What design considerations set the frequency bounds for superconducting qubits?

Superconducting qubits generally have frequencies within the range of 4 - 8 GHz. What design considerations give the upper and lower bounds for what is a feasible design. I.e., why can't they be higher or lower in frequency? [and from a comment]: "... what about the lower bound?"

The size of the components, their spacing, and materials determine a reasonable operating frequency range, while the minimum wavelength is twice the lattice spacing.

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From: "Microwave photonics with superconducting quantum circuits" (Nov 2017), by Xiu Gua, Anton Frisk Kockum, Adam Miranowicz, Yu-xi Liu, and Franco Nori:

"A basic requirement for the SQCs to function as artificial atoms is the anharmonicity of their energy-level spacing. Josephson junctions play an important role in SQCs because of the strong nonlinearity they provide, which is the key to making superconducting artificial atoms anharmonic.

...

A Josephson junction is made of two superconducting electrodes, separated by a thin insulating film (with typical thickness of 1–3 nm) in a sandwich structure, giving rise to an intrinsic capacitance. Cooper pairs, formed by two bound electrons, with opposite spins, in the superconducting electrodes, can tunnel coherently through the insulating barrier one by one.

...

In a given SQC, we can thus select the two lowest-energy levels from the non-equally spaced energy spectrum. These two levels form a quantum bit (qubit) for quantum-information processing.

When an ac voltage is applied to the two electrodes of the Josephson junction, the supercurrent $I$ is periodically modulated as $I=I_csin(ωt+ϕ)$ with the Josephson frequency $ν=ω∕(2π)=2eV∕h$. This is called the ac Josephson effect. The energy $hv$ equals the energy change of a Cooper pair transferred across the junction. The voltage applied to a Josephson junction is typically on the order of a few microvolts. Thus, SQCs are usually operating at frequencies in the microwave regime.".

From: "Superconducting Qubits: Current State of Play" (July 26 2019), by Morten Kjaergaard, Mollie E. Schwartz, Jochen Braumüller, Philip Krantz, Joel I-Jan Wang, Simon Gustavsson, and William D. Oliver, on page 5:

"... being only a few percent of the qubit level spacing

$$omega_q/2pi equiv omega_{01}/2pi sim 5 text{GHz} $$

For single-junction transmons (see Fig. 2(b)), this frequency is set by the size of the shunt capacitor and the critical current $I_c$ of the Josephson junction, determined by design and fabrication parameters such as materials choice, junction area, and insulator thickness. Replacing the single Josephson junction by a superconducting loop with two junctions in parallel – a dc-SQUID – enables one to tune the effective critical current of the Josephson junction (and hence the qubit frequency) via a magnetic field applied to the dc-SQUID loop. The trade-off for this additional control knob is that the qubit becomes susceptible to magnetic flux noise. Transmon qubits can be coupled capacitively – either directly or as mediated by a resonator “bus” – which, in the natural eigenbasis of the transmon qubits, lead to a two-qubit interaction term ...".

Wikipedia: Harmonic oscillators lattice: phonons:

"The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the $(N + 1)th$ atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

$$k=k_{n}={2npi over Na}quad {hbox{for}} n=0,pm 1,pm 2,ldots ,pm {N over 2}.$$

The upper bound to $n$ comes from the minimum wavelength, which is twice the lattice spacing $a$, as discussed above.".

Additional Information:

• Wikipedia - "Superconducting quantum computing - technology", "Hamiltonian and energy eigenstates" and "Anharmonicity".