Approximate Counting and Quantum Computation

Motivated by the result that an `approximate’ evaluation of the Jones polynomial of a braid at a $5^{th}$ root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes #P and GapP have such an approximation scheme under certain natural normalisations. However we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work.

💡 Research Summary

The paper investigates a bridge between quantum computation (the class BQP) and classical counting complexity (the classes #P and GapP) by introducing a novel notion of additive approximation. The motivation stems from the known result that an approximate evaluation of the Jones polynomial of a braid at a fifth root of unity can be used to simulate the quantum part of any BQP algorithm. This observation suggests that certain counting problems may capture the power of quantum computation, prompting the authors to ask whether a general additive approximation framework can subsume BQP‑computable functions.

The authors define an additive approximation scheme for a function f as a randomized algorithm A that, given an input x and a normalisation factor C(x) that grows at most polynomially with |x|, outputs a value (\tilde{v}) such that
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