Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics. For each self-adjoint, trace-class operator
O we define a set
Λn⊂R, and we show that it converges to the spectrum of
O in the Hausdorff metric under mild conditions. Our set
Λn only depends on the first
n moments of
O. We show that it can be effectively calculated for physically relevant operators, and it approximates the spectrum well without diagonalization.
We prove that using the above method we can converge to the minimal and maximal eigenvalues with super-exponential speed.
We also construct monotone increasing lower bounds
qn for the minimal eigenvalue (or decreasing upper bounds for the maximal eigenvalue). This sequence only depends on the moments of
O and a concrete upper estimate of its
1-norm; we also demonstrate that
qn can be effectively calculated for a large class of physically relevant operators. This rigorous lower bound
qn tends to the minimal eigenvalue with super-exponential speed provided that
O is not positive semidefinite. As a by-product, we obtain computable upper bounds for the
1-norm of
O, too.
Numerical examples demonstrate the relevance of our approximation in estimating entropy and negativity, which is useful, among others, in quantum optical and in open quantum system models. The results can be directly applicable to problems in quantum information, statistical mechanics, and quantum thermodynamics, where using traditional techniques based on diagonalization is impractical.
This note is devoted to discussing multivariate approximation of continuous functions on
[0,1]d with analytic Korobov kernels in the worst and average case settings. We only consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We study EC-
(s,t)-weak tractability under the absolute or normalized error criterion, and obtain necessary and sufficient conditions for
0<min(s,t)<1 and
max(s,t)≤1 in the worst case setting and for
s,t>0 in the average case setting.
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 asymptotic behavior for t→±∞ of S(t)=exp(−i H t)Sexp(+i H t) and its time average S̄ (t)=t−1∫0tdu S(u) is discussed. Here is S an element of the Banach space B1, constituted by the trace class of operators on the (separable or nonseparable) Hilbert space H, and H is the Hamiltonian, i.e., a bounded or unbounded self-adjoint operator on H. Necessary and sufficient conditions are given for the existence of the limits S̄(± ∞) and S(± ∞) with respect to the weak topology on B1, for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, ρ̄(t) has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].
We study a class of Friedrichs Hamiltonians, that is, operators describing an excited state coupled to a bath through a rank-one perturbation, in the case where the bath Hamiltonian has purely discrete spectrum. We consider sequences of such Hamiltonians for which the spectral measures associated to the coupling functions by the bath Hamiltonians converge weakly to a limiting measure that is absolutely continuous with a Hölder continuous density near the excited energy. Under this assumption, we show that the survival probability of the excited state decays in an approximately exponential manner on suitable time scales and under favourable conditions, with a decay rate given by Fermi's golden rule for the limiting measure. The error is estimated in terms of the coupling strength and the Lévy distance between the discrete spectral measures and the limiting measure. As an application, we treat a two-level atom coupled to a massless bosonic field confined to a large cavity in the rotating-wave approximation.