
The threshold theorem of quantum error correction ensures faulttolerant
quantum
computing, if every component can be controlled within a constant threshold
accuracy. However, the threshold is so demanding that the realizability
problem
of scalable quantum computers is reduced to the problem of controllability
under
such a stringent accuracy requirement rather than solved in principle. In
order
to figure out how fundamental laws set a limit for the elementary gate
operations, we consider here the accuracy limit induced by conservation
laws. The
idea that conservation laws limit quantum control goes back to the works of
Wigner, Araki, and Yanase (the WAY theorem) stating that observables
notcommuting with additively conserved quantity cannot be measured
precisely
[1]. Recently, the WAY theorem has been reformulated in the modern framework
of
measurement theory to obtain various quantitative generalizations [2, 3]
derived
from the universally valid reformulation of the uncertainty principle on the
noisedisturbance tradeoff [4, 5, 6], and applied to quantum limits of the
accuracy of elementary gate operations under the angular momentum
conservation
law obeyed by the interaction between the computational qubits and the
controller, including the atomfield interaction described by a
JaynesCummings
model. The inevitable error probability has been shown to be inversely
proportional to the variance of the controller's conserved quantity for the
CNOT gate [7, 8], the Hadamard gate [3], and the NOT gate [9], while the SWAP
gate obeys no constraint. In this talk, these considerations will be extended to
multiqubit gates such as the Toffoli gate, the Fredkin gate, and general
controlled unitary gates.
REFERENCES:
[1] H. Araki and M. M. Yanase, Phys. Rev. 120, 622 (1960).
[2] M. Ozawa, Phys. Rev. Lett. 88, 050402 (2002).
[3] M. Ozawa, Int. J. Quant. Inf. 1, 569 (2003).
[4] M. Ozawa, Phys. Rev. A 67, 042105 (2003).
[5] M. Ozawa, Ann. Phys. (N.Y.) 311, 350 (2004).
[6] M. Ozawa, J. Opt. B: Quantum Semiclass. Opt. 7, S672 (2005).
[7] M. Ozawa, Phys. Rev. Lett. 89, 057902 (2002).
[8] M. Ozawa, Phys. Rev. Lett. 91, 089802 (2003).
[9] T. Karasawa and M. Ozawa, Phys. Rev. A 75, 032324 (2007)..
