Information Complexity of Quantum Gates

This paper considers the realizability of quantum gates from the perspective of information complexity. Since the gate is a physical device that must be controlled classically, it is subject to random error. We define the complexity of gate operation in terms of the difference between the entropy of the variables associated with initial and final states of the computation. We argue that the gate operations are irreversible if there is a difference in the accuracy associated with input and output variables. It is shown that under some conditions the gate operation may be associated with unbounded entropy, implying impossibility of implementation. Introduction In this paper, we consider complexity and realizability of quantum gates from the point of view of information theory. A gate is a physical system that is controlled by varying some input variables, which are classical. In principle, such a physical system could implement a variety of operators based on the control variables. The gate functions may be also implemented by a single physical system that operates sequentially on the qubits in the quantum register. The complexity of the gate will be defined in terms of the entropy associated with its control. From a practical point of view, one is interested in asking how easy it is to control a gate. As no analog system can have infinite precision, we investigate what happens if the precision levels at the input and the output are different. The complexity of the gate, defined in terms of entropy, will be examined for the rotation and cnot gates in certain circuits. Information processing by gate One aspect of gate performance is its accuracy. Researchers on quantum information science have given much attention to the question of errors and their correction [1-3] by drawing upon parallels with classical information. Quantum error-correction coding works like classical error-correction to correct some large errors. But the framework of quantum information is distinct from that of classical information. In the classical case, it is implicitly assumed that there occurs an automatic correction of errors that are smaller than a threshold by means of clipping or by the use of a decision circuit. In the case of quantum information, the input data is nominally discrete, but in reality its precision cannot be absolute in any actual realization. Furthermore, unknown small errors in quantum information cannot be corrected [4-5]. Consequently, proposals for error correction and fault tolerance (such as [6-8]) remain unrealistic. Classical analog computation and quantum processing do have parallels. In general, fixed errors in gate operation could become irreversible due to actual small nonlinearity of nominally linear elements. Analog computing is not practical to implement because noise cannot be separated from useful signal and it accumulates, degrading the system performance in an uncorrectable manner. If there were no noise, the practicality of analog computing would depend on the feasibility of the gate implementation over the expected input-output range. This feasibility must be checked in the context of the limitations on information processing by the gate. Consider the gate G of Figure 1. It may be assumed that it is a physical system which is controlled by means of some variable. This control is implemented by choosing a setting on an instrument, and this choice is associated with random error. If one views the circuit operations to be implemented by the same device transitioning through various states in sequence, then one can determine the distribution of the control variable states, and compute its entropy. This entropy, when determined for the entire computing circuit, may be taken to represent its complexity. Information is preserved, therefore one can define the following relationship for the entropy expressions for the inputX, the gate control information C, and the output Y : H(Y ) = H(X) +H(C). (1)