Algebraic and logical emulations of quantum circuits

Quantum circuits exhibit several features of large-scale distributed systems. They have a concise design formalism but behavior that is challenging to represent let alone predict. Issues of scalability—both in the yet-to-be-engineered quantum hardware and in classical simulators—are paramount. They require sparse representations for efficient modeling. Whereas simulators represent both the system’s current state and its operations directly, emulators manipulate the images of system states under a mapping to a different formalism. We describe three such formalisms for quantum circuits. The first two extend the polynomial construction of Dawson et al. [1] to (i) work for any set of quantum gates obeying a certain “balance” condition and (ii) produce a single polynomial over any sufficiently structured field or ring. The third appears novel and employs only simple Boolean formulas, optionally limited to a form we call “parity-of-AND” equations. Especially the third can combine with off-the-shelf state-of-the-art third-party software, namely model counters and # SAT solvers, that we show capable of vast improvements in the emulation time in natural instances. We have programmed all three constructions to proof-of-concept level and report some preliminary tests and applications. These include algebraic analysis of special quantum circuits and the possibility of a new classical attack on the factoring problem. Preliminary comparisons are made with the libquantum simulator [2–4].