Quantifying quantum states' complexity is a key problem in various subfields of science, from computing to black-hole physics. We prove prominent conjecture by Brown and Susskind about how random circuits' increases. Consider constructing unitary Haar-random two-qubit gates. Implementing the exactly requires circuit some minimal number gates - unitary's exact complexity. that this grows linearl...

We propose a complexity model of quantum circuits analogous to the standard (acyclic) Boolean circuit model. It is shown that any function computable in polynomial time by a quantum Turing machine has a polynomial-size quantum circuit. This result also enables us to construct a universal quantum computer which can simulate, with a polynomial factor slowdown, a broader class of quantum machines ...

We present a framework for studying circuit complexity that is inspired by techniques that are used for analyzing the complexity of CSPs. We prove that the circuit complexity of a Boolean function f is characterized by the partial polymorphisms of f ’s truth table. Moreover, the non-deterministic circuit complexity of f is characterized by the polymorphisms of f ’s truth table.

We show that Shuffle(x, y, w), the problem of determining whether a string w can be composed from an order preserving shuffle of strings x and y, is not in AC, but it is in AC. The fact that shuffle is not in AC is shown by a reduction of parity to shuffle and invoking the seminal result [FSS84], while the fact that it is in AC is implicit in the results of [Man82a]. Together, the two results p...

In this paper a new approach for design of class-AB current-mode square-root domain filters based on the electronically simulated translinear loop principle is presented. FG-MOS transistors that operate in strong inverted saturation region are employed for implementation of translinear loop circuit. The filter achieves immunity to body effect, low circuit complexity, large input dynamic ran...

a log-domain current-mode true rms-to-dc converter based on a novel synthesis of a simplified current-mode low pass filter and a two-quadrant squarer/divider is presented. the circuit employs floating gate mos (fg-mos) transistors operating in weak inversion region. the converter features low power(<1.5uw), low supply voltage (0.9v), two quadrant input current, immunity from body effect, low ci...

2 Existential Lower Bounds Theorem 1 (Lupanov, Shannon) For each ε > 0 and for each sufficiently large n, ∃f : {0, 1}n → {0, 1} with circuit complexity (over the full basis B2) at least (1− ε)2n n . Proof: We will prove the required result using a counting argument. In particular, we will show that the number of functions mapping {0, 1}n to {0, 1} is much larger than the number of ’distinct’ ci...