The task of logic synthesis is to map a technology-independent representation of an application to hardware-specific operations, taking into account various constraints and trading off different costs associated with the implementation. Constraints may include the target gate library and feasible connectivity, whereas the costs capture, for example, the required chip area and delay. Here we propose to use parallel tempering Monte Carlo to find low-level implementations of a given Boolean function. In contrast to previous work leveraging Markov chain Monte Carlo methods such as simulated annealing, we do not start with an initially correct (but suboptimal) implementation that is then further optimized. Instead, our approach starts with a random logic network that is subsequently modified in order to reduce the error to zero. We apply our method to the task of synthesizing Majority-$n$ in terms of Majority-$3$ gates with and without inverters, aiming for a low Majority-$3$ count. Our method is able to find solutions that are on par or better than previously known solutions. For example, for $n\in\{9,11,13\}$ our approach successfully finds inverter-free implementations using between 7% and 42% fewer Majority-$3$ gates.

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