An official website of the United States government
Abstract We perform path-integral molecular dynamics (PIMD), ring-polymer MD (RPMD), and classical MD simulations of H$$_2$$ O and D$$_2$$ O using the q-TIP4P/F water model over a wide range of temperatures and pressures. The density$$\rho (T)$$ , isothermal compressibility$$\kappa _T(T)$$ , and self-diffusion coefficientsD(T) of H$$_2$$ O and D$$_2$$ O are in excellent agreement with available experimental data; the isobaric heat capacity$$C_P(T)$$ obtained from PIMD and MD simulations agree qualitatively well with the experiments. Some of these thermodynamic properties exhibit anomalous maxima upon isobaric cooling, consistent with recent experiments and with the possibility that H$$_2$$ O and D$$_2$$ O exhibit a liquid-liquid critical point (LLCP) at low temperatures and positive pressures. The data from PIMD/MD for H$$_2$$ O and D$$_2$$ O can be fitted remarkably well using the Two-State-Equation-of-State (TSEOS). Using the TSEOS, we estimate that the LLCP for q-TIP4P/F H$$_2$$ O, from PIMD simulations, is located at$$P_c = 167 \pm 9$$ MPa,$$T_c = 159 \pm 6$$ K, and$$\rho _c = 1.02 \pm 0.01$$ g/cm$$^3$$ . Isotope substitution effects are important; the LLCP location in q-TIP4P/F D$$_2$$ O is estimated to be$$P_c = 176 \pm 4$$ MPa,$$T_c = 177 \pm 2$$ K, and$$\rho _c = 1.13 \pm 0.01$$ g/cm$$^3$$ . Interestingly, for the water model studied, differences in the LLCP location from PIMD and MD simulations suggest that nuclear quantum effects (i.e., atoms delocalization) play an important role in the thermodynamics of water around the LLCP (from the MD simulations of q-TIP4P/F water,$$P_c = 203 \pm 4$$ MPa,$$T_c = 175 \pm 2$$ K, and$$\rho _c = 1.03 \pm 0.01$$ g/cm$$^3$$ ). Overall, our results strongly support the LLPT scenario to explain water anomalous behavior, independently of the fundamental differences between classical MD and PIMD techniques. The reported values of$$T_c$$ for D$$_2$$ O and, particularly, H$$_2$$ O suggest that improved water models are needed for the study of supercooled water.
more »
« less
A universal variational quantum eigensolver for non-Hermitian systems
Abstract Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional$$O(log_{2}{N})$$ quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications.
more »
« less
- Award ID(s):
- 2134840
- PAR ID:
- 10480002
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Scientific Reports
- Volume:
- 13
- Issue:
- 1
- ISSN:
- 2045-2322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset$$\{x_i\}_{i=1}^n$$ and a set of noisy labels$$\{y_i\}_{i=1}^n\subset \mathbb {R}$$ we let$$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$ be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When$$y_i = g(x_i)+\xi _i$$ , for iid noise$$\xi _i$$ , and using the geometric random graph, we identify (with high probability) the rate of convergence of$$u_n$$ togin the large data limit$$n\rightarrow \infty $$ . Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.more » « less
-
Abstract The quantum simulation of quantum chemistry is a promising application of quantum computers. However, forNmolecular orbitals, the$${\mathcal{O}}({N}^{4})$$ gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a two-step low-rank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with$${\mathcal{O}}({N}^{3})$$ gate complexity in small simulations, which reduces to$${\mathcal{O}}({N}^{2})$$ gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with$${\mathcal{O}}({N}^{3})$$ gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have$${\mathcal{O}}({N}^{2})$$ depth on a linearly connected array, an improvement over the$${\mathcal{O}}({N}^{3})$$ scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4000 layers of parallel nearest-neighbor two-qubit gates, consisting of fewer than 105non-Clifford rotations. We also apply our algorithm to iron–sulfur clusters relevant for elucidating the mode of action of metalloenzymes.more » « less
-
Abstract We consider the problem of building non-invertible quantum symmetries (as characterized by actions of unitary fusion categories) on noncommutative tori. We introduce a general method to construct actions of fusion categories on inductive limit C*-algberas using finite dimenionsal data, and then apply it to obtain AT-actions of arbitrary Haagerup-Izumi categories on noncommutative 2-tori, of the even part of the$$E_{8}$$ subfactor on a noncommutative 3-torus, and of$$\text {PSU}(2)_{15}$$ on a noncommutative 4-torus.more » « less
-
Abstract The free multiplicative Brownian motion$$b_{t}$$ is the large-Nlimit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$ in the sense of$$*$$ -distributions. The natural candidate for the large-Nlimit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$ . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$\Sigma _{t}$$ that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$W_{t}$$ on$$\overline{\Sigma }_{t},$$ which is strictly positive and real analytic on$$\Sigma _{t}$$ . This density has a simple form in polar coordinates:$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ where$$w_{t}$$ is an analytic function determined by the geometry of the region$$\Sigma _{t}$$ . We show also that the spectral measure of free unitary Brownian motion$$u_{t}$$ is a “shadow” of the Brown measure of$$b_{t}$$ , precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.more » « less
