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 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...

Quantum limits of measurements induced by multiplicative conservation laws: Extension of the Wigner-Araki-Yanase theorem

“…For additive conservation laws, there has been an extensive literature [1,2,3,4,5,6,7,8,9] on the conservation-law-induced quantum limits on the performance of elementary quantum gates; see also [29,30,31,32,33,34,35] for the model-dependent approach to the limitations of quantum gate operations realized by the atom-field interaction, which has turned out to be consistent with the model-independent approach based on conservation laws. Our method will be also expected to contribute to the problem of programmable quantum processors [36,37,38] and related subjects [39,40,41] in future investigations.…”

“…In recent investigations [1,2,3,4,5,6,7,8,9] it has been established that conservation laws put a precision limit or compensating resource requirements on quantum information processing. Quantitative analysis suggests that conservation laws lead to undesirable entanglement of the object system with the control system, such as atom qubits controlled by the Jaynes-Cummings interaction with an electromagnetic field, causing decoherence of the object system, even if the environment induced decoherence is completely suppressed.…”

Wigner-Araki-Yanase theorem beyond conservation laws

“…Although the details are beyond the scope of this paper and will be left as a topic for separate investigation, this approach could potentially lead to WAY-type limitations on quantum logic gates and computation that are more straightforward and general than those reported in Refs. [10][11][12].…”

“…For example, limitations on the realisability of quantum logic gates due to the WAYtheorem have been discussed in Refs. [9][10][11][12].…”

Theoretical constraints on implementations of arbitrary single-qubit gates under conservation laws

“…Recently, this general approach has been extended to investigate error bounds for quantum-NOT gate using the gate trace distance between the desired unitary map and the completely positive map of the implemented gate operations [12]. Evaluating the lower bound of the gate trance distance, we can obtain error bounds, which could be tighter than ones expected from the previous estimates based on the commutation relation.…”