Abstract

We explore the power of the unbounded Fan-Out gate and the Global Tunable gates generated by Ising-type Hamiltonians in constructing constant-depth quantum circuits, with particular attention to quantum memory devices. We propose two types of constant-depth constructions for implementing Uniformly Controlled Gates. These gates include the Fan-In gates (a generalization of the oracles for Boolean functions). The first of our constructions is based on computing the one-hot encoding of the input register, while the second is based on Boolean analysis and exploits different representations of f such as its Fourier expansion. Via these constructions, we obtain constant-depth circuits for the quantum counterparts of read-only and read-write memory devices – Quantum Random Access Memory (QRAM) and Quantum Random Access Gate (QRAG) – of memory size n. The implementation based on one-hot encoding requires either O(nlogn(loglogn)) ancillae and O(nlogn) Fan-Out gates or O(nlogn) ancillae and 6 Global Tunable gates. On the other hand, the implementation based on Boolean analysis requires only 2 Global Tunable gates at the expense of O(n^2) ancillae.