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

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.

38 pages, 2 figures

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• Universal limitations on implementing resourceful unitary evolutions
• Universal limitation of quantum information recovery: symmetry versus coherence
• Wigner-Araki-Yanase theorem for continuous and unbounded conserved observables
• Heisenberg's uncertainty relation: Violation and reformulation
• Measurement disturbance and conservation laws in quantum mechanics
• Quantum Limits of Measurements Induced by Multiplicative Conservation Laws: Extension of the Wigner-Araki-Yanase Theorem
• Coherence-variance uncertainty relation and coherence cost for quantum measurement under conservation laws
• WAY beyond conservation laws
• A Fundamental Upper Bound for Signal to Noise Ratio of Quantum Detectors
• Can we accurately read or write quantum data?