Towards a SAT Encoding for Quantum Circuits: A Journey From Classical Circuits to Clifford Circuits and Beyond

TL;DR

This paper explores the potential of SAT encoding for quantum circuits, focusing on Clifford circuits, and establishes criteria for feasibility while demonstrating practical application and limitations.

Contribution

It introduces a propositional SAT encoding for quantum circuits, identifies classes where it is feasible, and empirically tests it on Clifford circuits.

Findings

SAT encoding is feasible for Clifford circuits

Constructing a general SAT encoding for arbitrary quantum circuits is infeasible

Criteria for the feasibility of SAT encoding in quantum circuits are established

Abstract

Satisfiability Testing (SAT) techniques are well-established in classical computing where they are used to solve a broad variety of problems, e.g., in the design of classical circuits and systems. Analogous to the classical realm, quantum algorithms are usually modelled as circuits and similar design tasks need to be tackled. Thus, it is natural to pose the question whether these design tasks in the quantum realm can also be approached using SAT techniques. To the best of our knowledge, no SAT formulation for arbitrary quantum circuits exists and it is unknown whether such an approach is feasible at all. In this work, we define a propositional SAT encoding that, in principle, can be applied to arbitrary quantum circuits. However, we show that due to the inherent complexity of representing quantum states, constructing such an encoding is not feasible in general. Therefore, we establish general criteria for determining the feasibility of the proposed encoding and identify classes of quantum circuits fulfilling these criteria. We explicitly demonstrate how the proposed encoding can be applied to the class of Clifford circuits as a representative. Finally, we empirically demonstrate the applicability and efficiency of the proposed encoding for Clifford circuits. With these results, we lay the foundation for continuing the ongoing success of SAT in classical circuit and systems design for quantum circuits.