Constant-depth circuits for Boolean functions and quantum memory devices using multi-qubit gates

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 defined by |x>|b> 7→ |x>|b f(x)> for x ∈ {0, 1}n and b ∈ {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(d) n log(d+1) n) ancillae and O(n log(d) n) Fan-Out gates or O(n log(d) n) ancillae and 16d − 10 Global Tunable gates, where d is any positive integer and log(d) n = log · · · log n is the d-times iterated logarithm. On the other hand, the implementation based on Boolean analysis requires 8d − 6 Global Tunable gates at the expense of O(n1/(1−2−d)) ancillae.