Stochastic Quantum Walks and Artificial Neural Networks

Schuld et al. propose using quantum walks to construct a quantum ANN algorithm, specifi- cally with an eye to demonstrate associative memory capabilities. This is a sensible idea, as both discrete-time and continuous-time quantum walks are universal for quantum computation. In associative memories, a previously-seen complete input is retrieved upon presentation of an incomplete or noisy input.


The quantum walker position represents the pattern of the “active” neurons (the firing pattern). That is, on an n-dimensional hypercube, if the walker is in a specific corner labelled with an n-bit string, then this string will have n corresponding neurons, each of which is “active” if the corresponding bit is 1. In a Hopfield network for a given input state x, the outputs are the minima of the energy function

E (x1,….,xn) = -1/2 Σni=1 Σnj=1 wijxixj + Σni=1 θixi

where xi is the state of the i-th neuron, wij is the strength of the inter-neuron link and θi is the activation threshold. Their idea is to construct a quantum walker such that one of these minima (dynamic attractor state) is the desired final state with high probability.

The paper examines two different approaches. First is the naïve case, where activation of a Hopfield network neuron is done using a biased coin. However they prove that this cannot work as the required neuron updating process is not unitary. Instead, a non-linearity is introduced through stochastic quantum walks (SQW) on a hypercube. To inject attractors in the walker’s hypercube graph, they remove all edges leading to/from the corners which represent them. This means that the coherent part of the walk can’t reach/leave these states, thus they become sink states of the graph. The decoherent part, represented by jump operators, adds paths leading to the sinks. A few successful simulations were run, illustrating the possibility of building an associative memory using SQW, and showing that the walker ends up in the sink in a time dependent on the decoherent dynamics. This might be a result in the right direction, but it is not a definitive answer to the ANN problem since Schuld et al. only demonstrate some associative memory properties of the walk. Their suggestion for further work is to explore open quantum walks for training feed-forward ANNs.

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