Using Hasse Diagrams to Synthesize Ternary Quantum Circuits

We present the results of application of Hasse diagrams to the problem of synthesizing ternary quantum circuits represented as input-output mapping vectors. The paper specifically focuses on ternary quantum circuits with relatively large number of variables where valid solutions exist in an exponentially expanding search space. Valid solutions represent the set of all input vector permutations (arrangements or sequences) which satisfy the circuit specification and are algorithmically convergent. We discovered that a) the ordering of the input vector has an impact on the size of the resulting circuit and, that b) only certain orderings are algorithmically convergent. Here we describe a method for systematically constructing such valid sequences using a ternary Hasse diagram and illustrate a detailed proof of the critical issue of algorithmic convergence. In essence, we illustrate the benefit of exploring many input sequences over limiting the synthesis to the natural binary order.