Quantum Approximate Counting with Additive Error: Hardness and Optimality

Quantum counting is the task of determining the dimension of the subspace of states that are accepted by a quantum verifier circuit. It is the quantum analog of counting the number of valid solutions to NP problems - a problem well-studied in theoretical computer science with far-reaching implications in computational complexity. The complexity of solving the class #BQP of quantum counting problems, either exactly or within suitable approximations, is related to the hardness of computing many-body physics quantities arising in algebraic combinatorics. Here, we address the complexity of quantum approximate counting under additive error. First, we show that computing additive approximations to #BQP problems to within an error exponential in the number of witness qubits in the corresponding verifier circuit is as powerful as polynomial-time quantum computation. Next, we show that returning an estimate within error that is any smaller is #BQP-hard. Finally, we show that additive approximations to a restricted class of #BQP problems are equivalent in computational hardness to the class DQC1. Our work parallels results on additively approximating #P and GapP functions.