Implementation of Boolean Functions

Boyan Xu, Lvzhou Li

Whether $\mathsf{QAC}^0$ can compute $\mathtt{PARITY}_n$ remains open. Computing $\mathtt{PARITY}_n$ is equivalent to implementing $\mathtt{FANOUT}_n$ under $\mathsf{QAC}^0$ reductions. This raises a more general question: for an arbitrary symmetric Boolean function $f:\{0,1\}^n\to\{0,1\}$, what fanout size is necessary and sufficient for computing $f$ in $\mathsf{QAC}^0$? We show that the answer is exactly the transition radius $ρ(f)$: computing $f$ and implementing $\mathtt{FANOUT}_{ρ(f)}$ are equivalent under $\mathsf{QAC}^0$ reductions. In particular, if $ρ(f)\ge n^δ$ for some constant $δ>0$, then computing $f$ is $\mathsf{QAC}^0_{\mathrm{f}}$-complete. Combined with Paturi's theorem, our characterization implies that if $\mathtt{PARITY}_n \notin \mathsf{QAC}^0$, then any Boolean function in $\mathsf{QAC}^0$ of approximate degree $n^{1/2+Ω(1)}$ must be nonsymmetric.

Noga Alon, Amit Weinstein

Masanori Ohya, Igor Vasil'evich Volovich

Jie Sun, Songfeng Lu, Fang Liu

In this paper, we propose a modified construction of the quantum adiabatic algorithm for Boolean functions studied by M. Andrecut et al. [13, 14]. Our algorithm has the time complexity [Formula: see text] for the evaluation of Boolean functions, without additional computational cost of implementing the driving Hamiltonian, which is required by the adiabatic evolution described in [13, 14].

Alwin Zulehner, Robert Wille

Yorick Hardy, Willi-Hans Steeb

We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum gate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.