nips nips2003 nips2003-187 knowledge-graph by maker-knowledge-mining

187 nips-2003-Training a Quantum Neural Network

Source: pdf

Author: Bob Ricks, Dan Ventura

Abstract: Most proposals for quantum neural networks have skipped over the problem of how to train the networks. The mechanics of quantum computing are different enough from classical computing that the issue of training should be treated in detail. We propose a simple quantum neural network and a training method for it. It can be shown that this algorithm works in quantum systems. Results on several real-world data sets show that this algorithm can train the proposed quantum neural networks, and that it has some advantages over classical learning algorithms. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Most proposals for quantum neural networks have skipped over the problem of how to train the networks. [sent-5, score-0.96]

2 The mechanics of quantum computing are different enough from classical computing that the issue of training should be treated in detail. [sent-6, score-1.079]

3 We propose a simple quantum neural network and a training method for it. [sent-7, score-1.057]

4 It can be shown that this algorithm works in quantum systems. [sent-8, score-0.904]

5 Results on several real-world data sets show that this algorithm can train the proposed quantum neural networks, and that it has some advantages over classical learning algorithms. [sent-9, score-0.982]

6 1 Introduction Many quantum neural networks have been proposed [1], but very few of these proposals have attempted to provide an in-depth method of training them. [sent-10, score-1.032]

7 Most either do not mention how the network will be trained or simply state that they use a standard gradient descent algorithm. [sent-11, score-0.15]

8 This assumes that training a quantum neural network will be straightforward and analogous to classical methods. [sent-12, score-1.133]

9 While some quantum neural networks seem quite similar to classical networks [2], others have proposed quantum networks that are vastly different [3, 4, 5]. [sent-13, score-1.976]

10 Several of these networks also employ methods which are speculative or difﬁcult to do in quantum systems [7, 8]. [sent-15, score-0.922]

11 These signiﬁcant differences between classical networks and quantum neural networks, as well as the problems associated with quantum computation itself, require us to look more deeply at the issue of training quantum neural networks. [sent-16, score-2.882]

12 It is an open question what advantages a quantum neural network (QNN) would have over a classical network. [sent-18, score-1.061]

13 Other results have shown that QNNs may work best with some classical components as well as quantum components [2]. [sent-20, score-0.962]

14 This paper details such a network and how training could be done on it. [sent-23, score-0.151]

15 2 Quantum Computation Several necessary ideas that form the basis for the study of quantum computation are brieﬂy reviewed here. [sent-25, score-0.903]

16 The Hilbert space has a set of states, |φi , that form a basis, and the system is described by a quantum state |ψ = i ci |φi . [sent-30, score-0.938]

17 |ψ is said to be coherent or to be in a linear superposition of the basis states |φi , and in general the coefﬁcients ci are complex. [sent-31, score-0.194]

18 A postulate of quantum mechanics is that if a coherent system interacts in any way with its environment (by being measured, for example), the superposition is destroyed. [sent-32, score-1.059]

19 A two-state quantum system is used as the basic unit of quantum computation. [sent-37, score-1.772]

20 Such a system is referred to as a quantum bit or qubit and naming the two states |0 and |1 , it is easy to see why this is so. [sent-38, score-0.959]

21 2 Operators Operators on a Hilbert space describe how one wave function is changed into another and they may be represented as matrices acting on vectors (the notation |· indicates a column vector and the ·| a [complex conjugate] row vector). [sent-40, score-0.153]

22 In the quantum formalism, all properties are represented as operators whose eigenstates are the basis for the Hilbert space associated with that property and whose eigenvalues are the quantum allowed values for that property. [sent-44, score-1.89]

23 It is important to note that operators in quantum mechanics must be linear operators and further that they must be unitary. [sent-45, score-1.039]

24 This is a phenomenon common to all kinds of wave mechanics from water waves to optics. [sent-49, score-0.164]

25 The well known double slit experiment demonstrates empirically that at the quantum level interference also applies to the probability waves of quantum mechanics. [sent-50, score-1.839]

26 The wave function interferes with itself through the action of an operator – the different parts of the wave function interfere constructively or destructively according to their relative phases just like any other kind of wave. [sent-51, score-0.238]

27 4 Entanglement Entanglement is the potential for quantum systems to exhibit correlations that cannot be accounted for classically. [sent-53, score-0.902]

28 From a computational standpoint, entanglement seems intuitive enough – it is simply the fact that correlations can exist between different qubits – for example if one qubit is in the |1 state, another will be in the |1 state. [sent-54, score-0.228]

29 However, from a physical standpoint, entanglement is little understood. [sent-55, score-0.154]

30 What makes it so powerful (and so little understood) is the fact that since quantum states exist as superpositions, these correlations exist in superposition as well. [sent-57, score-1.032]

31 There are different degrees of entanglement and much work has been done on better understanding and quantifying it [10, 11]. [sent-67, score-0.148]

32 Finally, it should be mentioned that while interference is a quantum property that has a classical cousin, entanglement is a completely quantum phenomenon for which there is no classical analog. [sent-68, score-2.113]

33 5 An Example – Quantum Search One of the best known quantum algorithms searches an unordered database quadratically faster than any classical method [12, 13]. [sent-72, score-1.005]

34 The algorithm begins with a superposition of all N data items and depends upon an oracle that can recognize the target of the search. [sent-73, score-0.247]

35 Classically, searching such a database requires O(N ) oracle calls; however, on a quan√ tum computer, the task requires only O( N ) oracle calls. [sent-74, score-0.252]

36 Each oracle call consists of a quantum operator that inverts the phase of the search target. [sent-75, score-1.099]

37 3 A Simple Quantum Neural Network We would like a QNN with features that make it easy for us to model, yet powerful enough to leverage quantum physics. [sent-78, score-0.907]

38 The output layer does the same thing as the hidden layer(s), except that it also checks its accuracy against the target output of the network. [sent-83, score-0.241]

39 The network as a whole computes a function by checking which output bit is high. [sent-84, score-0.182]

40 The output layer works similarly, taking a weighted sum of the hidden nodes and checking against a threshold. [sent-92, score-0.224]

41 At the quantum gate level, the network will require O(blm + m2 ) gates for each node of the network. [sent-95, score-1.059]

42 Here b is the number of bits used for ﬂoating point arithmetic in |β , l is the number of bits for each weight and m is the number of inputs to the node [14]-[15]. [sent-96, score-0.307]

43 The overall network works as follows on a training set. [sent-97, score-0.169]

44 In our example, the network has two input parameters, so all n training examples will have two input registers. [sent-98, score-0.173]

45 Each hidden or output node has a weight vector, represented by |ψ i , each vector containing weights for each of its inputs. [sent-101, score-0.35]

46 After classifying a training example, the registers |ϕ 1 and |ϕ 2 reﬂect the networks ability to classify that the training example. [sent-102, score-0.269]

47 When all training examples have Figure 2: QNN Training been classiﬁed, |ρ will be the sum of the output nodes that have the correct answer throughout the training set and will range between zero and the number of training examples times the number of output nodes. [sent-104, score-0.417]

48 4 Using Quantum Search to Learn Network Weights One possibility for training this kind of a network is to search through the possible weight vectors for one which is consistent with the training data. [sent-105, score-0.464]

49 Quantum searches have been used already in quantum learning [16] and many of the problems associated with them have already been explored [17]. [sent-106, score-0.913]

50 We would like to ﬁnd a solution which classiﬁes all training examples correctly; in other words we would like |ρ = n ∗ m where n is the number of training examples and m is the number of output nodes. [sent-107, score-0.238]

51 Since we generally do not know how many weight vectors will do this, we use a generalization of the original search algorithm [18], intended for problems where the number of solutions t is unknown. [sent-108, score-0.241]

52 The basic idea is that we will put |ψ into a superposition of all possible weight vectors and search for one which classiﬁes all training examples correctly. [sent-109, score-0.446]

53 We start out with |ψ as a superposition of all possible weight vectors. [sent-110, score-0.234]

54 By using a superposition we classify the training examples with respect to every possible weight vector simultaneously. [sent-113, score-0.38]

55 Each weight vector is now entangled with |ρ in such a way that |ρ corresponds with how well every weight vector classiﬁes all the training data. [sent-114, score-0.413]

56 In this case, the oracle for the quantum search is |ρ = n ∗ m, which corresponds to searching for a weight vector which correctly classiﬁes the entire set. [sent-115, score-1.284]

57 Unfortunately, searching the weight vectors while entangled with |ρ would cause unwanted weight vectors to grow that would be entangled with the performance metric we are looking for. [sent-116, score-0.47]

58 The solution is to disentangle |ψ from the other registers after inverting the phase of those weights which match the search criteria, based on |ρ . [sent-117, score-0.18]

59 This means that the network will need to be recomputed each time we make an oracle call and after each measurement. [sent-119, score-0.177]

60 First, not all training data will have any solution networks that correctly classify all training instances. [sent-121, score-0.248]

61 This means that nothing will be marked by the search oracle, so every weight vector will have an equal chance of being measured. [sent-122, score-0.225]

62 Second, the amount of time needed to ﬁnd a vector which correctly classiﬁes the training set is O( 2b /t), which has exponential complexity with respect to the number of bits in the weight vector. [sent-124, score-0.306]

63 One way to deal with the ﬁrst problem is to search until we ﬁnd a solution which covers an acceptable percentage, p, of the training data. [sent-125, score-0.155]

64 In other words, the search oracle is modiﬁed to be |ρ ≥ n ∗ m ∗ p. [sent-126, score-0.181]

65 5 Piecewise Weight Learning Our quantum search algorithm gives us a good polynomial speed-up to the exponential task of ﬁnding a solution to the QNN. [sent-128, score-0.991]

66 This algorithm does not scale well, in fact it is exponential in the total number of weights in the network and the bits per weight. [sent-129, score-0.161]

67 Therefore, we propose a randomized training algorithm which searches each node’s weight vector independently. [sent-130, score-0.295]

68 The network starts off, once again, with training examples in |α , the corresponding answers in |Ω , and zeros in all the other registers. [sent-131, score-0.173]

69 A node is randomly selected and its weight vector, |ψ i , is put into superposition. [sent-132, score-0.197]

70 All other weight vectors start with random classical initial weights. [sent-133, score-0.234]

71 We then search for a weight vector for this node that causes the entire network to classify a certain percentage, p, of the training examples correctly. [sent-134, score-0.505]

72 That weight is ﬁxed classically and the process is repeated randomly for the other nodes. [sent-136, score-0.148]

73 Searching each node’s weight vector separately is, in effect, a random search through the weight space where we select weight vectors which give a good level of performance for each node. [sent-137, score-0.506]

74 Each node takes on weight vectors that tend to increase performance with some amount of randomness that helps keep it out of local minima. [sent-138, score-0.253]

75 First, to insure that hidden nodes ﬁnd weight vectors that compute something useful, a small performance penalty is added to weight vectors which cause a hidden node to output the same value for all training examples. [sent-141, score-0.63]

76 This helps select weight vectors which contain useful information for the output nodes. [sent-142, score-0.229]

77 Since each output node’s performance is independent of the performance or all output nodes, the algorithm only considers the accuracy of the output node being trained when training an output node. [sent-143, score-0.346]

78 Each of the hidden and the output nodes are thresholded nodes with three weights, one for each input and one for the threshold. [sent-145, score-0.166]

79 After an average of 58 weight updates, the algorithm was able to correctly classify the training data. [sent-150, score-0.263]

80 Since this is a randomized algorithm both in the number of iterations of the search algorithm before measuring and in the order which nodes update their weight vectors, the standard deviation for this method was much higher, but still reasonable. [sent-151, score-0.298]

81 In the randomized search algorithm, an epoch refers to ﬁnding and ﬁxing the weight of a single node. [sent-152, score-0.26]

82 We also tried the randomized search algorithm for a few real-world machine learning problems: lenses, Hayes-Roth and the iris datasets [19]. [sent-153, score-0.176]

83 The lenses data set is a data set that tries to predict whether people will need soft contact lenses, hard contact lenses or no contacts. [sent-154, score-0.214]

84 98% Backprop 96% 92% 83% Table 1: Training Results The lenses data set can be solved with a network that has three hidden nodes. [sent-163, score-0.204]

85 The randomized search algorithm found the correct target for 97. [sent-168, score-0.175]

86 We used four hidden nodes with two bit weights for the hidden nodes. [sent-171, score-0.169]

87 We had to normalize the inputs to range from zero to one once again so the larger inputs would not dominate the weight vectors. [sent-172, score-0.159]

88 86% of the training examples correctly since we are checking each output node for accuracy on each training example. [sent-176, score-0.352]

89 7 Conclusions and Future Work This paper proposes a simple quantum neural network and a method of training it which works well in quantum systems. [sent-179, score-1.961]

90 By using a quantum search we are able to use a wellknown algorithm for quantum systems which has already been used for quantum learning. [sent-180, score-2.741]

91 The algorithm is able to search for solutions that cover an arbitrary percentage of the training set. [sent-181, score-0.172]

92 This algorithm is exponential in the number of qubits of each node’s weight vector instead of in the composite weight vector of the entire network. [sent-185, score-0.38]

93 There may be quantum methods which could be used to improve current gradient descent and other learning algorithms. [sent-189, score-0.935]

94 It may also be possible to combine some of these with a quantum search. [sent-190, score-0.886]

95 An example would be to use gradient descent to try and reﬁne a composite weight vector found by quantum search. [sent-191, score-1.094]

96 Conversely, a quantum search could start with the weight vector of a gradient descent search. [sent-192, score-1.16]

97 This would allow the search to start with an accurate weight vector and search locally for weight vectors which improve overall performance. [sent-193, score-0.466]

98 Other types of QNNs may be able to use a quantum search as well since the algorithm only requires a weight space which can be searched in superposition. [sent-195, score-1.092]

99 In addition, more traditional gradient descent techniques might beneﬁt from a quantum speed-up themselves. [sent-196, score-0.935]

100 Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. [sent-299, score-0.886]

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