Quantum Limits of Measurement and Computing Induced by Conservation Laws and Uncertainty Relations

Errors in quantum computers can be classified into two classes: the static errors—decoherence in qubits arising from the interaction between computational qubits and the environment—and the dynamical errors—imperfection of logic operations arising from the interaction between computational qubits and controllers of quantum gates. The current theory of fault-tolerant quantum computing concludes that if the imperfection is below a certain threshold, the decoherence can be corrected in an arbitrarily large quantum computing. Thus, the fundamental question as to whether quantum computers are physically realizable or not can be reduced to such questions as whether fundamental physical laws lead to an unavoidable imperfection of quantum logic operations or not. On the other hand, it has been known in measurement theory that conservation laws limit the accuracy of measurements, as stated by the Wigner-Araki-Yanase (WAY) theorem: Observables which do not commute with bounded additive conserved quantities have no precise, nondisturbing measurements. During computation, a quantum computer usually needs to “measure” internal qubits to perform the branching program. Such “measurements” are carried out by the so-called controlled-not (CNOT) gates. Thus, it is natural to ask whether the WAY theorem leads to a conflict between the accuracy of quantum logic operations and conservation laws. In this paper, we give a quantitative expression of the WAY theorem to obtain a lower bound on the error and the disturbance. Then, we apply this bound to the implementations of CNOT gates and obtain the following results: If the computational basis is represented by a component of spin and physical implementations obey the angular momentum conservation law, any physically realizable unitary operations on the two computational qubits plus the S − 2 ancilla qubits cannot implement the CNOT gate within