Nearly-optimal state preparation for quantum simulations of lattice gauge theories

We present several improvements to the recently developed ground state preparation algorithm based on the Quantum Eigenvalue Transformation for Unitary Matrices (QETU), apply this algorithm to a lattice formulation of U(1) gauge theory in 2+1D, as well as propose a novel application of QETU, a highly efficient preparation of Gaussian distributions. The QETU technique has been originally proposed as an algorithm for nearly-optimal ground state preparation and ground state energy estimation on early fault-tolerant devices. It uses the time-evolution input model, which can potentially overcome the large overall prefactor in the asymptotic gate cost arising in similar algorithms based on the Hamiltonian input model. We present modifications to the original QETU algorithm that significantly reduce the cost for the cases of both exact and Trotterized implementation of the time evolution circuit. We use QETU to prepare the ground state of a U(1) lattice gauge theory in 2 spatial dimensions, explore the dependence of computational resources on the desired precision and system parameters, and discuss the applicability of our results to general lattice gauge theories. We also demonstrate how the QETU technique can be utilized for preparing Gaussian distributions and wave packets in a way which outperforms existing algorithms for as little as $n_q > 2-3$ qubits.Comment: 29 pages, 4 appendices, 19 figure

A hybrid method for quantum dynamics simulation

We propose a hybrid approach to simulate quantum many body dynamics by combining Trotter based quantum algorithm with classical dynamic mode decomposition. The interest often lies in estimating observables rather than explicitly obtaining the wave function's form. Our method predicts observables of a quantum state in the long time by using data from a set of short time measurements from a quantum computer. The upper bound for the global error of our method scales as $O(t^{3/2})$ with a fixed set of the measurement. We apply our method to quench dynamics in Hubbard model and nearest neighbor spin systems and show that the observable properties can be predicted up to a reasonable error by controlling the number of data points obtained from the quantum measurements.Comment: 9 pages, 4 figure

Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations

Scalable quantum algorithms for the simulation of quantum many-body systems in thermal equilibrium are important for predicting properties of quantum matter at finite temperatures. Here we describe and benchmark a quantum computing version of the minimally entangled typical thermal states (METTS) algorithm for which we adopt an adaptive variational approach to perform the required quantum imaginary time evolution. The algorithm, which we name AVQMETTS, dynamically generates compact and problem-specific quantum circuits, which are suitable for noisy intermediate-scale quantum (NISQ) hardware. We benchmark AVQMETTS on statevector simulators and perform thermal energy calculations of integrable and nonintegrable quantum spin models in one and two dimensions and demonstrate an approximately linear system-size scaling of the circuit complexity. We further map out the finite-temperature phase transition line of the two-dimensional transverse field Ising model. Finally, we study the impact of noise on AVQMETTS calculations using a phenomenological noise model.Comment: 13 pages, 6 figure

Efficient Step-Merged Quantum Imaginary Time Evolution Algorithm for Quantum Chemistry

We develop a resource-efficient step-merged quantum imaginary time evolution approach (smQITE) to solve for the ground state of a Hamiltonian on quantum computers. This heuristic method features a fixed shallow quantum circuit depth along the state evolution path. We use this algorithm to determine the binding energy curves of a set of molecules, including H2, H4, H6, LiH, HF, H2O, and BeH2, and find highly accurate results. The required quantum resources of smQITE calculations can be further reduced by adopting the circuit form of the variational quantum eigensolver (VQE) technique, such as the unitary coupled cluster ansatz. We demonstrate that smQITE achieves a similar computational accuracy as VQE at the same fixed-circuit ansatz, without requiring a generally complicated high-dimensional nonconvex optimization. Finally, smQITE calculations are carried out on Rigetti quantum processing units, demonstrating that the approach is readily applicable on current noisy intermediate-scale quantum devices

Adaptive Variational Quantum Dynamics Simulations

We propose a general-purpose, self-adaptive approach to construct variational wavefunction ans\"atze for highly accurate quantum dynamics simulations based on McLachlan's variational principle. The key idea is to dynamically expand the variational ansatz along the time-evolution path such that the ``McLachlan distance'', which is a measure of the simulation accuracy, remains below a set threshold. We apply this adaptive variational quantum dynamics simulation (AVQDS) approach to the integrable Lieb-Schultz-Mattis spin chain and the nonintegrable mixed-field Ising model, where it captures both finite-rate and sudden post-quench dynamics with high fidelity. The AVQDS quantum circuits that prepare the time-evolved state are much shallower than those obtained from first-order Trotterization and contain up to two orders of magnitude fewer CNOT gate operations. We envision that a wide range of dynamical simulations of quantum many-body systems on near-term quantum computing devices will be made possible through the AVQDS framework.Comment: 12 pages, 7 figure

Shallow-circuit variational quantum eigensolver based on symmetry-inspired Hilbert space partitioning for quantum chemical calculations

Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing. Based on the variational quantum eigensolver (VQE) with unitary coupled cluster ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace. In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit. With these strategies, VQE calculations on a noiseless quantum simulator achieve energies within a few meVs of those obtained with the full UCCSD ansatz for $\mathrm{H}_4$ square, $\mathrm{H}_4$ chain and $\mathrm{H}_6$ hexagon molecules; while the number of controlled-NOT (CNOT) gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35. Furthermore, we introduced an efficient "score" parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first. Using $\mathrm{H}_4$ square and $\mathrm{H}_4$ chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational ansatz

Penilaian Kinerja Keuangan Koperasi di Kabupaten Pelalawan

This paper describe development and financial performance of cooperative in District Pelalawan among 2007 - 2008. Studies on primary and secondary cooperative in 12 sub-districts. Method in this stady use performance measuring of productivity, efficiency, growth, liquidity, and solvability of cooperative. Productivity of cooperative in Pelalawan was highly but efficiency still low. Profit and income were highly, even liquidity of cooperative very high, and solvability was good

Juxtaposing BTE and ATE – on the role of the European insurance industry in funding civil litigation

One of the ways in which legal services are financed, and indeed shaped, is through private insurance arrangement. Two contrasting types of legal expenses insurance contracts (LEI) seem to dominate in Europe: before the event (BTE) and after the event (ATE) legal expenses insurance. Notwithstanding institutional differences between different legal systems, BTE and ATE insurance arrangements may be instrumental if government policy is geared towards strengthening a market-oriented system of financing access to justice for individuals and business. At the same time, emphasizing the role of a private industry as a keeper of the gates to justice raises issues of accountability and transparency, not readily reconcilable with demands of competition. Moreover, multiple actors (clients, lawyers, courts, insurers) are involved, causing behavioural dynamics which are not easily predicted or influenced. Against this background, this paper looks into BTE and ATE arrangements by analysing the particularities of BTE and ATE arrangements currently available in some European jurisdictions and by painting a picture of their respective markets and legal contexts. This allows for some reflection on the performance of BTE and ATE providers as both financiers and keepers. Two issues emerge from the analysis that are worthy of some further reflection. Firstly, there is the problematic long-term sustainability of some ATE products. Secondly, the challenges faced by policymakers that would like to nudge consumers into voluntarily taking out BTE LEI

Severe early onset preeclampsia: short and long term clinical, psychosocial and biochemical aspects

Preeclampsia is a pregnancy specific disorder commonly defined as de novo hypertension and proteinuria after 20 weeks gestational age. It occurs in approximately 3-5% of pregnancies and it is still a major cause of both foetal and maternal morbidity and mortality worldwide1. As extensive research has not yet elucidated the aetiology of preeclampsia, there are no rational preventive or therapeutic interventions available. The only rational treatment is delivery, which benefits the mother but is not in the interest of the foetus, if remote from term. Early onset preeclampsia (<32 weeks’ gestational age) occurs in less than 1% of pregnancies. It is, however often associated with maternal morbidity as the risk of progression to severe maternal disease is inversely related with gestational age at onset2. Resulting prematurity is therefore the main cause of neonatal mortality and morbidity in patients with severe preeclampsia3. Although the discussion is ongoing, perinatal survival is suggested to be increased in patients with preterm preeclampsia by expectant, non-interventional management. This temporising treatment option to lengthen pregnancy includes the use of antihypertensive medication to control hypertension, magnesium sulphate to prevent eclampsia and corticosteroids to enhance foetal lung maturity4. With optimal maternal haemodynamic status and reassuring foetal condition this results on average in an extension of 2 weeks. Prolongation of these pregnancies is a great challenge for clinicians to balance between potential maternal risks on one the eve hand and possible foetal benefits on the other. Clinical controversies regarding prolongation of preterm preeclamptic pregnancies still exist – also taking into account that preeclampsia is the leading cause of maternal mortality in the Netherlands5 - a debate which is even more pronounced in very preterm pregnancies with questionable foetal viability6-9. Do maternal risks of prolongation of these very early pregnancies outweigh the chances of neonatal survival? Counselling of women with very early onset preeclampsia not only comprises of knowledge of the outcome of those particular pregnancies, but also knowledge of outcomes of future pregnancies of these women is of major clinical importance. This thesis opens with a review of the literature on identifiable risk factors of preeclampsia