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

Zhengfeng Ji, Tongyang Li, Changpeng Shao, Xinzhao Wang, Yuxin Zhang

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.

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

Abstract

We study the computational complexity of estimating the normalized trace for a log-local Hamiltonian acting on qubits. This problem arises naturally in the DQC1 model, yet its complexity is only understood for a limited class of functions . We show that if is a continuous function with approximate degree , then estimating up to constant additive error is DQC1-complete, under a technical condition on the polynomial approximation error of . This condition holds for a broad class of functions, including exponentials, trigonometric functions, logarithms, and inverse-type functions. We further prove that when 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 -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.

Paper Structure

This paper contains 13 sections, 12 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Technical overview
  4. Comparison with previous work edenhofer2025dequantization
  5. Preliminaries
  6. One clean qubit model
  7. Periodic Jacobi matrices
  8. Chebyshev Equioscillation Theorem and approximate degree
  9. Constructing periodic Jacobi matrices from prescribed spectra
  10. Proof of Theorem \ref{['thm:intro-main']}
  11. Lower bound on classical query complexity
  12. Conclusions
  13. Proof of Proposition \ref{['prop for LSIP']}

Key Result

Theorem 1.2

Assume that $f(x):[a,b] \subseteq [-1,1]\rightarrow [-1,1]$ is a continuous function. Suppose there exists a constant $\varepsilon<1$ such that $d:=\widetilde{\deg}_\varepsilon (f)=\Omega(\mathop{\mathrm{poly}}\nolimits(n))$ and $E_{d}/E_{d-1}<1/2$. Then the "normalized trace estimation problem" is

Figures (3)

  • Figure 1: The DQC1 computational model.
  • Figure 2: Illustration of the Chebyshev Equioscillation Theorem.
  • Figure 3: A $k$-query DQC1 algorithm.

Theorems & Definitions (24)

  • Theorem 1.2: DQC1-completeness
  • Proposition 1.3
  • Conjecture 1.4
  • Theorem 1.5: Classical lower bound on query complexity (conditional)
  • Definition 2.1: DQC1 complexity class
  • Proposition 2.2: Theorem 2 of knill1998power
  • Proposition 2.3: Corollary 6 of cade
  • Theorem 2.4: Chebyshev Equioscillation Theorem golomb1962lectures
  • Remark 2.5
  • Definition 2.6: Approximate degree
  • ...and 14 more