NP-hardness of circuit minimization for multi-output functions

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions

THE FREDKIN GATE IN REVERSIBLE AND QUANTUM ENVIRONMENTS

Reversible Computing circuits are characterized by low power consumption and their proximity to circuits for quantum computing. The Fredkin gate was one of the earliest proposed controlled reversible circuits, which however, was soon superseded by the Toffoli gate, the NOT, and CNOT gates, which constituting a flexible functionally complete set could also realize the Fredkin gate as a building block. In quantum computing circuits, the Fredkin gate (under the name controlled SWAP) plays an important role regarding the superposition of states. The present paper studies extensions of the Fredkin gate in terms of mixed polarity in the reversible domain and an application in quantum computing