Limited Parallelization in Gate Operations Leads to Higher Space Overhead and Lower Noise Threshold

In a modern error corrected quantum memory or circuit, parallelization of gate operations is severely restricted due to issues like cross-talk. Hence, there are enough idle qubits not undergoing gate operations either during the computation phase or during the error correction phase, which suffer further decoherence while waiting. Thus, in reality, the space overhead and the noise threshold would depend on the level of gate parallelization. In this paper, we obtain an analytical lower bound on the required space overhead in terms of the level of parallelization for an error correction framework that has more error correction capability than the existing ones. We consider two types of errors: i.i.d. erasure and depolarization. In comparison to the known lower bounds which assume full gate parallelization, our bound is provably strictly larger despite allowing more capability to the error correction framework. This shows the steep price to be paid for lack of gate parallelization. An implication of the bound is that the noise or decoherence threshold, i.e., the noise beyond which no fault-tolerant memory or circuit can be realized, vanishes if the number of parallel gate operations does not scale linearly with the number of physical qubits.