DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians

TL;DR

This paper investigates the complexity of estimating normalized traces of functions of log-local Hamiltonians, establishing DQC1-completeness for functions with high approximate degree and linking quantum and classical hardness.

Contribution

It demonstrates that the approximate degree of a function determines the quantum and classical complexity of normalized trace estimation, revealing an exponential quantum-classical separation.

Findings

Estimating trace of functions with high approximate degree is DQC1-complete.

Classical query complexity is exponential in approximate degree under certain conjectures.

The results unify quantum and classical complexity insights through polynomial approximation theory.

Abstract

We study the computational complexity of estimating the normalized trace $2^{-n}Tr[f(A)]$ for a log-local Hamiltonian $A$ acting on $n$ qubits. This problem arises naturally in the DQC1 model, yet its complexity is only understood for a limited class of functions $f(x)$. We show that if $f(x)$ is a continuous function with approximate degree $\Omega({\rm poly}(n))$, then estimating $2^{-n}Tr[f(A)]$ up to constant additive error is DQC1-complete, under a technical condition on the polynomial approximation error of $f(x)$. This condition holds for a broad class of functions, including exponentials, trigonometric functions, logarithms, and inverse-type functions. We further prove that when $A$ is sparse, the classical query complexity of this problem is exponential in the approximate degree, assuming a conjectured lower bound for a trace variant of the $k$-Forrelation problem in the DQC1 query model. Together, these results identify the approximate degree as the key parameter governing the complexity of normalized trace estimation: it characterizes both the quantum complexity (via efficient DQC1 algorithms) and, conditionally, the classical hardness, yielding an exponential quantum-classical separation. Our proof develops a unified framework that cleanly combines circuit-to-Hamiltonian constructions, periodic Jacobi operators, and tools from polynomial approximation theory, including the Chebyshev equioscillation theorem.