Jonathan Allcock

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 defined by x>|b> —> |x>|b+ f(x)> for x in 0,1^n and b in 0,1, where f is a Boolean function. The first of our constructions is based on computing the one-hot encoding of the control register |x>, 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(n log(n)łogłog(n)) ancillae and O(n log(n)) Fan-Out gates or O(n log(n)) 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.