Scaling of Decoherence for a Decoupled Multi-spin System

We study the decoherence of $n$ confined but decoupled electron spin qubits by examining the state fidelity for various initial states under the influence of hyperfine interaction with local environmental nuclear spins. We find that $n$-qubit inhomogeneous broadening time $T_2^*(n)$ and the narrowed-state free induction decay time $T_2(n)$ have the same scaling behaviors. For a superposed state whose product basis states are all from the same Zeeman manifold, both $T_2^*(n)$ and $T_2(n)$ are scale-free with respect to $n$ and the number of basis states, $m$. For a superposed state whose product basis states are selected from different Zeeman manifolds, both $T_2(n)$ and $T_2^*(n)$ are roughly inversely proportional to $\sqrt{n}$. Our results can be extended to other decoherence mechanisms, and to decoherence mechanisms in the presence of dynamical decoupling, such as decay of spin echo. This analysis should allow a more meaningful discussion on the scalability of any spin-based solid state quantum technology.