We study the computational complexity of estimating the normalized trace $2^{-n}\mathrm{Tr}[f(A)]$ 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 $Ω(\mathrm{poly}(n))$, then estimating $2^{-n}\mathrm{Tr}[f(A)]$ 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. 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 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.
Comments
With the assistance of GPT-5.6 Sol, we have further relaxed this "technical condition'' to the assumption that $f(x)$ is Lipschitz continuous. To appear in FOCS 2026
We study the computational complexity of estimating the normalized trace 2−nTr[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(poly(n)), then estimating 2−nTr[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. 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 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.
The local Hamiltonian (LH) problem is the canonical QMA-complete problem introduced by Kitaev. In this paper, we show its hardness in a very strong sense: we show that the 3-local Hamiltonian problem on n qubits cannot be solved classically in time O(2(1−ε)n) for any ε>0 under the Strong Exponential-Time Hypothesis (SETH), and cannot be solved quantumly in time O(2(1−ε)n/2) for any ε>0 under the Quantum Strong Exponential-Time Hypothesis (QSETH). These lower bounds give evidence that the currently known classical and quantum algorithms for LH cannot be significantly improved.
Furthermore, we are able to demonstrate fine-grained complexity lower bounds for approximating the quantum partition function (QPF) with an arbitrary constant relative error. Approximating QPF with relative error is known to be equivalent to approximately counting the dimension of the solution subspace of QMA problems. We show the SETH and QSETH hardness to estimate QPF with constant relative error. We then provide a quantum algorithm that runs in O(2n) time for an arbitrary 1/poly(n) relative error, matching our lower bounds and improving the state-of-the-art algorithm by Bravyi, Chowdhury, Gosset, and Wocjan (Nature Physics 2022) in the low-temperature regime.
To prove our fine-grained lower bounds, we introduce the first size-preserving circuit-to-Hamiltonian construction that encodes the computation of a T-time quantum circuit acting on N qubits into a (d+1)-local Hamiltonian acting on N+O(T1/d) qubits. This improves the standard construction based on the unary clock, which uses N+O(T) qubits.
We investigate the dividing line between classical and quantum computational power in estimating properties of matrix functions. More precisely, we study the computational complexity of two primitive problems: given a function f and a Hermitian matrix A , we compute a matrix element of f ( A ) or a local measurement on f ( A ) | 0 ⟩ ⊗ n , with | 0 ⟩ ⊗ n being an n -qubit reference state vector, in both cases up to additive approximation error. We consider four functions—monomials, Chebyshev polynomials, the time evolution function, and the inverse function—and probe the complexity across a broad landscape covering different problem input regimes. Namely, we consider two types of matrix inputs (sparse and Pauli access), matrix properties (norm and sparsity), the approximation error, and function-specific parameters. We identify BQP- forms of both problems for each function and then toggle the problem parameters to easier regimes to see where hardness remains, or where the problem becomes classically easy. As part of our results, we make concrete a hierarchy of hardness across the functions; in parameter regimes where we have classically efficient algorithms for monomials, all three other functions remain robustly BQP-, or hard under usual computational complexity assumptions. In identifying classically easy regimes, among others, we show that for any polynomial of degree poly ( n ) , both problems can be efficiently classically simulated when A has O ( log n ) nonzero coefficients in the Pauli basis. This contrasts with the fact that the problems are BQP- in the sparse access model even for constant row sparsity, whereas the stated Pauli access efficiently constructs sparse access with row sparsity O ( log n ) . Our work provides a catalog of efficient quantum and classical algorithms for fundamental linear-algebra tasks.
We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form Lφ(ρ)=tr(φ(dρ))/d, where ρ is an unknown d-dimensional quantum state and φ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating Lφ(ρ) from properties of φ, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting.
The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is Θ(εlog2(d)max{1,εlog2(d)}d2+ε2log2(d)), and the sample complexity of spectrum estimation is Θ(ε2log2(d)max{1,ε2log2(d)}d2). In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.
We study the randomized classical query complexity of cyclic Forrelation, a cyclic analogue of k-Forrelation [AA15, STOC'15; BS21, STOC'21]. Given k Boolean functions f1,…,fk:F2n→{−1,1}, the goal is to estimate Tk=N−1Tr(Df1H⊗n⋯DfkH⊗n), where Dfi=diag(fi(x):x∈F2n), H is the Hadamard gate, and N=2n. This normalized-trace problem can be solved with k quantum queries in the DQC1 model. Its classical query complexity was left open in [JLS+26, FOCS'26], which conjectured a lower bound of Ω(N1−2/k).
We establish a nearly complete characterization of the nonadaptive classical query complexity of cyclic Forrelation. For even k≥4, we prove a tight bound of Θk(N1−2/k) for constant additive error. For odd k≥5, we obtain a lower bound of Ωk(N1−2/(k−1)) and an upper bound of Ok(N1−2/k). Surprisingly, k=3 admits a polylogarithmic-query nonadaptive classical algorithm, disproving the conjecture in this case. These results suggest an unexpected parity-dependent structure: the nonadaptive complexity may be governed by the independence number of the k-cycle, leading to the exponent 1−1/⌊k/2⌋, rather than the previously conjectured 1−2/k, which was motivated by a Fourier-growth heuristic. Finally, allowing adaptivity, we prove a lower bound of Ω(N1/4) for every k≥4.
Although formally close to k-Forrelation, the cyclic structure leads to rather different classical behavior and requires different techniques. Our lower bounds are based on hidden-shift collision constructions, and our upper bounds follow from randomized sampling together with cycle-specific conditional-moment estimates. Our results provide a systematic understanding of its query advantages over classical computation.
3-Local Hamiltonian Problem and Constant Relative Error Quantum Partition Function Approximation: O(22n) Algorithm Is Nearly Optimal under QSETH
We investigate the computational complexity of the Local Hamiltonian (LH) problem and the approximation of the Quantum Partition Function (QPF), two central problems in quantum many-body physics and quantum complexity theory. Both problems are known to be QMA-hard, and under the widely believed assumption that BQP=QMA, no efficient quantum algorithm exits. The best known quantum algorithm for LH runs in O(22n(1−o(1))) time, while for QPF, the state-of-the-art algorithm achieves relative error δ in O∗(δ1Z2n) time, where Z denotes the value of the partition function. A nature open question is whether more efficient algorithms exist for both problems. In this work, we establish tight conditional lower bounds showing that these algorithms are nearly optimal. Under the plausible Quantum Strong Exponential Time Hypothesis (QSETH), we prove that no quantum algorithm can solve either LH or approximate QPF significantly faster than O(2n/2), even for 3-local Hamiltonians. In particular, we show: 1) 3-local LH cannot be solved in time O(22n(1−ε)) for any ε>0 under QSETH; 2) 3-local QPF cannot be approximated up to any constant relative error in O(22n(1−ε)) time for any ε>0 under QSETH; and 3) we present a quantum algorithm that approximates QPF up to relative error 1/2+1/poly(n) in O∗(2n/2) time, matching our conditional lower bound. Notably, our results provide the first fine-grained lower bounds for both LH and QPF with fixed locality. This stands in sharp contrast to QSETH and the trivial fine-grained lower bounds for LH, where the locality of the SAT instance and the Hamiltonian depends on the parameter ε in the O(22n(1−ε)) running time.
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.