Identifying quantum coherent memristive and resistive transport

<p dir="ltr">Quantum memristors are at the forefront of cutting-edge research, with growing promise to combine quantum information processing with advances in neuromorphic computing. They extend the classical idea of a memristor, a resistor with memory, into the quantum regime. Quantum memristors exhibit many key behaviours that will be beneficial in the growing race for energy-efficient AI operating in high-dimensional space; chief among them is the unity of incorporating quantum spiking dynamics with 'memory' behaviours.</p><p dir="ltr">The scientific community has traditionally overlooked quantum resistance and memristance due to the perceived necessity for energy relaxation. Here, we argue that this conclusion is too hasty; only momentum relaxation is strictly necessary for resistive flow. Coherence is locally maintained when the coherence length exceeds the elastic mean-free path length. This thesis investigates the possibility of quantum coherent resistive and memristive transport in nanoscale electronic systems. A device is proposed where transport through distinct circuit arms is in a coherent superposition of momentum relaxations, effectively realising a coherent resistor. Such a component would extend the circuit model of quantum electronics by introducing a resistive element that preserves coherence, complementing existing inductive and capacitive counterparts.</p><p dir="ltr">In chapter 3, we propose a Two-Level System as a quantum coherent resistor and take into account some third-order corrections to the Green's function. This project lays the groundwork to complete the set of quantum fundamental circuit elements. The use of Green's functions paves the way for resistive (and potentially memristive) corrections to be made in the form of the scattering matrix (S-matrix), which would allow direct applications in quantum circuit analysis.</p><p dir="ltr">In chapter 4, we theoretically consider experimental validation, developing a protocol for testing not only the proposed resistor, but also a whole class of coherent memristors or resistors. Simulations of the device reveal unique frequency mixing, consistent with coherent transport. This testing scheme provides a route to experimentally validate quantum-coherent resistive/memristive dynamics, addressing a central challenge in quantum device characterisation. </p><p dir="ltr">These results highlight the differences in behaviour between coherent and incoherent resistive and memristive devices and help clarify the distinctive features of a fully quantum memristor. As fundamental building blocks, these devices offer new opportunities for engineered dissipation, hybrid quantum-classical architectures, and potential applications in quantum neuromorphic computing. More broadly, this work establishes a framework for probing the quantum nature of resistive transport, with implications for the next generation of quantum devices and information technologies.</p>