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The industrial manufacturing of chemicals consumes a significant amount of energy and raw materials. In principle, the development of new catalysts could greatly improve the efficiency of chemical production. However, the discovery of viable catalysts can be exceedingly challenging because it is difficult to know the efficacy of a candidate without experimentally synthesizing and characterizing it. This study explores the feasibility of using fault-tolerant quantum computers to accelerate the discovery of homogeneous catalysts for nitrogen fixation, an industrially important chemical process. It introduces a set of ground-state energy estimation problems representative of calculations needed for the discovery of homogeneous catalysts and analyzes them on three dimensions: economic utility, classical hardness, and quantum resource requirements. For the highest utility problem considered, two steps of a catalytic cycle for the generation of cyanate anion from dinitrogen, the economic utility of running these computations is estimated to be $200,000, and the required runtime for double-factorized phase estimation on a fault-tolerant superconducting device is estimated under conservative assumptions to be 139,000 QPU-hours. The computational cost of an equivalent DMRG calculation is estimated to be about 400,000 CPU-hours. These results suggest that, with continued development, it will be feasible for fault-tolerant quantum computers to accelerate the discovery of homogeneous catalysts., Comment: 27 pages, 11 tables, 8 figures plus appendix

We consider the question of how correlated the system hardness is between classical algorithms of electronic structure theory in ground state estimation and quantum algorithms. To define the system hardness for classical algorithms we employ empirical criterion based on the deviation of electronic energies produced by coupled cluster and configuration interaction methods from the exact ones along multiple bonds dissociation in a set of molecular systems. For quantum algorithms, we have selected the Variational Quantum Eigensolver (VQE) and Quantum Phase Estimation (QPE) methods. As characteristics of the system hardness for quantum methods, we analyzed circuit depths for the state preparation, the number of quantum measurements needed for the energy expectation value, and various cost characteristics for the Hamiltonian encodings via Trotter approximation and linear combination of unitaries (LCU). Our results show that the quantum resource requirements are mostly unaffected by classical hardness, with the only exception being the state preparation part, which contributes to both VQE and QPE algorithm costs. However, there are clear indications that constructing the initial state with a significant overlap with the true ground state (>10%) is easier than obtaining the state with an energy expectation value within chemical precision. These results support optimism regarding the identification of a molecular system where a quantum algorithm excels over its classical counterpart, as quantum methods can maintain efficiency in classically challenging systems., Comment: 11 pages

Accurately solving the electronic structure problem through the variational quantum eigensolver (VQE) is hindered by the available quantum resources of current and near-term devices. One approach to relieving the circuit depth requirements for VQE is to "pre-process" the electronic Hamiltonian by a similarity transformation incorporating some degree of electronic correlation, with the remaining correlation left to be addressed by the circuit ansatz. This often comes at the price of a substantial increase in the number of terms to measure in the similarity transformed Hamiltonian. In this work, we propose an efficient approach to sampling elements from the complete Pauli group for $N$ qubits which minimize the onset of new terms in the transformed Hamiltonian, while facilitating substantial energy lowering. We benchmark the growth-mitigating generator selection technique for ground state energy estimations applied to models of the H$_4$, N$_2$ and H$_2$O molecular systems. It is found that utilizing a selection procedure which obtains the growth-minimizing generator from the set of operators with maximal energy gradient is the most competitive approach to reducing the onset of Hamiltonian terms while achieving systematic energy lowering of the reference state., Comment: 14 pages, 4 figures

This work describes the theoretical foundation for all quantum chemistry functionality in PennyLane, a quantum computing software library specializing in quantum differentiable programming. We provide an overview of fundamental concepts in quantum chemistry, including the basic principles of the Hartree-Fock method. A flagship feature in PennyLane is the differentiable Hartree-Fock solver, allowing users to compute exact gradients of molecular Hamiltonians with respect to nuclear coordinates and basis set parameters. PennyLane provides specialized operations for quantum chemistry, including excitation gates as Givens rotations and templates for quantum chemistry circuits. Moreover, built-in simulators exploit sparse matrix techniques for representing molecular Hamiltonians that lead to fast simulation for quantum chemistry applications. In combination with PennyLane's existing methods for constructing, optimizing, and executing circuits, these methods allow users to implement a wide range of quantum algorithms for quantum chemistry. We discuss how PennyLane can be used to implement variational algorithms for calculating ground-state energies, excited-state energies, and energy derivatives, all of which can be differentiated with respect to both circuit and Hamiltonian parameters. We provide an example workflow describing how to jointly optimize circuit parameters, nuclear coordinates, and basis set parameters for quantum chemistry algorithms. We discuss a functionality for reducing the number of qubits by using symmetries and explain how PennyLane can be used to estimate quantum resources needed to implement several quantum algorithms. By combining insights from quantum computing, computational chemistry, and machine learning, PennyLane is the first library for differentiable quantum computational chemistry.

Optimization of unitary transformations in Variational Quantum Algorithms benefits highly from efficient evaluation of cost function gradients with respect to amplitudes of unitary generators. We propose several extensions of the parametric-shift-rule to formulating these gradients as linear combinations of expectation values for generators with general eigen-spectrum (i.e. with more than two eigenvalues). Our approaches are exact and do not use any auxiliary qubits, instead they rely on a generator eigen-spectrum analysis. Two main directions in the parametric-shift-rule extensions are 1) polynomial expansion of the exponential unitary operator based on a limited number of different eigenvalues in the generator and 2) decomposition of the generator as a linear combination of low-eigenvalue operators (e.g. operators with only 2 or 3 eigenvalues). These techniques have a range of scalings for the number of needed expectation values with the number of generator eigenvalues from quadratic (for polynomial expansion) to linear and even $\log_2$ (for generator decompositions). This allowed us to propose efficient differentiation schemes superior to previous approaches for commonly used 2-qubit transformations (e.g. match-gates, transmon and fSim gates) and $\hat S^2$-conserving fermionic operators for the variational quantum eigensolver.

Origami design principles are scale invariant and enable direct miniaturization of origami structures provided the sheets used for folding have equal thickness to length ratios. Recently, seminal steps have been taken to fabricate microscale origami using unidirectionally actuated sheets with nanoscale thickness. Here, we extend the full power of origami-inspired fabrication to nanoscale sheets by engineering bidirectional folding with 4 nm thick atomic layer deposition (ALD) SiNx-SiO2 bilayer films. Strain differentials within these bilayers result in bending, producing microscopic radii of curvature. We lithographically pattern these bilayers and localize the bending using rigid panels to fabricate a variety of complex micro-origami devices. Upon release, these devices self-fold according to prescribed patterns. Our approach combines planar semiconductor microfabrication methods with computerized origami design, making it easy to fabricate and deploy such microstructures en masse. These devices represent an important step forward in the fabrication and assembly of deployable micromechanical systems that can interact with and manipulate micro- and nanoscale environments., Comment: This document is the unedited Author's version of a Submitted Work that was subsequently accepted for publication in Nano Letters, copyright American Chemical Society after peer review. To access the final edited and published work see: https://pubs.acs.org/articlesonrequest/AOR-JJHKJZVTJF4JFWSYZ2NW

One of the main challenges in the Variational Quantum Eigensolver (VQE) framework is construction of the unitary transformation. The dimensionality of the space for unitary rotations of $N$ qubits is $4^N-1$, which makes the choice of a polynomial subset of generators exponentially difficult process. Moreover, due to non-commutativity of generators, the order in which they are used strongly affects results. Choosing the optimal order in a particular subset of generators requires testing the factorial number of combinations. We propose an approach based on the Lie algebra - Lie group connection and corresponding closure relations that systematically eliminates the order problem.

Application of current and near-term quantum hardware to the electronic structure problem is highly limited by qubit counts, coherence times, and gate fidelities. To address these restrictions within the variational quantum eigensolver (VQE) framework, many recent contributions have suggested dressing the electronic Hamiltonian to include a part of electron correlation, leaving the rest to be accounted by VQE state preparation. We present a new dressing scheme that combines preservation of the Hamiltonian hermiticity and an exact quadratic truncation of the Baker-Campbell-Hausdorff expansion. The new transformation is constructed as the exponent of an involutory linear combination (ILC) of anti-commuting Pauli products. It incorporates important strong correlation effects in the dressed Hamiltonian and can be viewed as a classical preprocessing step alleviating the resource requirements of the subsequent VQE application. The assessment of the new computational scheme for electronic structure of the LiH, H$_2$O, and N$_2$ molecules shows significant increase in efficiency compared to conventional qubit coupled cluster dressings.

To obtain estimates of electronic energies, the Variational Quantum Eigensolver (VQE) technique performs separate measurements for multiple parts of the system Hamiltonian. Current quantum hardware is restricted to projective single-qubit measurements, and thus, only parts of the Hamiltonian which form mutually qubit-wise commuting groups can be measured simultaneously. The number of such groups in the electronic structure Hamiltonians grows as $N^4$, where $N$ is the number of qubits, and thus puts serious restrictions on the size of the systems that can be studied. Using a partitioning of the system Hamiltonian as a linear combination of unitary operators we found a circuit formulation of the VQE algorithm that allows one to measure a group of fully anti-commuting terms of the Hamiltonian in a single series of single-qubit measurements. Numerical comparison of the unitary partitioning to previously used grouping of Hamiltonian terms based on their qubit-wise commutativity shows an $N$-fold reduction in the number of measurable groups.

An iterative version of the qubit coupled cluster (QCC) method [I.G. Ryabinkin et al., J. Chem. Theory Comput. 14, 6317 (2019)] is proposed. The new method seeks to find ground electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices. Each iteration involves a canonical transformation of the Hamiltonian and employs constant-size quantum circuits at the expense of increasing the Hamiltonian size. We numerically studied the convergence of the method on ground-state calculations for LiH, H$_2$O, and N$_2$ molecules and found that the exact ground-state energies can be systematically approached only if the generators of the QCC ansatz are sampled from a specific set of operators. We report an algorithm for constructing this set that scales linearly with the size of a Hamiltonian., Comment: 10 pages; 6 figures, largely rewritten

Solving the electronic structure problem on a universal-gate quantum computer within the variational quantum eigensolver (VQE) methodology requires constraining the search procedure to a subspace defined by relevant physical symmetries. Ignoring symmetries results in convergence to the lowest eigenstate of the Fock space for the second quantized electronic Hamiltonian. Moreover, this eigenstate can be symmetry broken due to limitations of the wavefunction ansatz. To address this VQE problem, we introduce and assess methods of exact and approximate projection operators to irreducible eigen-subspaces of available physical symmetries. Feasibility of symmetry projection operators in the VQE framework is discussed, and their efficiency is compared with symmetry constraint optimization procedures. Generally, projectors introduce higher numbers of terms for VQE measurement compared to the constraint approach. On the other hand, the projection formalism improves accuracy of the variational wavefunction ansatz without introducing additional unitary transformations, which is beneficial for reducing depths of quantum circuits.

PennyLane is a Python 3 software framework for differentiable programming of quantum computers. The library provides a unified architecture for near-term quantum computing devices, supporting both qubit and continuous-variable paradigms. PennyLane's core feature is the ability to compute gradients of variational quantum circuits in a way that is compatible with classical techniques such as backpropagation. PennyLane thus extends the automatic differentiation algorithms common in optimization and machine learning to include quantum and hybrid computations. A plugin system makes the framework compatible with any gate-based quantum simulator or hardware. We provide plugins for hardware providers including the Xanadu Cloud, Amazon Braket, and IBM Quantum, allowing PennyLane optimizations to be run on publicly accessible quantum devices. On the classical front, PennyLane interfaces with accelerated machine learning libraries such as TensorFlow, PyTorch, JAX, and Autograd. PennyLane can be used for the optimization of variational quantum eigensolvers, quantum approximate optimization, quantum machine learning models, and many other applications., Comment: Code available at https://github.com/XanaduAI/pennylane/ . Significant contributions to the code (new features, new plugins, etc.) will be recognized by the opportunity to be a co-author on this paper

Deployable mechanical systems such as space solar panels rely on the intricate stowage of passive modules, and sophisticated deployment using a network of motorized actuators. As a result, a significant portion of the stowed mass and volume are occupied by these support systems. An autonomous solar panel array deployed using the inherent material behavior remains elusive. In this work, we develop an autonomous self-deploying solar panel array that is programmed to activate in response to changes in the surrounding temperature. We study an elastic "flasher" origami sheet embedded in a circle of scissor mechanisms, both printed with shape memory polymers. The scissor mechanisms are optimized to provide the maximum expansion ratio while delivering the necessary force for deployment. The origami sheet is also optimized to carry the maximum number of solar panels given space constraints. We show how the folding of the "flasher" origami exhibits a bifurcation behavior resulting in either a cone or disk shape both numerically and in experiments. A folding strategy is devised to avoid the undesired cone shape. The resulting design is entirely 3D printed, achieves an expansion ratio of 1000% in under 40 seconds, and shows excellent agreement with simulation prediction both in the stowed and deployed configurations., Comment: 12 pages, 12 figures

The geometric, aesthetic, and mathematical elegance of origami is being recognized as a powerful pathway to self-assembly of micro and nano-scale machines with programmable mechanical properties. The typical approach to designing the mechanical response of an ideal origami machine is to include mechanisms where mechanical constraints transform applied forces into a desired motion along a narrow set of degrees of freedom. In fact, to date, most design approaches focus on building up complex mechanisms from simple ones in ways that preserve each individual mechanism's degree of freedom (DOF), with examples ranging from simple robotic arms to homogenous arrays of identical vertices, such as the well-known Miura-ori. However, such approaches typically require tight fabrication tolerances, and often suffer from parasitic compliance. In this work, we demonstrate a technique in which high-degree-of-freedom mechanisms associated with single vertices are heterogeneously combined so that the coupled phase spaces of neighboring vertices are pared down to a controlled range of motions. This approach has the advantage that it produces mechanisms that retain the DOF at each vertex, are robust against fabrication tolerances and parasitic compliance, but nevertheless effectively constrain the range of motion of the entire machine. We demonstrate the utility of this approach by mapping out the configuration space for the modified Miura-ori vertex of degree 6, and show that when strung together, their combined configuration spaces create mechanisms that isolate deformations, constrain the configuration topology of neighboring vertices, or lead to sequential bistable folding throughout the entire origami sheet., Comment: 10 pages, 4 figures

We develop an intrinsic necessary and sufficient condition for single-vertex origami crease patterns to be able to fold rigidly. We classify such patterns in the case where the creases are pre-assigned to be mountains and valleys as well as in the unassigned case. We also illustrate the utility of this result by applying it to the new concept of minimal forcing sets for rigid origami models, which are the smallest collection of creases that, when folded, will force all the other creases to fold in a prescribed way.

In this work the shear viscosity of strongly interacting matter is calculated within a two-flavor Nambu--Jona-Lasinio model as a function of temperature and chemical potential. The general Kubo formula is applied, incorporating the full Dirac structure of the thermal quark spectral function and avoiding commonly used on-shell approximations. Mesonic fluctuations contributing via Fock diagrams provide the dominant dissipative processes. The resulting ratio $\eta/s$ (shear viscosity over entropy density) decreases with temperature and chemical potential. Interpolating between our NJL results at low temperatures and hard-thermal-loop results at high temperatures a minimum slightly above the AdS/CFT benchmark $\eta/s=1/4\pi$ is obtained., Comment: 15 pages, 11 figures. Revision with minor corrections matches published version

A large-$N_{\rm c}$ expansion is combined with the Kubo formalism to study the shear viscosity $\eta$ of strongly interacting matter in the two-flavor NJL model. We discuss analytical and numerical approaches to $\eta$ and investigate systematically its strong dependence on the spectral width and the momentum-space cutoff. Thermal effects on the constituent quark mass from spontaneous chiral symmetry breaking are included. The ratio $\eta/s$ and its thermal dependence are derived for different parameterizations of the spectral width and for an explicit one-loop calculation including mesonic modes within the NJL model., Comment: 14 pages, 11 figures. Revision includes additionally the spectral width at one-loop level (chapter V) and Appendix A. Matches published version

The shear viscosity of an interacting pion gas is studied using the Kubo formalism as a microscopic description of thermal systems close to global equilibrium. We implement the skeleton expansion in order to approximate the retarded correlator of the viscous part of the energy-momentum tensor. After exploring this in $g\phi^4$ theory we show how the skeleton expansion can be consistently applied to pions in chiral perturbation theory. The shear viscosity $\eta$ is determined by the spectral width, or equivalently, the mean free path of pions in the heat bath. We derive a new analytical result for the mean free path which is well-conditioned for numerical evaluation and discuss the temperature and pion-mass dependence of the mean free path and the shear viscosity. The ratio $\eta/s$ of the interacting pion gas exceeds the lower bound $1/4\pi$ from AdS/CFT correspondence., Comment: 12 pages, 7 figures. Revision includes additional Appendix B. Matches published version

We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems. On the positive side, we show that any set of circles of total area 1 can be packed into a square of size 4/\sqrt{pi}=2.2567... These results are motivated by problems arising in the context of origami design., Comment: 17 pages, 13 figures. An abstract was presented at the 5th International Conference on Origami in Science, Mathematics and Education (5OSME). Updated to fix a numerical typo related to Figure 13, and a new result

Many observations indicate that dark matter dominates the extra-galactic Universe, yet no totally dark structure of galactic proportions has ever been convincingly identified. Previously we have suggested that VIRGOHI 21, a 21-cm source we found in the Virgo Cluster using Jodrell Bank, was a possible dark galaxy because of its broad line-width (~200 km/s) unaccompanied by any visible gravitational source to account for it. We have now imaged VIRGOHI 21 in the neutral-hydrogen line and find what could be a dark, edge-on, spinning disk with the mass and diameter of a typical spiral galaxy. Moreover, VIRGOHI 21 has unquestionably been involved in an interaction with NGC 4254, a luminous spiral with an odd one-armed morphology, but lacking the massive interactor normally linked with such a feature. Numerical models of NGC 4254 call for a close interaction ~10^8 years ago with a perturber of ~10^11 solar masses. This we take as additional evidence for the massive nature of VIRGOHI 21 as there does not appear to be any other viable candidate. We have also used the Hubble Space Telescope to search for stars associated with the HI and find none down to an I band surface brightness limit of 31.1 +/- 0.2 mag/sq. arcsec., Comment: 8 pages, accepted to ApJ, uses emulateapj.cls. Mpeg animation (Fig. 2) available at ftp://ftp.naic.edu/pub/publications/minchin/video2.mpg

VIRGOHI21 is an HI source detected in the Virgo Cluster survey of Davies et al. (2004) which has a neutral hydrogen mass of 10^8 M_solar and a velocity width of Delta V_20 = 220 km/s. From the Tully-Fisher relation, a galaxy with this velocity width would be expected to be 12th magnitude or brighter; however deep CCD imaging has failed to turn up a counterpart down to a surface-brightness level of 27.5 B mag/sq. arcsec. The HI observations show that it is extended over at least 16 kpc which, if the system is bound, gives it a minimum dynamical mass of ~10^11 M_solar and a mass to light ratio of M_dyn/L_B > 500 M_solar/L_solar. If it is tidal debris then the putative parents have vanished; the remaining viable explanation is that VIRGOHI21 is a dark halo that does not contain the expected bright galaxy. This object was found because of the low column density limit of our survey, a limit much lower than that achieved by all-sky surveys such as HIPASS. Further such sensitive surveys might turn up a significant number of the dark matter halos predicted by Dark Matter models., Comment: Accepted by ApJL