Research Interest
Entropy and Quantum Computing
Entropies of Schrödinger’s Cat States During Digitized Adiabatic Evolution
In applications of adiabatic quantum computing, a quantum system is initialized in the known ground state of a simple Hamiltonian HI, and then evolved under a slowly varying perturbation to obtain the ground state of a problem Hamiltonian Hp.
H(s) = s*Hp + (1-s) HI
where s = t/T, ranging from 0 to 1. By the adiabatic theorem of Born and Fock , for a large T such that the variation of HI to Hp is carried out sufficiently slowly, the system will remain in the instantaneous ground state throughout the evolution. Adiabatic quantum computations are universal and equivalent to the circuit model. Adiabatic quantum computing is robust to errors due to decoherence and certain random unitary perturbations. Thus, it is compatible with current noisy-intermediate scale quantum (NISQ) devices.
In this study, Schrödinger’s cat states were prepared by starting in an eigenstate for noninteracting spins in a magnetic field in the x direction, and then converting the Hamiltonian to an Ising-model Hamiltonian with nearest-neighbor coupling of the z components of the spins.
We have explored both the forward and backward digitized evolution using Qiskit, with Trotterization of the time-evolution operator. Simulations were done on both a noiseless simulator and a fake backend, which has decoherence based on calibration data of IBMQ’s Manila.
Differences in the Shannon entropy during forward and backward evolution can be used to quantify the level of adiabaticity of the evolution, since truly adiabatic evolution should proceed through identical states, in either direction.
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Shannon Entropy of Schrödinger's Cat States
Schrödinger's cat states are maximally entangled n-qubit quantum states, which are a generalization of the 3 spin 1/2 particle GHZ state. I am interested in the Shannon and von Neumann entropy, of the qubits in these coupled states. By entangling qubits into large "cat" states.
We have seen that as the number of entangled qubits grows, there is a corresponding linear growth in Shannon entropy of the measurements. I use IBM's publicly accessible quantum computers and Qiskit to create the entangled states. It is seen that the magnitude of the slope corresponds to the quality of the measured states. It is hoped that this finding can lead to a physically motivated classification of quality entanglement and provide a benchmark complementary to Quantum Volume .
This work has been published in PCCP: https://doi.org/10.1039/D1CP05255A
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Intermolecular Forces
Dynamic Intermolecular Interactions in the Alkane/Perfluoroalkane Dimers: Can transient dipoles offer an intuitive explanation of the unexpected phase separation?
The hydrophobic effect is commonly summarized as “like dissolves like”. The usual explanation is that different forces hold polar vs nonpolar liquids together; molecules of hydrocarbons and other nonpolar compounds enjoy mutual attraction primarily via van der Waals (AKA dispersion) interactions, whereas electrostatics (dipole-dipole and hydrogen bonding) dominate among polar species such as water. Thus, “oil and water don’t mix”, aggregating instead into separate liquid layers. This phase separation is routinely exploited in the purification and isolation of reaction products.
One long known outlier of this trend is the phase separation between perfluorocarbons and hydrocarbons. Both are nonpolar and hydrophobic in nature, but when perfluorinated alkanes (PFAS) and alkanes are mixed, they form separate hydrocarbon and “fluorous phase” layers. Theoretical analysis1,2 ascribes the PFAS’ mutual attraction to electron correlation and dispersive forces, specifically among the fluorine atoms. Why, then, do nonpolar perfluorocarbons, held together by dispersion, not mix with similarly nonpolar, dispersion-bound hydrocarbons? The above theoretical treatment does not paint a complete physical picture, as it focuses on ground state equilibrium geometries and static interactions between species.
We compare the computed infrared (IR) spectra of isolated CH4 and CF4 molecules to those of their dimer counterparts. The ~30 cm-1 splitting in the CF4…CF4 asymmetric stretching modes points to their strong coupling, suggesting a dynamic attractive interaction between the molecules. In contrast, the partners in the CH4…CH4 and CF4…CH4 dimers “feel” each other less, showing much lower splittings of 8 and 3 cm-1 respectively. Thus, beyond dispersion (electron motion), the strongly polarized C-F bonds in CF4 enable mutual interaction via the transient dipoles and quadrupoles created by vibrational motion. In the splitting, the lowered vibrations show displacements that produce mutually attractive dipoles between the CF4 fragments, whereas in the raised frequencies, they are repulsive. These dynamic dipole-dipole interactions may help explain why alkanes and perfluorinated alkanes phase separate.
Origin of Dispersion Forces on the Nuclei in H2 in the Triplet State
Dispersion forces are universal forces of attraction between all atoms and molecules in their ground electronic states. The forces are purely quantum mechanical in origin because they result from correlation between the electrons in the interacting molecules. Fritz London first explained these forces in 1930 via perturbation theory. Though dispersion is a very weak intermolecular force, falling off as R-7 in the distance R between the molecular centers of mass, the attractive interaction can become significant for large nonpolar molecules, including proteins, and in bulk materials.
In 1939, Richard Feynman suggested that the dispersion forces on the nuclei of atoms in S states can be attributed to the attraction of each nucleus to its “own” charge distribution, polarized by correlation effects. In 1990, Hunt proved Feynman’s statement analytically within the polarization approximation.3 She also proved that the interpretation of dispersion forces generalizes to molecules of arbitrary symmetry. If one or both of the interacting molecules lack a center of symmetry, then the dipole moment induced by dispersion varies as R-6 in the separation between the centers of mass, as shown by Galatry and Hardisson4, while the dispersion force still varies as R-7. In this case, the rationale given by Feynman breaks down, yet Feynman’s explanation of the nature of dispersion forces on the nuclei still holds.
We have carried out ab initio calculations to probe the origins of the dispersion forces on the hydrogen nuclei in the H2 molecule in its lowest triplet state to determine whether Feynman’s interpretation still holds when exchange is included in the analysis.
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Interaction-Induced Dipole Moment
Molecular hydrogen and other centrosymmetric diatomic species lack a permanent dipole, thus they are infrared inactive. In dense gasses, fly-by collisions of the species can create instantaneous dipoles, allowing for roto-vibrational transitions that are generally forbidden. This phenomena, known as collision induced absorption, allows for absorption in the infrared and microwave regions.
In my research I perform advanced methods of computing the system's dipole, using finite field technique and spherical tensor analysis. With these techniques we are able to theoretically calculate the spectra of the pairs, to further probe astrophysical phenomena such as determining the temperatures of the planetary atmospheres of Jupiter or Saturn.
Currently I have run over 20,000 CCSD(T) calculations at 28 orientations and 24 intermolecular distances, and 36 bond lengths. These dipole values are fit to spherical harmonics to determine the tensor coefficients. The coefficients allow us to determine theoretical spectra and create 3D dipole surfaces such as the bottom left figure.