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用生物科技製造量子電腦！
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quantum dots是很少量的原子所組成，其電子能量態密度介於原子與材質之間，由quantum
dots效應,會出現像原子的不連續電子能階發生
Quantum learning without quantum memory
A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum mechanics for any size of the training set. This result is shown to be robust under (an arbitrary amount of) noise and under (statistical) variations in the composition of the training set, provided it is large enough. This machine can be used an arbitrary number of times without retraining. Its required classical memory grows only logarithmically with the number of training qubits, while its excess risk decreases as the inverse of this number, and twice as fast as the excess risk of an “estimate-and-discriminate” machine, which estimates the states of the training qubits and classifies the data qubit with a discrimination protocol tailored to the obtained estimates.
Subject terms:Introduction
Quantum computers are expected to perform some (classical) computational tasks of practical interest, e.g., large integer factorization, with unprecedented efficiency. Quantum simulators, on the other hand, perform tasks of a more “quantum nature”, which cannot be efficiently carried out by a classical computer. Namely, they have the ability to simulate complex quantum dynamical systems of interest. The need to perform tasks of genuine quantum nature is emerging as individual quantum systems play a more prominent role in labs (and, eventually, in everyday life). Examples include: quantum teleportation, dynamical control of quantum systems, or quantum state identification. Quantum information techniques are already being developed in order to execute these tasks efficiently.
This paper is concerned with a simple, yet fundamental instance of quantum state identification. A source produces two unknown pure qubit states with equal probability. A human expert (who knows the source specifications, for instance) classifies a number of 2n states produced by this source into two sets of size roughly n (statistical fluctuations of order  should be expected) and attaches the labels 0 and 1 to them. We view these 2n states as a training sample, and we set ourselves to find a universal machine that uses this sample to assign the right label to a new unknown state produced by the same source. We refer to this task as quantum classification for short.Quantum classification can be understood as a supervised quantum learning problem, as has been noticed by Guta and Kotlowski in their recent work1 (though they use a slightly different setting). Learning theory, more properly named machine learning theory, is a very active and broad field which roughly speaking deals with algorithms capable of learning from experience2. Its quantum counterpart3, 4, 5, 6, 7 not only provides improvements over some classical learning problems but also has a wider range of applicability, which includes the problem at hand. Quantum learning has also strong links with quantum control theory and is becoming a significant element of the quantum information processing toolbox. An absolute limit on the minimum error in quantum classification is provided by the so called optimal programmable discrimination machine8, 9, 10, 11, 12, 13. In this context, to ensure optimality one assumes that a fully general two-outcome joint measurement is performed on both the 2n training qubits and the qubit we wish to classify, where the observed outcome determines which of the two labels, 0 or 1, is assigned to the latter qubit. Thus, in principle, this assumption implies that in a learning scenario a quantum memory is needed to store the training sample till the very moment we wish to classify the unknown qubit. The issue of whether or not the joint measurement assumption can be relaxed has not yet been addressed. Nor has the issue of how the information left after the joint measurement can be used to classify a second unknown qubit produced by the same source, unless a fresh new training set (TS) is provided (which may seem unnatural in a learning context). The aim of this paper is to show that for a sizable TS (asymptotically large n) the lower bound on the probability of misclassifying the unknown qubit set by programmable discrimination can be attained by first performing a suitable measurement on the TS followed by a Stern-Gerlach type of measurement on the unknown qubit, where forward classical communication is used to control the parameters of the second measurement. The whole protocol can thus be undersood as a learning machine (LM), which requires much less demanding assumptions while still having the same accuracy as the optimal programmable discrimination machine. All the relevant information about the TS needed to control the Stern-Gerlach measurement is kept in a classical memory, thus classification can be executed any time after the learning process is completed. Once trained, this machine can be subsequently used an arbitrary number of times to classify states produced by the same source. Moreover, this optimal LM is robust under noise, i.e., it still attains optimal performance if the states produced by the source undergo depolarization to any degree. Interestingly enough, in the ideal scenario where the qubit states are pure and the TS consists in exactly the same number of copies of each of the two types 0/1 (no statistical fluctuations are allowed) this LM attains the optimal programmable discrimination bound for any size 2n of the TS, not necessarily asymptotically large. At this point it should be noted that LMs without quantum memory can be naturally assembled from two quantum information primitives: state estimation and state discrimination. We will refer to these specific constructions as “estimate-and-discriminate” (E&D) machines. The protocol they execute is as follows: by performing, e.g., an optimal covariant measurement on the n qubits in the TS labeled 0, their state |ψ0〉 is estimated with some accuracy, and likewise the state |ψ1〉 of the other n qubits that carry the label 1 is characterized. This classical information is stored and subsequently used to discriminate an unknown qubit state. It will be shown that the excess risk (i.e., excess average error over classification when the states |ψ0〉 and |ψ1〉 are perfectly known) of this protocol is twice that of the optimal LM. The fact that the E&D machine is suboptimal means that the kind of information retrieved from the TS and stored in the classical memory of the optimal LM is specific to the classification problem at hand, and that the machine itself is more than the mere assemblage of well known protocols. We will first present our results for the ideal scenario where states are pure and no statistical fluctuation in the number of copies of each type of state is allowed. The effect of these fluctuations and the robustness of the LM optimality against noise will be postponed to the end of the section. |
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