Conservation-Law-Induced Quantum Limits for Physical Realizations of the Quantum NOT Gate

TL;DR

This paper introduces a new method to determine fundamental quantum limits on implementing the quantum NOT gate, revealing more stringent error bounds imposed by conservation laws, which impact quantum computing accuracy.

Contribution

The authors develop a novel approach using orthogonal polynomials to derive lower bounds on error probability for the quantum NOT gate under conservation laws.

Findings

Lower bounds are more stringent than previous estimates.

The method applies to systems with conserved total angular momentum.

Results improve understanding of physical constraints on quantum gate implementations.

Abstract

In recent investigations, it has been found that conservation laws generally lead to precision limits on quantum computing. Lower bounds of the error probability have been obtained for various logic operations from the commutation relation between the noise operator and the conserved quantity or from the recently developed universal uncertainty principle for the noise-disturbance trade-off in general measurements. However, the problem of obtaining the precision limit to realizing the quantum NOT gate has eluded a solution from these approaches. Here, we develop a new method for this problem based on analyzing the trace distance between the output state from the realization under consideration and the one from the ideal gate. Using the mathematical apparatus of orthogonal polynomials, we obtain a general lower bound on the error probability for the realization of the quantum NOT gate in terms of the number of qubits in the control system under the conservation of the total angular momentum of the computational qubit plus the the control system along the direction used to encode the computational basis. The lower bound turns out to be more stringent than one might expect from previous results. The new method is expected to lead to more accurate estimates for physical realizations of various types of quantum computations under conservation laws, and to contribute to related problems such as the accuracy of programmable quantum processors.