the proposed molecular quantum simulator can be realized by a simple dimer of a spin 1/2 and a spin $S$ transition metal ion, solely controlled by microwave pulses. The spin $S$ ion is exploited to encode the photon field in a flexible architecture, which enables the digital simulation of a wide range of spin-boson models much more efficiently than by using a multi-qubit register. The effectiveness of our proposal is demonstrated by numerical simulations using realistic molecular parameters, whose prerequisites delineating possible chemical approaches are also discussed., Comment: 9 pages, 3 figures