We consider a model of quantum computation we call “varying $Z$” ($\mathrm{V}Z$), defined by applying controllable $Z$-diagonal Hamiltonians in the presence of a uniform and constant external $X$ field, and prove that it is universal, even in one dimension. Universality is demonstrated by construction of a universal gate set with $O\left(1\right)$ depth overhead. We then use this construction to describe a circuit whose output distribution cannot be classically simulated unless the polynomial hierarchy collapses, with the goal of providing a low-resource method of demonstrating quantum supremacy. The $\mathrm{V}Z$ model can achieve quantum supremacy in $O\left(n\right)$ depth in one dimension, equivalent to the random circuit sampling models despite a higher degree of homogeneity: it requires no individually addressed $X$ control.

Quantum phase transitions are a ubiquitous many-body phenomenon that occurs in a wide range of physical systems, including superconductors, quantum spin liquids, and topological materials. However, investigations of quantum critical systems also represent one of the most challenging problems in physics, since highly correlated many-body systems rarely allow for an analytic and tractable description. Here, we present a Lee-Yang theory of quantum phase transitions including a method to determine quantum critical points which readily can be implemented within the tensor network formalism and even in realistic experimental setups. We apply our method to a quantum Ising chain and the anisotropic quantum Heisenberg model and show how the critical behavior can be predicted by calculating or measuring the high cumulants of properly defined operators. Our approach provides a powerful formalism to analyze quantum phase transitions using tensor networks, and it paves the way for systematic investigations of quantum criticality in two-dimensional systems.

We study the 3D Kitaev and Kitaev-Heisenberg models, respectively, on the hyperhoneycomb and hyperoctagon lattices, both at zero and finite-temperature, in the thermodynamic limit. Our analysis relies on advanced tensor network (TN) simulations based on graph projected entangled-pair states (gPEPS). We map out the TN phase diagrams of the models and characterize their underlying gapped and gapless phases both at zero and finite temperature. In particular, we demonstrate how cooling down the hyperhoneycomb system from high temperature leads to fractionalization of spins to Majorana fermions and gauge fields that occurs in two separate temperature regimes, leaving their fingerprint on specific heat as a double-peak feature as well as on other quantities such as the thermal entropy, spin-spin correlations, and bond entropy. Using the Majorana representation of the Kitaev model, we further show that the low-temperature thermal transition to the Kitaev quantum spin liquid (QSL) phase is associated with the nontrivial Majorana band topology and the presence of Weyl nodes, which manifests itself via nonvanishing Chern number and finite thermal Hall conductivity. Beyond the pure Kitaev limit, we study the 3D Kitaev-Heisenberg (KH) model on the hyperoctagon lattice and extract the full phase diagram for different Heisenberg couplings. We further explore the thermodynamic properties of the magnetically-ordered regions in the KH model and show that, in contrast to the QSL phase, here the thermal phase transition follows the standard Landau symmetry-breaking theory.

Analysis of higher-order correlation functions has become a powerful tool for investigating interacting many-body systems in quantum simulators, such as quantum gas microscopes. Experimental measurements of mixed spin-charge correlation functions in the 2D Hubbard have been used to study equilibrium properties of magnetic polarons and to identify coherent and incoherent regimes of their dynamics. In this paper we consider theoretically an extension of this technique to systems which use a pinning potential to reduce the mobility of a single dopant in the Mott insulating regime of the 2D Hubbard model. We find that localization of the dopant has a dramatic effect on its magnetic dressing. The connected third order spin correlations are weakened in the case of a mobile hole but strengthened near an immobile hole. In the case of the fifth-order correlation function, we find that its bare value has opposite signs in cases of the mobile and of fully pinned dopant, whereas the connected part is similar for both cases. We study suppression of higher-order correlators by thermal fluctuations and demonstrate that they can be observed up to temperatures comparable to the spin-exchange energy $J$. We discuss implications of our results for understanding the interplay of spin and charge in doped Mott insulators.

We propose a method to construct a classical analog of an open quantum system, namely, a single quantum particle confined in a potential well and immersed in a thermal bath. The classical analog is made out of a collection of identical wells where classical particles of mass $m$ are trapped. The distribution $n(x,t)$ of the classical positions is used to reconstruct the quantum Bohm potential $V_{\mathrm{Bohm}}=-\frac{{\hslash }^2}{2m}\frac{\mathrm{\Delta }\sqrt{n}}{\sqrt{n}}$, which in turn acts on the shape of the potential wells. As a result, the classical particles experience an effective “quantum” force. This protocol is tested with numerical simulations using single- and double-well potentials, evidencing typical quantum effects such as long-lasting correlations and quantum tunneling. For harmonic confinement, the analogy is implemented experimentally using micron-sized dielectric beads optically trapped by a laser beam.

We predict an optical effect associated with systems, which exhibit topologically protected states separated by a finite distance. We develop a tight-binding model to calculate the optical selection rules in linear chains of atoms of different lengths, and show the crucial importance of edge states. For long enough molecules the interband transitions involving these edge states are in the highly sought-after THz frequency range. Although we have specifically considered finite carbon chains terminated by gold nanoparticles, the main results of our paper can be generalized to various systems, which exhibit topologically protected states separated by a finite distance.

Eigenstate thermalization hypothesis (ETH) has played a pivotal role in understanding ergodicity and its breaking in isolated quantum many-body systems. The recent experiment on 51-atom Rydberg quantum simulator by Bernien et al. [Nature (London) 551, 579 (2017)] and subsequent theoretical analysis have shown that hardcore kinetic constraint can lead to weak ergodicity breaking. In this work, we demonstrate, using $1d$ spin-1 PXP models, that miscellaneous type of ergodicity can be realized by adjusting the hardcore constraints between different components of nearest-neighbor spins. This includes ETH violation due to emergent shattering of Hilbert space into exponentially many subsectors of various sizes, a novel form of nonintegrability with an extensive number of local conserved quantities and strong ergodicity. We analyze these different forms of ergodicity and study their impact on the nonequilibrium dynamics of a ${\mathbb{Z}}_2$ initial state. We use forward-scattering approximation to understand the amount of ${\mathbb{Z}}_2$ oscillation present in these models. Our work shows that not only ergodicity breaking but an appropriate choice of constraints can lead to restoration of ergodicity as well.

Currently available quantum hardware allows for small-scale implementations of quantum machine learning algorithms. Such experiments aid the search for applications of quantum computers by benchmarking the near-term feasibility of candidate algorithms. Here we demonstrate the quantum learning of a two-qubit unitary by a sequence of three parameterized quantum circuits containing a total of 21 variational parameters. Moreover, we variationally diagonalize the unitary to learn its spectral decomposition, i.e., its eigenvalues and eigenvectors. We illustrate how this can be used as a subroutine to compress the depth of dynamical quantum simulations. One can view our implementation as a demonstration of entanglement-enhanced machine learning, as only a single (entangled) training data pair is required to learn a $4\times 4$ unitary matrix.

We use the quantum Fisher information (QFI) to diagnose a dynamical phase transition (DPT) in a closed quantum system, which is usually defined in terms of nonanalytic behavior of a time-averaged order parameter. Employing the Lipkin-Meshkov-Glick model as an illustrative example, we find that the DPT correlates with a peak in the QFI that can be explained by a generic connection to an underlying excited-state quantum phase transition that also enables us to relate the scaling of the QFI with the behavior of the order parameter. Motivated by the QFI as a quantifier of metrologically useful correlations and entanglement, we also present a robust interferometric protocol that can enable DPTs as a platform for quantum-enhanced sensing.

Orbital-free density-functional theory (OFDFT) offers a challenging way of electronic-structure calculations scaled as $\mathcal{O}\left(N\right)$ computation for system size $N$. We here develop a scheme of the OFDFT calculations based on the accurate and transferrable kinetic-energy density-functional (KEDF), which is created in an unprecedented way using appropriately constructed neural network (NN). We show that our OFDFT scheme reproduces the electron density obtained in the state-of-the-art DFT calculations and then provides accurate structural properties of 24 different systems, ranging from atoms, molecules, metals, semiconductors, and an ionic material. The accuracy and the transferability of our KEDF is achieved by our NN training system in which the kinetic-energy functional derivative (KEFD) at each real-space grid point is used. The choice of the KEFD as a set of training data is essentially important, because first it appears directly in the Euler equation, which one should solve, and second, its learning assists in reproducing the physical quantity expressed as the first derivative of the total energy. More generally, the present development of KEDF $T\left[\rho \right]$ is in the line of systematic expansion in terms of the functional derivatives ${\delta }^{{\ell }_1}T/\delta {\rho }^{{\ell }_1}$ through progressive increase of ${\ell }_1$. The present numerical success demonstrates the validity of this approach. The computational cost of the present OFDFT scheme indeed shows the $\mathcal{O}\left(N\right)$ scaling, as is evidenced by the computations of the semiconductor SiC used in power electronics.

Smoothing is a technique that is used to estimate the state of a system using measurement information both prior and posterior to the estimation time. Two notable examples of this technique are the Rauch-Tung-Striebel and Mayne-Fraser-Potter smoothing techniques for linear Gaussian systems, both resulting in the optimal smoothed estimate of the state. However, when considering a quantum system, classical smoothing techniques can result in an estimate that is not a valid quantum state. Consequently, a different smoothing theory was developed explicitly for quantum systems. This theory has since been applied to the special case of linear Gaussian quantum (LGQ) systems, where, in deriving the LGQ state smoothing equations, the Mayne-Fraser-Potter technique was utilized. As a result, the final equations describing the smoothed state are closely related to the classical Mayne-Fraser-Potter smoothing equations. In this paper, I derive the equivalent Rauch-Tung-Striebel form of the quantum state smoothing equations, which further simplify the calculation for the smoothed quantum state in LGQ systems, and I provide insight into the dynamics of the smoothed quantum state. This form of the LGQ smoothing equations brings to light a property of the smoothed quantum state that was hidden in the Mayne-Fraser-Potter form, namely the nondifferentiability of the smoothed mean. By identifying the nondifferentiable part of the smoothed mean, I am then able to derive a necessary and sufficient condition for the quantum smoothed mean to be differentiable in the steady-state regime.

One limitation of the variational quantum eigensolver algorithm is the large number of measurement steps required to estimate different terms in the Hamiltonian of interest. Unitary partitioning reduces this overhead by transforming the problem Hamiltonian into one containing fewer terms. We explore two different circuit constructions of the transformation required—one built by a sequence of rotations and the other built by a linear combination of unitaries (LCU). To assess performance, we simulated chemical Hamiltonians and studied the ground states of ${\mathrm{H}}_2$ and $\mathrm{LiH}$. Both implementations are successful even in the presence of noise. The sequence-of-rotations realization offers the greatest benefit to calculations, whereas the probabilistic nature of LCU reduces its effectiveness. This work also demonstrates an experimental implementation of LCU on quantum hardware.

A quantum system $\mathcal{Q}$ is characterized by a single potential $v$ and its eigenstates. While $v$ is usually postulated for a given physical problem, it represents the interaction with an implicit environment $\mathcal{E}$. We use the exact factorization to show how $v$ emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem $\mathcal{S}=\mathcal{Q}\cup \mathcal{E}$ corresponds to a different potential $v_j$ and state ${\chi }_j$ of $\mathcal{Q}$. Such a state ${\chi }_j$ typically has no nodes and is the ground state of $v_j$, even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting $\mathcal{Q}$ is much smaller than for exciting $\mathcal{E}$, the potentials $v_j$ are similar in shape and differ only by sharp spikes. An excitation of $\mathcal{S}$ can then be viewed as an excitation of $\mathcal{Q}$ with its environment being unaffected, and $\mathcal{Q}$ is approximately described by a single spikeless potential $v$ and its eigenstates. Second, ${\chi }_j$ can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of ${\chi }_j$ being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems.

Despite the significance of short-term memory in cognitive function, the process of encoding and sustaining the input information in neural activity dynamics remains elusive. Herein, we unveiled the significance of transient neural dynamics to short-term memory. By training recurrent neural networks to short-term memory tasks and analyzing the dynamics, the characteristics of the short-term memory mechanism were obtained in which the input information was encoded in the amplitude of transient oscillations, rather than the stationary neural activities. This transient trajectory was attracted to a slow manifold, which permitted the discarding of irrelevant information. Additionally, we investigated the process by which the dynamics acquire robustness to noise. In this transient oscillation, the robustness to noise was obtained by a strong contraction of the neural states after perturbation onto the manifold. This mechanism works for several neural network models and tasks, which implies its relevance to neural information processing in general.

We report on the structural and microscopic superconducting properties of the ${\mathrm{Ca}}_{1-x}{\mathrm{Sr}}_x\mathrm{AlSi}$ solid solution. Specifically, we have realized the continuous solid solution, which for all members, other than $x=0$ (CaAlSi), crystallizes in the ${\mathrm{AlB}}_2$-type structure. For CaAlSi, we present an improved structural model where all Al/Si layers are buckled, leading to a 6-folded structure along the crystallographic $c$ direction. We, furthermore, find indications for the structural instability in the parent compound CaAlSi to enhance the superconductivity across the solid solution. Our investigation of the magnetic penetration depths by means of muon-spin rotation experiments reveals that CaAlSi is a two-gap superconductor, that SrAlSi is a single-gap superconductor, and that there is a continuous transition from one electronic state to the other across the solid solution. Hence, we show that the ${\mathrm{Ca}}_{1-x}{\mathrm{Sr}}_x\mathrm{AlSi}$ solid solution is a highly tunable two-gap to single-gap superconducting system on a honeycomb lattice, where the superconductivity is strongly connected to a structural instability, i.e., the buckling of the Al/Si layers.

${\mathrm{D}}_2$ molecules, excited by linearly cross-polarized femtosecond extreme ultraviolet (XUV) and near-infrared (NIR) light pulses, reveal highly structured ${\mathrm{D}}^{+}$ ion fragment momenta and angular distributions that originate from two different four-step dissociative ionization pathways after four-photon absorption (one XUV $+$ three NIR). We show that, even for very low dissociation kinetic energy release $\le$ 240 meV, specific electronic excitation pathways can be identified and isolated in the final ion momentum distributions. With the aid of ab initio electronic structure and time-dependent Schrödinger equation calculations, angular momentum, energy, and parity conservation are used to identify the excited neutral molecular states and molecular orientations relative to the polarization vectors in these different photoexcitation and dissociation sequences of the neutral ${\mathrm{D}}_2$ molecule and its ${\mathrm{D}}_2^{+}$ cation. In one sequential photodissociation pathway, molecules aligned along either of the two light polarization vectors are excluded, while another pathway selects molecules aligned parallel to the light propagation direction. The evolution of the nuclear wave packet on the intermediate $B{}^1{\mathrm{\Sigma }}_u^{+}$ electronic state of the neutral ${\mathrm{D}}_2$ molecule is also probed in real time.