Constant-depth circuits for Uniformly Controlled Gates and Boolean functions with application to quantum memory circuits
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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∈{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(nlognloglogn) 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.
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