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Magic is a property of quantum states that enables universal fault-tolerant quantum computing using simple sets of gate operations. Understanding the mechanisms by which magic is created or destroyed is, therefore, a crucial step towards efficient and practical fault-tolerant computation. We observe that a random stabilizer code subject to coherent errors exhibits a phase transition in magic, which we characterize through analytic, numeric and experimental probes.Below a critical error rate, stabilizer syndrome measurements remove the accumulated magic in the circuit, effectively protecting against coherent errors; above the critical error rate syndrome measurements concentrate magic. A better understanding of such rich behavior in the resource theory of magic could shed more light on origins of quantum speedup and pave pathways for more efficient magic state generation.&nbsp

Quantum computers have the potential to advance material design and drug discovery by performing costly electronic structure calculations. A critical aspect of this application requires optimizing the limited resources of the quantum hardware. Here, we experimentally demonstrate an end-to-end pipeline that focuses on minimizing quantum resources while maintaining accuracy. Using density matrix embedding theory as a problem decomposition technique, and an ion-trap quantum computer, we simulate a ring of 10 hydrogen atoms without freezing any electrons. The originally 20-qubit system is decomposed into 10 two-qubit problems, making it amenable to currently available hardware. Combining this decomposition with a qubit coupled cluster circuit ansatz, circuit optimization, and density matrix purification, we accurately reproduce the potential energy curve in agreement with the full configuration interaction energy in the minimal basis set. Our experimental results are an early demonstration of the potential for problem decomposition to accurately simulate large molecules on quantum hardware. Problem decomposition methods may help to overcome the size limitations of quantum hardware and allow largescale electronic structure simulations. Here, a method to simulate a ten-atom Hydrogen ring by decomposing it into smaller fragments that are amenable to a currently available trapped ion quantum computer is demonstrated experimentally.

Quantum computers with tens to hundreds of noisy qubits are being developed today. To be useful for real-world applications, we believe that these near-term systems cannot simply be scaled-down non-error-corrected versions of future fault-tolerant large-scale quantum computers. These near-term systems require specific architecture and design attributes to realize their full potential. To efficiently execute an algorithm, the quantum coprocessor must be designed to scale with respect to qubit number and to maximize useful computation within the qubits' decoherence bounds. In this work, we employ an application-system-qubit co-design methodology to architect a near-term quantum coprocessor. To support algorithms from the real-world application area of simulating the quantum dynamics of a material system, we design a (parameterized) arbitrary single-qubit rotation instruction and a two-qubit entangling controlled-Z instruction. We introduce dynamic gate set and paging mechanisms to implement the instructions. To evaluate the functionality and performance of these two instructions, we implement a two-qubit version of an algorithm to study a disorder-induced metal-insulator transition and run 60 random instances of it, each of which realizes one disorder configuration and contains 40 two-qubit instructions (or gates) and 104 single-qubit instructions. We observe the expected quantum dynamics of the time-evolution of this system., Comment: Received August 15, 2019; revised December 9, 2019; accepted December 13, 2019; date of publication January 28, 2020; date of current version February 14, 2020

Amplitude estimation is a fundamental quantum algorithmic primitive that enables quantum computers to achieve quadratic speedups for a large class of statistical estimation problems, including Monte Carlo methods. The main drawback from the perspective of near term hardware implementations is that the amplitude estimation algorithm requires very deep quantum circuits. Recent works have succeeded in somewhat reducing the necessary resources for such algorithms, by trading off some of the speedup for lower depth circuits, but high quality qubits are still needed for demonstrating such algorithms. Here, we report the results of an experimental demonstration of amplitude estimation on a state-of-the-art trapped ion quantum computer. The amplitude estimation algorithms were used to estimate the inner product of randomly chosen four-dimensional unit vectors, and were based on the maximum likelihood estimation (MLE) and the Chinese remainder theorem (CRT) techniques. Significant improvements in accuracy were observed for the MLE based approach when deeper quantum circuits were taken into account, including circuits with more than ninety two-qubit gates and depth sixty, achieving a mean additive estimation error on the order of $10^{-2}$. The CRT based approach was found to provide accurate estimates for many of the data points but was less robust against noise on average. Last, we analyze two more amplitude estimation algorithms that take into account the specifics of the hardware noise to further improve the results.

Modeling joint probability distributions is an important task in a wide variety of fields. One popular technique for this employs a family of multivariate distributions with uniform marginals called copulas. While the theory of modeling joint distributions via copulas is well understood, it gets practically challenging to accurately model real data with many variables. In this work, we design quantum machine learning algorithms to model copulas. We show that any copula can be naturally mapped to a multipartite maximally entangled state. A variational ansatz we christen as a `qopula' creates arbitrary correlations between variables while maintaining the copula structure starting from a set of Bell pairs for two variables, or GHZ states for multiple variables. As an application, we train a Quantum Generative Adversarial Network (QGAN) and a Quantum Circuit Born Machine (QCBM) using this variational ansatz to generate samples from joint distributions of two variables for historical data from the stock market. We demonstrate our generative learning algorithms on trapped ion quantum computers from IonQ for up to 8 qubits and show that our results outperform those obtained through equivalent classical generative learning. Further, we present theoretical arguments for exponential advantage in our model's expressivity over classical models based on communication and computational complexity arguments., 19 pages, 11 figures. v2: published version

Quantum machine learning has seen considerable theoretical and practical developments in recent years and has become a promising area for finding real world applications of quantum computers. In pursuit of this goal, here we combine state-of-the-art algorithms and quantum hardware to provide an experimental demonstration of a quantum machine learning application with provable guarantees for its performance and efficiency. In particular, we design a quantum Nearest Centroid classifier, using techniques for efficiently loading classical data into quantum states and performing distance estimations, and experimentally demonstrate it on a 11-qubit trapped-ion quantum machine, matching the accuracy of classical nearest centroid classifiers for the MNIST handwritten digits dataset and achieving up to 100% accuracy for 8-dimensional synthetic data.

Quantum architects, with knowledge of both physics and computer engineering, are key to developing scalable quantum computers.

A disordered quantum system of interacting particles exhibits localized behavior when the disorder is large compared to the interaction strength. Studying this phenomenon on a quantum computer with no, or limited, error correction is challenging because even weak coupling to a thermal environment destroys most signatures of localization. Fortunately, spectral functions of local operators are known to contain features that can survive the presence of noise. In these spectra, discrete peaks and a soft gap at low frequencies compared to the thermal phase indicate localization. Here, we present the computation of spectral functions on a trapped-ion quantum computer for a one-dimensional Heisenberg model with disorder. Further, we design an error-mitigation technique which is effective at removing the noise from the measurement allowing clear signatures of localization to emerge as the disorder increases. Thus, we show that spectral functions can serve as a robust and scalable diagnostic of many-body localization on current and future generations of quantum computers.

Quantum computers have the potential to perform accurate and efficient electronic structure calculations, enabling the simulation of properties of materials. However, today’s noisy, intermediate-scale quantum (NISQ) devices have a limited number of qubits and gate operations due to the presence of errors. Here, we propose a systematically improvable end-to-end pipeline to alleviate these limitations. Our proposed resource-efficient pipeline combines problem decomposition techniques for compact molecular representations, circuit optimization methods for compilation, solving the eigenvalue problem on advanced quantum hardware, and error-mitigation techniques in post-processing the results. Using the density matrix embedding theory for compact representation, and an ion-trap quantum computer, we simulate a ring of 10 hydrogen atoms taking into account all electrons equally and explicitly in the electronic structure calculation. In our experiment, we simulated the largest molecular system on a quantum computer within chemical accuracy with respect to total molecular energy calculated by the full CI method. Our methods reduce the number of qubits required for high-accuracy quantum simulations by an order of magnitude in the present work, enabling the simulation of larger, more industrially relevant molecules using NISQ devices. They are further systematically improvable as devices’ computational capacity continues to grow.

In this work we introduce an open source suite of quantum application-oriented performance benchmarks that is designed to measure the effectiveness of quantum computing hardware at executing quantum applications. These benchmarks probe a quantum computer's performance on various algorithms and small applications as the problem size is varied, by mapping out the fidelity of the results as a function of circuit width and depth using the framework of volumetric benchmarking. In addition to estimating the fidelity of results generated by quantum execution, the suite is designed to benchmark certain aspects of the execution pipeline in order to provide end-users with a practical measure of both the quality of and the time to solution. Our methodology is constructed to anticipate advances in quantum computing hardware that are likely to emerge in the next five years. This benchmarking suite is designed to be readily accessible to a broad audience of users and provides benchmarks that correspond to many well-known quantum computing algorithms., Comment: 33 pages, 36 figures; Added to Section VI about Impact of Compiler Optimization Techniques; Updated 20230105 for clarity after review feedback

Quantum computers can study thermodynamics by finding zeros of functions in the complex plane., Partition functions are ubiquitous in physics: They are important in determining the thermodynamic properties of many-body systems and in understanding their phase transitions. As shown by Lee and Yang, analytically continuing the partition function to the complex plane allows us to obtain its zeros and thus the entire function. Moreover, the scaling and nature of these zeros can elucidate phase transitions. Here, we show how to find partition function zeros on noisy intermediate-scale trapped-ion quantum computers in a scalable manner, using the XXZ spin chain model as a prototype, and observe their transition from XY-like behavior to Ising-like behavior as a function of the anisotropy. While quantum computers cannot yet scale to the thermodynamic limit, our work provides a pathway to do so as hardware improves, allowing the future calculation of critical phenomena for systems beyond classical computing limits.

Recently, there has been much interest in the efficient preparation of complex quantum states using low-depth quantum circuits, such as Quantum Approximate Optimization Algorithm (QAOA). While it has been numerically shown that such algorithms prepare certain correlated states of quantum spins with surprising accuracy, a systematic way of quantifying the efficiency of QAOA in general classes of models has been lacking. Here, we propose that the success of QAOA in preparing ordered states is related to the interaction distance of the target state, which measures how close that state is to the manifold of all Gaussian states in an arbitrary basis of single-particle modes. We numerically verify this for several examples of non-integrable quantum models, including Ising models with two- and three-spin interactions and the cluster model in an external field. Our results suggest that the structure of the entanglement spectrum, as witnessed by the interaction distance, correlates with the success of QAOA state preparation, and that this correlation also contains information about different phases present in the model. We conclude that QAOA typically finds a solution that perturbs around the closest free-fermion state., 13 pages, 9 figures

Significance Our experiment prepares two types of nontrivial quantum states on a trapped ion quantum computer: the thermofield double state of the transverse-field Ising model at arbitrary temperature and the quantum critical state of the zero-temperature model. We use techniques motivated by the quantum approximate optimization algorithm, and we implement a hybrid quantum–classical optimization loop to prepare the quantum critical state. Our results pave the way for exploring strongly correlated models at finite temperature and teleportation protocols inspired by black hole physics., Finite-temperature phases of many-body quantum systems are fundamental to phenomena ranging from condensed-matter physics to cosmology, yet they are generally difficult to simulate. Using an ion trap quantum computer and protocols motivated by the quantum approximate optimization algorithm (QAOA), we generate nontrivial thermal quantum states of the transverse-field Ising model (TFIM) by preparing thermofield double states at a variety of temperatures. We also prepare the critical state of the TFIM at zero temperature using quantum–classical hybrid optimization. The entanglement structure of thermofield double and critical states plays a key role in the study of black holes, and our work simulates such nontrivial structures on a quantum computer. Moreover, we find that the variational quantum circuits exhibit noise thresholds above which the lowest-depth QAOA circuits provide the best results.

A many-body Hamiltonian can be block-diagonalized by expressing it in terms of symmetry-adapted basis states. Finding the group orbit representatives of these basis states and their corresponding symmetries is currently a memory/computational bottleneck on classical computers during exact diagonalization. We apply Grover's search in the form of a minimization procedure to solve this problem. Our quantum solution provides an exponential reduction in memory, and a quadratic speedup in time over classical methods. We discuss explicitly the full circuit implementation of Grover minimization as applied to this problem, finding that the oracle only scales as polylog in the size of the group, which acts as the search space. Further, we design an error mitigation scheme that, with no additional qubits, reduces the impact of bit-flip errors on the computation, with the magnitude of mitigation directly correlated with the error rate, improving the utility of the algorithm in the Noisy Intermediate Scale Quantum era., 18 pages, 14 figures, added some additional analysis of the AEM error mitigation

Simulation of quantum systems is expected to be one of the most important applications of quantum computing, with much of the theoretical work so far having focused on fermionic and spin-$\frac{1}{2}$ systems. Here, we instead consider encodings of $d$-level (i.e. qudit) quantum operators into multi-qubit operators, studying resource requirements for approximating operator exponentials by Trotterization. We primarily focus on spin-$s$ and truncated bosonic operators in second quantization, observing desirable properties for approaches based on the Gray code, which to our knowledge has not been used in this context previously. After outlining a methodology for implementing an arbitrary encoding, we investigate the interplay between Hamming distances, sparsity patterns, bosonic truncation, and other properties of local operators. Finally, we obtain resource counts for five common Hamiltonian classes used in physics and chemistry, while modeling the possibility of converting between encodings within a Trotter step. The most efficient encoding choice is heavily dependent on the application and highly sensitive to $d$, although clear trends are present. These operation count reductions are relevant for running algorithms on near-term quantum hardware because the savings effectively decrease the required circuit depth. Results and procedures outlined in this work may be useful for simulating a broad class of Hamiltonians on qubit-based digital quantum computers., Comment: Edits for final NPJ-QI version

Quantum computing hardware is undergoing rapid development from proof-of-principle devices to scalable machines that could eventually challenge classical supercomputers on specific tasks. On platforms with local connectivity, the transition from one- to two-dimensional arrays of qubits is seen as a natural technological step to increase the density of computing power and to reduce the routing cost of limited connectivity. Here we map and schedule representative algorithmic workloads - the Quantum Fourier Transform (QFT) relevant to factoring, the Grover diffusion operator relevant to quantum search, and Jordan-Wigner parity rotations relevant to simulations of quantum chemistry and materials science - to qubit arrays with varying connectivity. In particular we investigate the impact of restricting the ideal all-to-all connectivity to a square grid, a ladder and a linear array of qubits. Our schedule for the QFT on a ladder results in running time close to that of a system with all-to-all connectivity. Our results suggest that some common quantum algorithm primitives can be optimized to have execution times on systems with limited connectivities, such as a ladder and linear array, that are competitive with systems that have all-to-all connectivity

We present a quantum algorithm to compute the entanglement spectrum of arbitrary quantum states. The interesting universal part of the entanglement spectrum is typically contained in the largest eigenvalues of the density matrix which can be obtained from the lower Renyi entropies through the Newton-Girard method. Obtaining the $p$ largest eigenvalues ($\lambda_1>\lambda_2\ldots>\lambda_p$) requires a parallel circuit depth of $\mathcal{O}(p(\lambda_1/\lambda_p)^p)$ and $\mathcal{O}(p\log(N))$ qubits where up to $p$ copies of the quantum state defined on a Hilbert space of size $N$ are needed as the input. We validate this procedure for the entanglement spectrum of the topologically-ordered Laughlin wave function corresponding to the quantum Hall state at filling factor $\nu=1/3$. Our scaling analysis exposes the tradeoffs between time and number of qubits for obtaining the entanglement spectrum in the thermodynamic limit using finite-size digital quantum computers. We also illustrate the utility of the second Renyi entropy in predicting a topological phase transition and in extracting the localization length in a many-body localized system.

The efficient simulation of correlated quantum systems is the most promising near-term application of quantum computers. Here, we present a measurement of the second Renyi entropy of the ground state of the two-site Fermi-Hubbard model on a $5$-qubit programmable quantum computer based on trapped ions. Our work illustrates the extraction of a non-linear characteristic of a quantum state using a controlled-swap gate acting on two copies of the state. This scalable measurement of entanglement on a universal quantum computer will, with more qubits, provide insights into many-body quantum systems that are impossible to simulate on classical computers., Comment: main text: 5 pages, 4 figures, 2 F-H systems, 2 qubits each, 1 ancilla; supplementary material: 4 pages, 7 figures

We report a systematic study of the fractional quantum Hall effect (FQHE) using the density-matrix renormalization group (DMRG) method on two different geometries: the sphere and the cylinder. We provide convergence benchmarks based on model Hamiltonians known to possess exact zero-energy ground states, as well as an analysis of the number of sweeps and basis elements that need to be kept in order to achieve the desired accuracy.The ground state energies of the Coulomb Hamiltonian at $\nu=1/3$ and $\nu=5/2$ filling are extracted and compared with the results obtained by previous DMRG implementations in the literature. A remarkably rapid convergence in the cylinder geometry is noted and suggests that this boundary condition is particularly suited for the application of the DMRG method to the FQHE., Comment: 5 pages, 7 figures

We use numerical exact diagonalization to analyze which aspects of the many-body localization phenomenon survive in an imperfectly isolated setting, when the system of interest is weakly coupled to a thermalizing environment. We show that widely used diagnostics (such as many-body level statistics and expectation values in exact eigenstates) cease to show signatures of many-body localization above a critical coupling that is exponentially small in the size of the environment. However, we also identify alternative diagnostics for many-body localization, in the spectral functions of local operators. Diagnostics include a discrete spectrum and a hierarchy of energy gaps, including a universal gap at zero frequency. These alternative diagnostics are shown to be robust, and continue to show signatures of many-body localization as long as the coupling to the bath is weaker than the characteristic energy scales in the system. We also examine how these signatures disappear when the coupling to the environment becomes larger than the characteristic energy scales of the system.

We describe a large-disorder renormalization group (LDRG) method for the Anderson model of localization in one dimension which decimates eigenstates based on the size of their wave functions rather than their energy. We show that our LDRG scheme flows to infinite disorder, and thus becomes asymptotically exact. We use it to obtain the disorder-averaged inverse participation ratio (IPR) and density of states (DOS) for the entire spectrum. A modified scheme is formulated for higher dimensions, which is found to be less efficient, but capable of improvement.

We determine the size of the elementary quasihole in $\ensuremath{\nu}=1/3$ and $\ensuremath{\nu}=7/3$ quantum Hall states via exact-diagonalization and density-matrix renormalization group calculations on the sphere and cylinder, using a variety of short- and long-range pinning potentials. The size of the quasihole at filling factor $\ensuremath{\nu}=1/3$ is estimated to be $\ensuremath{\approx}4{\ensuremath{\ell}}_{B}$, and that of $\ensuremath{\nu}=7/3$ is $\ensuremath{\approx}7{\ensuremath{\ell}}_{B}$, where ${\ensuremath{\ell}}_{B}$ is the magnetic length. In contrast, the size of the Laughlin quasihole, expected to capture the basic physics in these two states, is around $\ensuremath{\approx}2.5{\ensuremath{\ell}}_{B}$. Our work supports the earlier findings that the quasihole in the first excited Landau level is significantly larger than in the lowest Landau level.

We show that the probability of electric field induced interband tunneling in solid state systems is generically a non-monotonic (oscillatory) function of the applied field. This unexpected behavior can be understood as arising due to a common path interference between two distinct tunneling solutions. The phenomenon is insensitive to magnetic field, and arises whenever the low energy dispersion relation contains higher order terms in addition to the usual $p^2$ term. Such higher order terms are generically present, albeit with small co-efficient, so that the oscillatory Zener tunneling is a universal phenomenon. However, the first `Zener oscillation' occurs at a transmission probability which is exponentially small when the co-efficient of the higher order terms is small. This explains why this oscillatory aspect of Zener tunneling has been hitherto overlooked, despite its universality. The common path interference is also destroyed by the presence of odd powers of $p$ in the low energy dispersion relation. Since odd powers of $p$ are strictly absent only when the tunneling barrier lies along an axis of mirror symmetry, it follows that the robustness of the oscillatory behavior depends on the orientation of the tunneling barrier. Bilayer graphene is identified as a particularly good material for observation of common path interference, due to its unusual nearly isotropic dispersion relation, where the $p^4$ term makes the leading contribution.

We discuss the role of rare fluctuation effects in quantum condensed matter systems. In particular, we present recent numerical results of the effect of resonant states in Anderson's original model of electron localization. We find that such resonances give rise to anomalous behavior of eigenstates not just far in the Lifshitz tail, but rather for a substantial fraction of eigenstates, especially for intermediate disorder. The anomalous behavior includes non-analyticity in various properties as a characteristic. The effect of dimensionality on the singularity, which is present in all dimensions, is described, and the behavior for bounded and unbounded disorder is contrasted.

It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantum Hall states -- the filling $\nu=1/3$ Laughlin state. We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize the concept of coherent states to formulate the bulk off-diagonal long range order for the Laughlin state, and compute the deformations of the metric in the vicinity of the edge of the system. We introduce a "pair amplitude" operator and show that it can be used to numerically determine the intrinsic metric of the Laughlin state. These various probes are applied to several experimentally relevant settings that can expose the quantum geometry of the Laughlin state, in particular to systems with mass anisotropy and in the presence of an electric field gradient., Comment: 26 pages, 15 figures; to appear in NJP focus issue on "Topological physics: from condensed matter to cold atoms and optics"

We show analytically that the apparent non-analyticity discovered recently in the inverse participation ratio (IPR) of the eigenstates in Anderson's model of localization is also present in a simple two-site model, along with a concurrent non-analyticity in the density of states (DOS) at the same energy. We demonstrate its evolution from two sites to the thermodynamic limit by numerical methods. For the two site model, non-analyticity in higher derivatives of the DOS and IPR is also proven to exist for all bounded distributions of disorder.

We observe a singularity in the electronic properties of the Anderson Model of Localization with bounded diagonal disorder, which is clearly distinct from the well-established mobility edge (localization-delocalization transition) that occurs in dimensions $d>2$. We present results of numerical calculations for Anderson's original (box) distribution of onsite disorder in dimensions $d$ = 1, 2 and 3. To establish this hitherto unreported behavior, and to understand its evolution with disorder, we contrast the behavior of two different measures of the localization length of the electronic wavefunctions - the averaged inverse participation ratio and the Lyapunov exponent. Our data suggest that Anderson's model exhibits richer behavior than has been established so far., Correction to v1: Fig.3 caption now displayed

Extended Bose Hubbard models with nearest neighbour interaction describe minimally the effect of long range interaction on ultra cold atoms in deep optical lattices. Rotation of such optical lattices subject such neutral cold atoms to the effect of an artificial magnetic field. The modification of the phase boundaries of the density wave and Mott Insulator phases due to this rotation are shown to be related to the edge spectrum of spinorial and scalar Harper equation. Corresponding profiles of the checkerboard vortex states with sublattice modulated superfluid order parameter near density wave phase boundary are calculated., Comment: Slightly revised, Accepted in Physical Review A for publication