Quantum Approximate Counting with Additive Error: Hardness and Optimality

TL;DR

This paper investigates the computational complexity of quantum approximate counting with additive error, revealing its hardness and optimality, and establishing connections to classes like #BQP and DQC1.

Contribution

It characterizes the hardness of additive approximations in quantum counting, showing their equivalence to known quantum complexity classes and establishing optimal bounds.

Findings

Additive approximation within exponential error is as powerful as polynomial-time quantum computation.

Smaller additive errors are #BQP-hard to compute.

Certain restricted #BQP problems are equivalent in hardness to DQC1.

Abstract

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.