Thompson's model of very large scale integration computation relates the energy of a computation to the product of the circuit area and the number of clock cycles needed to carry out the computation. It is shown that for any sequence of increasing block-length decoder circuits implemented according to this model, if the probability of block error is asymptotically less than 1/2 then the energy of the computation scales at least as (n(log n) 1/2 ), and so the energy of decoding per bit must scale at least as (log n) 1/2 . This implies that the average energy per decoded bit must approach infinity for any sequence of decoders that approaches capacity. The analysis techniques used are then extended to show that for any sequence of increasing block-length serial decoders, if the asymptotic block error probability is less than 1/2 then the energy scales at least as fast as (n log n). In a very general case that allows for the number of output pins to vary with block length, it is shown that the energy must scale as (n(log n) 1/5 ). A simple example is provided of a class of circuits performing low-density parity-check decoding whose energy complexity scales as O(n 2 log log n).

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It is shown that in a sequence of randomly generated bipartite configurations with number of left nodes approaching infinity, the probability that a particular configuration in the sequence has a minimum bisection width proportional to the number of vertices in the configuration approaches 1 so long as a sufficient condition on the node degree distribution is satisfied. This graph theory result implies an almost sure Ω n 2 scaling rule for the energy of capacity-approaching LDPC decoder circuits that directly instantiate their Tanner Graphs and are generated according to a uniform configuration model, where n is the block length of the code. For a sequence of circuits that have a full set of check nodes but do not necessarily directly instantiate a Tanner graph, this implies an Ω n 1.5 scaling rule. In another theorem, it is shown that all (as opposed to almost all) capacity-approaching LDPC decoding circuits that directly implement their Tanner graphs must have energy that scales as Ω n (log n) 2 .These results further imply scaling rules for the energy of LDPC decoder circuits as a function of gap to capacity.