https://arxiv.org/abs/2105.11603
The inductive Grover oracular quantum neural network (IGO-QNN) is a novel paradigm of integration of Grover's unstructured search in a machine learning framework. The model defines a variational quantum circuit with hidden layers of parameterized quantum neurons densely connected via entangle synapses to encode a dynamic Grover's search oracle that can be trained from a set of database-hit training examples. This widens the range of problem applications of Grover's unstructured search algorithm to include the vast majority of problems lacking analytic descriptions of solution verifiers, allowing for quadratic speed-up in unstructured search for the set of search problems with relationships between input and output spaces that are tractably underivable deductively. This generalization of Grover's oracularization may prove particularly effective in deep reinforcement learning, computer vision, and, more generally, as a feature vector classifier at the top of an existing model.
The inductive Grover's oracular quantum neural network (IGO-QNN) model we propose can be abstracted as a directed quantum graph of quantum neural nodes and entangle synaptic edges. Quantum neurons are encoded as a
The neural network consists of input, output, and hidden quantum layers, an oracle channel register, and a classical measure register.
The input layer is a variational quantum register of
The
The neural network is functionally encoded as a variational quantum circuit implementing a parameterized representation of Grover's search. The quantum neurons in each hidden layer and the oracle qubit are put in superposition via Hadamard transforms. Then the neural weights are initialized and applied to each neuron channel. Deeper hidden layer neurons first apply their respective activation functions and then are sequentially densely connected with each other via entangle synapses, and the final hidden layer is densely entangled with the output layer to create the oracle which is then applied to the input database. The output state of the neurally-encoded checker is then mapped to the oracle qubit (oracle flagging). Uncomputation using the inverse gate of the oracularizer is applied to the input and output layers, and the diffuser is applied for hit-state amplitude amplification. This sequence of oracularization, oracle flagging, uncomputation, and diffusion operations are iteratively applied
The inductive Grover's oracular neural network is trained via a classical optimizer. At each training iteration, the variational (parameterized) circuit is provided input arguments (
The current paradigm for oracularization is the deductive representation of a decision problem (determining whether a proposed solution to a given problem is indeed a valid solution). As a consequence, an explicit solution to a given decision problem i.e. a solution-verification algorithm must be known in order to Grover's query a database for solutions to the associated search problem. This is the primary limitation of this deductive Grover's paradigm that narrows the range of applications to solely problems that contain closed form criteria for verification. However, of the set of all search problems, it is probability 1 that a given problem does not have a known deductive solution verifier. By means of the described quantum neural network architecture (IGO-QNN), we have described an inductive paradigm for the encoding of Grover's unstructured search oracles that allows for fast problem-space solution flagging without a closed form analytical framework for verification. This widens the range of problem applications to include that vast majority without deductively-described oracles allowing for quadratic speed-up in unstructured search for problems with complex relationships between input and output spaces. The only prerequisite for the application of an IGO-QNN is that sufficient database-hit training examples constitute the training set for the classical optimizer to effectively minimize oracularization loss.
This generalization of Grover's oracularization may prove particularly effective in deep reinforcement learning, computer vision, and, more generally, as a feature vector classifier at the top of an existing model. In the subdomain of reinforcement learning, a deep IGO-QNN can be given a database of potential future actions and the current state of the agent and flag the optimal decision. In the context of computer vision, an IGO-CNN can be given a database of images (likely semantically encoded into feature vectors) and be trained on matching image classification search hits to learn an oracle to flag images of a given classification. Any generalized set of feature vectors can be set as the input space of the network and the model can be trained to mark feature vectors that satisfy some condition for a general classification task. The network’s non-restrictive input/output space definition allows for it to be modularized and integrated as a layer in a larger machine learning model such as a hybrid quantum-classical context, allowing for the development of more robust models.
To create a new instance of a model,
from igo_qnn import InductiveGroversQNN
model = InductiveGroversQNN(2, [{'size': 2, 'type': 'pairwise-full'}, {'size': 2, 'type': 'pairwise-full'}])
To cast the model to an instance of qiskit_machine_learning.neural_networks.CircuitQNN
,
qnn = model.toCircuitQNN()