Concepts of convergence for a quantum law of large numbers

Wilhelm Ochs

Clement Ronfaut, Robin Ollive, Stephane Louise

In quantum computing, some of the main quantum operations (i.e., gates) cannot be simply implemented and require an approximation. Major quantum algorithm routines enable the implementation of these specific quantum gates by combining basic quantum circuits with an iterative structure. In this structure, the number of repetitions of the basic circuit pattern is associated with convergence parameters. The asymptotic convergence of the gate error with respect to the number of basic pattern repetitions is known as the query complexity. The underlying convergence law is bounded, but often lacks explicit constants for specific problem instances. Upper bounds are generally too pessimistic to be useful in practice. The actual convergence law contains constants that depend on the joint properties of the matrix encoded by the query and the initial state vector, which are difficult to compute classically. This paper proposes a strategy to study this convergence law directly from the Quantum Processing Unit (QPU) output. Given a convergence law, this protocol extracts the numerical values of the associated constants from the gate (i.e., operation) approximation at different accuracies (i.e., convergence parameters) constructed directly on the QPU. This protocol is called Numerical Error Extraction by Quantum Measurement Algorithm (NEEQMA). NEEQMA concepts are tested on specific instances of Quantum Signal Processing (QSP) and Hamiltonian Simulation via Trotterization. Knowing the convergence constants associated with a semi-norm of the error allows selecting the smallest convergence parameters that achieve the required gate approximation accuracy, thereby satisfying the quantum algorithm's requirements.

Fedor Herbut

The quantum probabilistic convergence in measurement, distinct from mathematical convergence, is derived for indeterminate probabilities from the weak quantum law of large numbers. This is presented in three theorems. The first establishes the necessary bridge between ensemble theory and experiment. The second analyzes the most important theoretical ensemble entity: the eigen-projector of the relative frequency operator. Its physical meaning is the experimental relative frequency. The third theorem formulates the quantum probabilistic convergence, which is the final result of this investigation.

Luigi Accardi, Igor Vasil'evich Volovich, Yun Gang Lu

Stanley Gudder

Imre Csiszár, Fumio Hiai, Dénes Petz

In a thermodynamic model, Diósi et al. [Int. J. Quantum Inf. 4, 99–104 (2006)] arrived at a conjecture stating that certain differences of von Neumann entropies converge to relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The analytic proof uses the quantum law of large numbers and the inequality between the Belavkin-Staszewski and Umegaki relative entropies. Moreover, the concept of channel capacity per unit cost is introduced for classical-quantum channels. For channels with binary input alphabet, this capacity is shown to equal the relative entropy. The result provides a second proof of the conjecture and a new interpretation. Both approaches lead to generalizations of the conjecture.