Upper and Lower Bounds on the Computational Complexity of Polar Encoding and Decoding

It is shown that all polar encoding schemes using a standard encoding matrix with rate R > 1/2 and block length N have energy within the Thompson circuit model that scales at least as E >= Omega (N-3/2). This lower bound is achievable up to polylogarithmic factors using a mesh network topology defined by Thompson and the encoding algorithm defined by Arikan. A general class of circuits that compute successive cancellation decoding adapted from Arikan's butterfly network algorithm is defined. It is shown that such decoders implemented on a rectangle grid for codes of rate R > 2/3 must take energy E >= Omega(N-3/2). The energy of a Mead memory architecture and a mesh network memory architecture are analyzed and it is shown that a processor architecture using these memory elements can reach the decoding energy lower bounds to within a polylogarithmic factor. Similar scaling rules are derived for polar list decoders and belief propagation decoders. Capacity approaching sequences of energy optimal polar encoders and decoders, as a function of reciprocal gap to capacity chi = (1-R/C)(-1) (where R is rate and C is channel capacity), have energy that scales as Omega(chi(5.3685)) <= E <= O (chi(7.071) log(4) (chi)). Known results in constant depth circuit complexity theory imply that no polynomial size classical circuits can compute polar encoding, but this is possible in quantum circuits that include a constant depth quantum fan-out gate.