It has been shown that, for even $n$, evolving $n$ qubits according to a Hamiltonian that is the sum of pairwise interactions between the particles, can be used to exactly implement an $(n+1)$-qubit fanout gate using a particular constant-depth circuit [arXiv:quant-ph/0309163]. However, the coupling coefficients in the Hamiltonian considered in that paper are assumed to be all equal. In this paper, we generalize these results and show that for all $n$, including odd $n$, one can exactly implement an $(n+1)$-qubit parity gate and hence, equivalently in constant depth an $(n+1)$-qubit fanout gate, using a similar Hamiltonian but with unequal couplings, and we give an exact characterization of which couplings are adequate to implement fanout via the same circuit. We also investigate pairwise couplings that satisfy an inverse square law, giving necessary and sufficient criteria for implementing fanout given spatial arrangements of identical qubits in two and three dimensions subject to this law. We use our criteria to give planar arrangements of four qubits that (together with a target qubit) are adequate to implement $5$-qubit fanout.

翻译：已有研究表明，对于偶数$n$，通过按照由粒子间两两相互作用之和构成的哈密顿量演化$n$个量子比特，可以借助特定常数深度的电路精确实现$(n+1)$量子比特扇出门[arXiv:quant-ph/0309163]。然而，该论文所考虑的哈密顿量中的耦合系数被假设为全部相等。本文推广了这些结果，证明对于所有$n$（包括奇数$n$），可以使用类似哈密顿量（但耦合系数不相等）精确实现$(n+1)$量子比特奇偶校验门，进而等价地在常数深度内实现$(n+1)$量子比特扇出门，并给出了通过相同电路实现扇出所需的耦合系数的精确刻画条件。我们还研究了满足平方反比律的两两耦合，给出了在二维和三维空间中服从该律的相同量子比特空间排布下实现扇出的充要条件。利用这些条件，我们给出了四个量子比特的平面排布方式（连同目标量子比特），足以实现5量子比特扇出。