which variables are fixed allows the inference problem to be changed dynamically as the circuit is running.]{}

![image](figures/f2-transitions.ai){height="\textheight"}

[ (A) The discrete-sample gate is a central building block for stochastic transition circuits, used to implement Gibbs transition operators that update a variable by sampling from its conditional distribution given the variables it interacts with. The gate renormalizes the input log probabilities it is given, converts them to probabilities (by exponentiation), and then samples from the resulting distribution. Input energies are specified via a custom fixed-point coding scheme. (B) Discrete-sample gates remain accurate even when implemented at extremely low bit-precision. Here we show the relative entropy between true distributions and their low-precision implementations, for millions of distributions over discrete sets with 1000 elements; accuracy loss is negligible even when only 8 bits of precision are used. (C) The accuracy of low-precision discrete-sample gates can be understood by considering multinomial distributions with high, medium and low entropy. High entropy distributions involve outcomes with very similar probability, insensitive to ratios, while low entropy distributions are dominated by the location of the most probable outcome. (D) Low-precision transition circuits save area as compared to high-precision floating point alternatives; these area savings make it possible 