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

• URL: http://arxiv.org/abs/2604.01519v1
• Date: Thu, 02 Apr 2026 01:15:12 GMT
• Title: DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians
• Abstract summary: We study the computational complexity of estimating the normalized trace $2-nTr[f(A)]$ for a log-local Hamiltonian $A$ acting on $n$ qubits.<n>This problem arises naturally in the DQC1 model, yet its complexity is only understood for a limited class of functions $f(x)$.<n>We show that if $f(x)$ is a continuous function with approximate degree $(rm poly(n))$, then estimating $2-nTr[f(A)]$ up to constant additive error is
• Score: 20.990700912031773
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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