arXiv subjects

Jeremy Canfield

Publications and source records attributed to Jeremy Canfield.

3 recordsLinked to original sources

arXiv2026

Low Depth Unitary Coupled Cluster Algorithm for Large Chemical Systems

The unitary coupled cluster (UCC) algorithm is one of the most promising implementations of the variational quantum eigensolver for quantum computers. However, for large systems, the number of UCC factors leads to deep quantum circuits, which are prohibitive for execution on quantum hardware. To address this, circuit depth can be reduced at the cost of more measurements with a Taylor series expansion of UCC factors with small angles, while treating the large-angle factors exactly. We implement this approach to quadratic order (qUCC) for systems with strong correlations and systems where conventional methods like coupled cluster (CC) with low excitation levels fail, but UCC and qUCC perform well. We study hydrogen chains and the BeH2 molecule that allow us to change the degree of strong correlation due to geometrical distortions. We show, via a dramatic increase in number of factors able to handle exactly, a systematic convergence of these results as more exact UCC factors are included in the calculations -- the hardest to converge regime is in the crossover from weak to strong coupling. In all cases the total number of UCC factors needed to be treated exactly is much less than the total number of UCC factors available (typically about one-third to one-half of the total number of factors).

quant-ph↗

arXiv2022

The Laplace method for energy eigenvalue problems in quantum mechanics

Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schroedinger equation for them is solved by using a generalized series solution for the bound states (using the Froebenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schroedinger when he solved for the wavefunctions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach for graduate students to learn and describe how it can be employed to solve all problems whose wavefunctions are represented in terms of confluent hypergeometric functions.

quant-ph↗

arXiv2020

Schrodinger's original quantum-mechanical solution for hydrogen

In 1926, Erwin Schrodinger wrote a series of papers that invented wave mechanics and set the foundation for much of the single-particle quantum mechanics that we teach today. In his first paper, he solved the Schrodinger equation using the Laplace method, which is a technique that is quite powerful, but rarely taught. This is unfortunate, because it opens the door to examining quantum mechanics from a complex-analysis perspective. Gaining this experience with complex analysis is a useful notion to consider when teaching quantum mechanics, as these techniques can be widely used outside of quantum mechanics, unlike the standard Frobenius summation method, which is normally taught, but rarely used elsewhere. The Laplace method strategy is subtle and no one has carefully gone through the arguments that Schrodinger did in this first paper, instead it is often just stated that the solution was adopted from Schlesinger's famous differential equation textbook. In this work, we show how the Laplace method can be used to solve for the quantum-mechanical energy eigenfunctions of the hydrogen atom, following Schrodinger's original solution, with all the necessary details, and illustrate how it can be taught in advanced instruction; it does require familiarity with intermediate-level complex analysis, which we also briefly review.

quant-ph↗