We develop a recursive algorithm to generalize the quantum SWAP test for an arbitrary number m of quantum states requiring O(m) controlled-swap (CSWAP) gates and O(log m) ancillary qubits. We construct a quantum circuit able to simultaneously measure overlaps | φ i , φ j | 2 of m arbitrary pure states |φ 1 . . . φ m . Our construction relies on a pairing unitary that generates a superposition state where every pair of input states is labelled by a basis state formed by the ancillaries.

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The history of empire is the history of organized violence. The ability to impose and manage violence is the most basic demand and highest priority for the survival of any empire. This chapter outlines the various forms of military organizations different empires have employed throughout history. It describes the internal and external functions of these military organizations in protecting and expanding the empire, as well as the problems faced by imperial regimes of financing and controlling military organizations that might otherwise turn against the very same regimes they were created to protect. Lastly, the chapter gives a historical walk-through of the “Revolutions in Military Affairs” (RMAs) that have propelled the advancement in military affairs through history, which has allowed empires to grow in scale and size.