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-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 spins 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.

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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.One of the striking differences between classical and quantum systems is the phenomenon of entanglement. Analyzing large entangled states is of considerable interest for quantum computing applications. This is particularly relevant to quantum chemistry and materials science simulations involving interacting fermions [1,2], small versions of which have been simulated on few-qubit quantum computers [3][4][5]. Recently, a quantum algorithm was developed to construct the entanglement spectrum of an arbitrary wave function prepared on a quantum computer via measurement of the Renyi entropies [6]. In this Letter we measure the second Renyi entropy in a 5-qubit circuit by implementing a controlled-swap (C-Swap) gate, and mitigate experimental errors by exploiting the symmetry properties of this gate. We note that previous measurements of the Renyi entropy such as [7] were not implemented on universal machines and may not be easily generalizable to arbitrary Hamiltonians or scalable to larger systems.For a many-body quantum system ideally described by the state |Ψ and composed of two subsystems A and B, the nth Renyi entropy is given by S n = 1 1−n log(R n ), whereis the trace of the nth power of the reduced density matrix ρ A = Tr B (|Ψ Ψ|). For non-zero entanglement we have R 2 < 1, which has the same universality properties as the von Neumann entropy S = − Tr(ρ A log(ρ A )).Both are measures of the entanglement between A and B, and provide valuable information about the underlying physics of the system. For example, the Renyi entropy can be used to distinguish many-body localized states from thermalized states [8-12] through their time dependence and dimensional scaling law [13], and to study topological order [14,15] and quantum critical systems [16].The system under investigation for this work is the two-site Fermi-Hubbard model, which describes interacting electrons on a lattice [17,18]. Despite its simplicity, it has been postulated as a model for complex phenomena such as high-temperature superconductivity. Since its behavior in the thermodynamic limit remains inaccessible to classical numerical techniques, it has become a prime candidate for simulation by quantum computers [19,20]. Our work consists of several co-designed theoretical and experimental steps. First, we find an efficient mapping from the electronic problem to the qubit space. Second, we develop a circuit for...

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.

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

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. CONTENTS

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.

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 1 3 n = 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. OPEN ACCESS RECEIVED