In order to make a unified treatment for estimation problems of a very small noise or a very weak signal in a quantum process, we introduce the notion of a low-noise quantum channel with one noise parameter. It is known in several examples that prior entanglement together with nonlocal output measurement improves the performance of the channel estimation. In this paper, we study this "ancilla-assisted enhancement" for estimation of the noise parameter in a general low-noise channel. For channels on two level systems we prove that the enhancement factor, the ratio of the Fisher information of the ancilla-assisted estimation to that of the original one, is always upper bounded by 3/2. Some conditions for the attainability are also given with illustrative examples.

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 tradeoff 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.Recently, there have been extensive research efforts to explore whether fundamental physical laws put any constraints on realizing scalable quantum computing. Soon after the discovery of Shor's algorithm [1], it was pointed out by several physicists [2,3,4] that the decoherence, the exponential decay of coherence in time, caused by the coupling between a quantum computer and the environment would cancel out the computational advantage of quantum computers. To overcome this difficulty, quantum error-correction was proposed [5,6], and the subsequent development has established the so-called threshold theorem: if the error caused by the decoherence in individual quantum gates is below a certain constant threshold, it is possible in principle to efficiently perform an arbitrary scale of fault-tolerant quantum computation with error-correction [7]. Thus, the error-correction reduces, in principle, the scalability problem to the accuracy problem requiring individual quantum logic gates to clear the error threshold, though being still quite demanding.In general, decoherence in quantum computer components can be classified into two classes: (i) static decoherence, arising from the interaction between computational qubits, typically in the memory, and the environment, and (ii) dynamical decoherence, arising from the interaction between computational qubits, typically in the register, and the control system of gate operations [8]. The static decoherence may be overcome by developing materials with long decoherence time. On the other hand, the dynamical decoherence poses a dilemma between contr...

Recent investigations show that conservation laws limit the accuracy of gate operations in quantum computing. The inevitable error under the angular momentum conservation law has been evaluated so far for the CNOT, Hadamard, and NOT gates for spin 1/2 qubits, while the SWAP gate has no constraint. Here, we extend the above results to general single-qubit gates. We obtain an upper bound of the gate fidelity of arbitrary single-qubit gates implemented under arbitrary conservation laws, determined by the geometry of the conservation law and the gate operation on the Bloch sphere as well as the size of the ancilla.

The notion of low-noise channels was recently proposed and analyzed in detail in order to describe noise-processes driven by environment [M. Hotta, T. Karasawa and M. Ozawa, Phys. Rev. A72, 052334 (2005) ]. An estimation theory of low-noise parameters of channels has also been developed. In this report, we address the low-noise parameter estimation problem for the N -body extension of low-noise channels. We perturbatively calculate the Fisher information of the output states in order to evaluate the lower-bound of the mean-square error of the parameter estimation. We show that the maximum of the Fisher information over all input states can be attained by a factorized input state in the leading order of the low-noise parameter. Thus, to achieve optimal estimation, it is not necessary for there to be entanglement of the N subsystems, as long as the true low-noise parameter is sufficiently small.

We consider the achievability problem of the Cramér-Rao bound for multiparameter estimations. In general, it is not achievable due to the noncommutativity of optimal measurements for the corresponding parameters. However, we show that, under certain conditions, it can always be attained up to the leading order in the parameters as long as D ഛ N − 1, where D and N denote the number of parameters and the dimension of the system, respectively. After proving that, we discuss the achievability in the context of channel estimation for a general channel called a low-noise channel, which is very useful for investigating parameter estimation in the leading order. This allows us to find an ancilla-assisted enhancement effect: if entangled input states with an ancilla system are utilized for the channel estimation together with collective measurements on those output states, the bound becomes achievable for D ഛ N 2 −1.