Quantum information density scaling and qubit operation time constraints of CMOS silicon based quantum computer architectures

Even the quantum simulation of simple molecules such as Fe$_2$S$_2$ requires more than 10$^6$ qubits. In order to assess such a multimillion scale of identical qubits and control lines, the silicon platform seems to be one of the most indicated routes as it provides the capability of nanometric, serial and industrial quality fabrication. The maximum amount of quantum information per unit surface and the consequent space constraints on qubit operations are key parameters towards fault-tolerant quantum information processing (QIP) with Si qubits. Such maximum density of quantum information is expressed for the compact exchange-only Si double quantum dot qubit architecture as a function of the CMOS technology node. The size scale optimizing both physical qubit operation time and quantum error correction (QEC) requirements is assessed by reviewing the physical and technological constraints. We determine the workable operation frequency range of a Si-CMOS quantum processor to be within 1 and 100 GHz, which limits its feasibility only to the most advanced nodes. The compatibility with classical control circuitry is discussed, focusing on the cryogenic CMOS operation required to bring the classical controller as close as possible to the quantum processor and to enable interfacing thousands of qubits on the same chip. The operation time range prospected for cryogenic control electronics is found to be compatible with the qubit operation time. By combining the forecast of technology development with operation time and classical circuitry constraints, we derive a maximum quantum information density for logical qubits of 2.8 and 4 Mqb/cm$^2$ for the 10-nm and 7-nm technology nodes respectively for the Steane code. The density is one and two orders of magnitude less for surface and concatenated codes respectively. Such values provide a benchmark for the development of Si-based fault tolerant QIP.