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Register a new userMethods of Modern Differential Geometry in Quantum Chemistry: TD Theories on Grassmann and Hartree-Fock Manifolds
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A. I. Panin
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Hamiltonian and Schrodinger evolution equations on finite-dimensional projective space are analyzed in detail. Hartree-Fock (HF) manifold is introduced as a submanifold of many electron projective space of states. Evolution equations, exact and linearized, on this manifold are studied. Comparison of matrices of linearized Schrodinger equations on many electron projective space and on the corresponding HF manifold reveals the appearance in the HF case a constraining matrix that involves matrix elements of many-electron Hamiltonian between HF state and double excited determinants. Character of dependence of transition energies on the matrix elements of constraining matrix is established by means of perturbation analysis. It is demonstrated that success of time-dependent HF theory in calculation of transition energies is mainly due to the wrong behavior of these energies as functions of matrix elements of constraining matrix as compared with the exact transition energies
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Quantum computational chemistry is a potential application of quantum computers that is expected to effectively solve several quantum-chemistry problems, particularly the electronic structure problem. Quantum computational chemistry can be compared to the conventional computational devices. This review comprehensively investigates the applications and overview of quantum computational chemistry, including a review of the Hartree-Fock method for quantum information scientists. Quantum algorithms, quantum phase estimation, and variational quantum eigensolver, have been applied to the post-Hartree-Fock method.
We prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have far-reaching consequences for the approximation of the eigenfunctions of the problems considered, and they can be used to prove a priori estimates on the numerical solution of such eigenvalue problems.
As the search continues for useful applications of noisy intermediate scale quantum devices, variational simulations of fermionic systems remain one of the most promising directions. Here, we perform a series of quantum simulations of chemistry the largest of which involved a dozen qubits, 78 two-qubit gates, and 114 one-qubit gates. We model the binding energy of ${rm H}_6$, ${rm H}_8$, ${rm H}_{10}$ and ${rm H}_{12}$ chains as well as the isomerization of diazene. We also demonstrate error-mitigation strategies based on $N$-representability which dramatically improve the effective fidelity of our experiments. Our parameterized ansatz circuits realize the Givens rotation approach to non-interacting fermion evolution, which we variationally optimize to prepare the Hartree-Fock wavefunction. This ubiquitous algorithmic primitive corresponds to a rotation of the orbital basis and is required by many proposals for correlated simulations of molecules and Hubbard models. Because non-interacting fermion evolutions are classically tractable to simulate, yet still generate highly entangled states over the computational basis, we use these experiments to benchmark the performance of our hardware while establishing a foundation for scaling up more complex correlated quantum simulations of chemistry.
The Hartree-Fock problem provides the conceptual and mathematical underpinning of a large portion of quantum chemistry. As efforts in quantum technology aim to enhance computational chemistry algorithms, the fundamental Hartree-Fock problem is a natural target. While quantum computers and quantum simulation offer many prospects for the future of modern chemistry, the Hartree-Fock problem is not a likely candidate. We highlight this fact from a number of perspectives including computational complexity, practical examples, and the full characterization of the energy landscapes for simple systems.
We compute the ${cal N}=2$ supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kahler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $mathbb{C}^2$. The evaluation of these residues is greatly simplified by using an abstruse duality that relates the residues at the poles of the one-loop and instanton parts of the $mathbb{C}^2$ partition function. As particular cases, our formulae compute the $SU(2)$ and $SU(3)$ {it equivariant} Donaldson invariants of $mathbb{P}^2$ and $mathbb{F}_n$ and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the $SU(2)$ case. Finally, we show that the $U(1)$ self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $mathcal{N}=2$ analog of the $mathcal{N}=4$ holomorphic anomaly equations.
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