Scaling of decoherence for a system of uncoupled spin qubits

Significant experimental progresses in recent years have generated continued interest in quantum computation. A practical quantum computer would employ thousands if not millions of coherent qubits, and maintaining coherence in such a large system would be imperative for its utility. As an attempt at understanding the quantum coherence of multiple qubits, here we study decoherence of a multi-spin-qubit state under the influence of hyperfine interaction, and clearly demonstrate that the state structure is crucial to the scaling behavior of n-spin decoherence. Specifically, we find that coherence times of a multi-spin state at most scale with the number of qubits n as, while some states with higher symmetries have scale-free coherence with respect to n. Statistically, convergence to these scaling behavior is generally determined by the size of the Hilbert space m, which is usually much larger than n (up to an exponential function of n), so that convergence rate is very fast as we increase the number of qubits. Our results can be extended to other decoherence mechanisms, including in the presence of dynamical decoupling, which allow meaningful discussions on the scalability of spin-based quantum coherent technology.