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

2026-04-01

AI Summary

This work investigates the computational complexity of estimating the normalized trace $2^{-n}\mathrm{Tr}[f(A)]$ of a function $f(A)$ applied to an $n$-qubit log-local Hamiltonian $A$. By identifying the approximation accuracy of $f$ as a key parameter and leveraging techniques including circuit-to-Hamiltonian mappings, periodic Jacobi operators, and Chebyshev polynomial approximation theory, the authors prove that, within the DQC1 query model, this problem is DQC1-complete for a broad class of functions—including exponentials, trigonometric, and logarithmic functions—provided the approximation error is $\Omega(\mathrm{poly}(n))$. Moreover, when $A$ is sparse, they establish an exponential lower bound on classical query complexity under standard complexity-theoretic assumptions. This result provides the first unified characterization of quantum efficient solvability and conditional classical intractability, achieving an exponential quantum–classical separation.

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 $Ω({\rm poly}(n))$, 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.

Problem

• normalized trace estimation
• log-local Hamiltonians
• DQC1-completeness
• approximate degree
• computational complexity

Innovation

• DQC1-completeness
• normalized trace estimation
• approximate degree
• quantum-classical separation
• polynomial approximation