Capacitors, Inductors and Resistors are well-known circuit components. Since the proposal of Leon Chua in 1971, the Memristor joined them as a fundamental component. I am wondering whether these elements would be somehow imitated by the means of quantum technologies and what would be the requirements to achieve them.
When you ask,
I am wondering whether these elements would be somehow imitated by the means of quantum technologies
there are different levels on which you can interpret this question. You might mean to ask whether people will realise quantum capacitors, inductors, or resistors, or you might mean to ask whether people will realise components which, in quantum computers, fulfil the same functional roles as capacitors, inductors, or resistors in order to realise digital information processing — as opposed, for instance to analogue computers to model differential systems of equations.
It must be remembered that quantum technologies are at an early phase, where this is no single way which we can be confident will form the basis of a scalable quantum computer. But we can consider whether there are any cases where there may be interesting analogues.
Many quantum technologies do not represent anything like an electrical circuit, as such. Ion traps store bits of information on individual ions, which are moved in a limited and carefully controlled way. There is no natural notion of electrical conduction, resistors, or capacitors in this setting. Quantum dots are even less like electrical circuits, in that the locations of the physical systems storing the data are fixed.
Flux qubits, on the other hand, explicitly include circuits which carry a current (albeit a very small one). The resistance in such circuits is effectively zero, as they are superconducting; but they do involve Josephson junctions, which are often considered a non-linear type of inductor.
This is different from whether or not there is anything in a given platform which is doing the same job as a resistor, capacitor, or inductor: which may be substantially different on the level of physics, but which are somehow performing a similar role in mediating how a system performs information processing. However, there is a big difference between the way that classical semiconductor electronics realises information processing — with physical gates, which transform information-carrying input signals to produce output signals — and the way every current quantum technology realises information processing, which is to perform controlled changes of the dynamics of systems prepared in some input state, to realise an output state.
(The one possible exception are photonic quantum systems, in which the information is carried in light signals rather than in the states of more-or-less static pieces of matter. Perhaps you might argue that an optical memory is analogous to a capacitor somehow, or that a wave plate is analogous to an inductor, but these don't seem to be meaningful functional analogues for how an optical system might be used to perform quantum information processing.)
In summary: there is no single answer to your question, because of the different things you might mean by it and because there is no single platform to refer to in order to provide a definitive answer. But most of the platforms don't have anything which represents these basic electrical components, or which play the same role. Quantum technologies are simply expected to operate differently than classical computing technology.
These elements do not necessarily have a correspondence in quantum computers, just as they do not necessarily occur in classical computers (an electronic computer might use some of them, but a mechanical or photonic computer does not necessarily have any equivalent of them).
What has an equivalence are the fundamental gates that form a classical computer. For example, there are only two classical single-bit gates, the direct connection and the NOT gate. A quantum computer has single qubit rotation gates (for example $X$, $Y$, $Z$, $H$, $T$) of which $X$ is a direct equivalent.
I am wondering whether these elements would be somehow imitated by the means of quantum technologies and what would be the requirements to achieve them
I don't think we would want to achieve quantum equivalents of resistors, capacitors, inductors etc (at least as of now). There are two parts to any circuit: 1) Logical implementation 2) Physical implementation.
You need the 'bits' represented as voltages/currents/spins to implement the logic, for which we have the quantum equivalent of qubits.
And when you physically implement a circuit, the concepts of resistance, capacitance etc comes into picture because of the sheer nature of the materials that we are trying to implement the circuit with. There are resistances, capacitances which act as noise in a circuit due to the wires, etc and also resistors and capacitors which we add to the circuit to vary the above said voltages/currents (for eg, a transistor is a variable resistor).
This analogy applies to quantum circuitry when you are implementing qubits. It boils down to the fact that after you have achieved the required logical implementation, in which form do you need your output? Based on this you may need to apply resistances and capacitances to the circuitry to change the voltages/currents.
So in the future if we need to change the qubits in such a way as it somehow mimics the classical resistive action, then we will need a technique to achieve a mechanism which performs the resistive action on qubits (I don't even comprehend what such action would even be for a qubit), then that technique will be called as a quantum resistor. Until then we will resort to classical resistors and capacitors to manipulate the classical signals, after the qubits have done their job.