Analyzing Algebraic Quantum Circuits Using Exponential Sums

We introduce and analyze circuits that are the quantum mechanical generalization of classical algebraic circuits. Using the algebraic operations of addition and multiplication, as well as the quantum Fourier transform, such circuits are well-defined for rings Z/mZ and finite fields Fq. The acceptance probabilities of such algebraic quantum circuits can be expressed as exponential sums ∑x exp(2πi f (x)/m) where the multivariate polynomial f is determined by the circuit, while it is independent of the ring or field over which we interpret the circuit. Dawson et al. [Quantum Information & Computation, 5(2), pp. 102–112 (2004)] introduced this “sum over paths” description as a discrete version of the path integral approach of standard quantum mechanics. From this perspective, the polynomial f should be interpreted as the “action” of a specific (classical) computational path between the input and output of the circuit. In this article we prove several properties of algebraic quantum circuits. Using the theory of exponential sums, we show that in the limit of large m or q, the acceptance probabilities of a circuit converge to zero or to one. Circuits that do not involve the multiplication operation are the algebraic generalization of Clifford circuits and we show how their acceptance probabilities can be calculated exactly in a classical efficient manner. For algebraic circuits that are defined over rings Z/prZ we derive a “least action principle” that shows how the behaviour of such circuits is determined by those computational paths whose action polynomials are extremal.