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“See the main text for an explanation of the terms. Regular languages domains*

Table 3 “See the main text for an explanation of the terms. Regular languages domains*

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A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255 Fig. 1. A diagram of a modern version Young’s double-slit experiment (see main text). The peaks on the detector screen when both slits are open (extreme right of diagram) represent the arrival of particles — each of which travels in the form of two waves (one from each slit) — either in phase or out of phase.
A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255 Fig. 1. A diagram of a modern version Young’s double-slit experiment (see main text). The peaks on the detector screen when both slits are open (extreme right of diagram) represent the arrival of particles — each of which travels in the form of two waves (one from each slit) — either in phase or out of phase.
A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255 Fig. 2. The pattern of arrival of individual photons at the detector screen with either one or both slits open can be explained in quantum mechanical terms through an appeal to ‘wave-—particle’ duality: the particle travels as a wave between the gun and detector screen and passes through one or both slits as a wave, but collapses to a particle (point) when the detection screen is encountered. The pattern of arrival now reflects whether the particle arrived at the screen at the peak of its wave (in phase) or at its trough (out of phase). Additionally, the phase of the waves can be shifted if appropriate ‘phase-shifters’ are introduced between slits and screen.
A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255 Fig. 2. The pattern of arrival of individual photons at the detector screen with either one or both slits open can be explained in quantum mechanical terms through an appeal to ‘wave-—particle’ duality: the particle travels as a wave between the gun and detector screen and passes through one or both slits as a wave, but collapses to a particle (point) when the detection screen is encountered. The pattern of arrival now reflects whether the particle arrived at the screen at the peak of its wave (in phase) or at its trough (out of phase). Additionally, the phase of the waves can be shifted if appropriate ‘phase-shifters’ are introduced between slits and screen.
A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255
A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255
Fig. 3. A quantum neural network can be modelled on the double slit experiment. The photon gun now becomes the input pattern, the slits the input neurons, the waves between the slits and the screen the connections between the input and output neurons, and the screen the output neurons. This differs from the Chrisley approach in that, here, the inputs and weights are kept distinct. Chrisley originally proposed that the slits could be weights themselves. However, in our approach, it is possible to amend the weights (through forms of phase-shifting) independently of modifying the slits. Such phase-shifting can be used to implement back-propagation, for instance (see Fig. 2).
Fig. 3. A quantum neural network can be modelled on the double slit experiment. The photon gun now becomes the input pattern, the slits the input neurons, the waves between the slits and the screen the connections between the input and output neurons, and the screen the output neurons. This differs from the Chrisley approach in that, here, the inputs and weights are kept distinct. Chrisley originally proposed that the slits could be weights themselves. However, in our approach, it is possible to amend the weights (through forms of phase-shifting) independently of modifying the slits. Such phase-shifting can be used to implement back-propagation, for instance (see Fig. 2).
Each type of network will be designated by [x1 x2 x3 x4] So, for instance, [qcoqe] = quantum input to hidden unit weights with classical hidden to output unit weights with quantum hidden unit activations with classical output unit activations
Each type of network will be designated by [x1 x2 x3 x4] So, for instance, [qcoqe] = quantum input to hidden unit weights with classical hidden to output unit weights with quantum hidden unit activations with classical output unit activations
Fig. 5. The 16 different types of network for a classical 2:2:1 (two input units, two hidden units, one output unit) formed from using combinations of (c)lassical and (q)uantum architectures for the input-to-hidden, hidden-to-output, hidden units and output units sections. The network types represented are for two training patterns of two bits each, hence two input neurons. The classical architecture (top left) consists of two input neurons, two hidden units and one output neuron. Quantum components involving connections are subsequently depicted by multiple connections between layers, and quantum components involving units by multiple nodes within a layer. A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255
Fig. 5. The 16 different types of network for a classical 2:2:1 (two input units, two hidden units, one output unit) formed from using combinations of (c)lassical and (q)uantum architectures for the input-to-hidden, hidden-to-output, hidden units and output units sections. The network types represented are for two training patterns of two bits each, hence two input neurons. The classical architecture (top left) consists of two input neurons, two hidden units and one output neuron. Quantum components involving connections are subsequently depicted by multiple connections between layers, and quantum components involving units by multiple nodes within a layer. A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255
The animate—inanimate classification domain* * This contains the best QUANN results, given by QUANN-type ceqe, together with the results of statistical tests. A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255
The animate—inanimate classification domain* * This contains the best QUANN results, given by QUANN-type ceqe, together with the results of statistical tests. A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255
The mushroom classification domain® * This contains the best QUANN results, given by QUANN-type eeqe, together with the results of statistical tests.
The mushroom classification domain® * This contains the best QUANN results, given by QUANN-type eeqe, together with the results of statistical tests.
Context-free and type-zero languages domains A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255
Context-free and type-zero languages domains A. Narayanan, T. Menneer | Information Sciences 128 (2000) 231-255
esults of statistical tests for the formal language domains and the percentage level of significance* * The mean results from Tables 3 and 4 are also included. Comparisons were made between those variations in which the CLANN and the QUANN gave their highest generalisation performance. The first result is for the training time and the second is for the generalisation. The QUANN results are for cccq-type unless otherwise stated. The number of hidden units (HU) is given for those experiments in which a varying number of hidden units was used.
esults of statistical tests for the formal language domains and the percentage level of significance* * The mean results from Tables 3 and 4 are also included. Comparisons were made between those variations in which the CLANN and the QUANN gave their highest generalisation performance. The first result is for the training time and the second is for the generalisation. The QUANN results are for cccq-type unless otherwise stated. The number of hidden units (HU) is given for those experiments in which a varying number of hidden units was used.
A comparison of the QUANN and control results for the language domains* “The highest generalisation of the QUANN, across variations, is given as a percentage of the highest generalisation achieved by the control. The QUANN training time for these variations is also given as a percentage of the control training time. A code is given after each value to indicate whether the difference between the control and the QUANN result was significant (s) or not (n); ‘—’ indicates no test was carried out. All values are for QUANN-type cceq.
A comparison of the QUANN and control results for the language domains* “The highest generalisation of the QUANN, across variations, is given as a percentage of the highest generalisation achieved by the control. The QUANN training time for these variations is also given as a percentage of the control training time. A code is given after each value to indicate whether the difference between the control and the QUANN result was significant (s) or not (n); ‘—’ indicates no test was carried out. All values are for QUANN-type cceq.
Fig. 7. The two best QUANN types, as evidenced by the simulations. alise in such circumstances. Another implication for ANN research of QUANN research is that it has explored methods for quantum implementation of ANNs. It is essential that ools such as ANNs are able to exploit the immense increase in efficiency of hese quantum computers over those currently used. In order to do this it is necessary for ANNs to be implementable on quantum computers by using methods based on quantum theoretical concepts in their operation. There were wo types of QUANN that gave the best generalisation. These were ceqe for the microfeatural domains and cecq for the formal language domains (see Fig. 7). This shows that different QUANN-types lend themselves to different classes of problem. These experiments have provided a first indication as to which QUANN-type to use for a given type of problem. More interestingly and perhaps controversially, the results provided here indicate that it is not nec- essary to have a fully quantum network to achieve improvements over their classical counterparts. This has several advantages. First, we do not have to hrow away everything we have so far learned about neural networks. We can keep the best classical components and add on the quantum components as necessary. That means that we do not have to generate a whole new theory of QUANNs. Instead, we generate new theories of individual quantum compo- nents and graft these onto existing (classical) knowledge of how ANNs work. That does not mean that, ultimately, some new total theory of QUANNs may not be developed which subsumes classical interpretations and theories of CLANNSs. However, our research indicates that it is not necessary to have an understanding of this whole theory of QUANNSs before experimenting with QUANNs with only one quantum component. Secondly, given the widespread scepticism that quantum hardware will ever
Fig. 7. The two best QUANN types, as evidenced by the simulations. alise in such circumstances. Another implication for ANN research of QUANN research is that it has explored methods for quantum implementation of ANNs. It is essential that ools such as ANNs are able to exploit the immense increase in efficiency of hese quantum computers over those currently used. In order to do this it is necessary for ANNs to be implementable on quantum computers by using methods based on quantum theoretical concepts in their operation. There were wo types of QUANN that gave the best generalisation. These were ceqe for the microfeatural domains and cecq for the formal language domains (see Fig. 7). This shows that different QUANN-types lend themselves to different classes of problem. These experiments have provided a first indication as to which QUANN-type to use for a given type of problem. More interestingly and perhaps controversially, the results provided here indicate that it is not nec- essary to have a fully quantum network to achieve improvements over their classical counterparts. This has several advantages. First, we do not have to hrow away everything we have so far learned about neural networks. We can keep the best classical components and add on the quantum components as necessary. That means that we do not have to generate a whole new theory of QUANNs. Instead, we generate new theories of individual quantum compo- nents and graft these onto existing (classical) knowledge of how ANNs work. That does not mean that, ultimately, some new total theory of QUANNs may not be developed which subsumes classical interpretations and theories of CLANNSs. However, our research indicates that it is not necessary to have an understanding of this whole theory of QUANNSs before experimenting with QUANNs with only one quantum component. Secondly, given the widespread scepticism that quantum hardware will ever
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