Quantum neural networks are computational
neural network models which are based on the principles of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
. The first ideas on quantum neural computation were published independently in 1995 by
Subhash Kak
Subhash Kak is an Indian-American computer scientist and historical revisionist. He is the Regents Professor of Computer Science Department at Oklahoma State University–Stillwater, an honorary visiting professor of engineering at Jawaharlal N ...
and Ron Chrisley, engaging with the theory of
quantum mind
The quantum mind or quantum consciousness is a group of hypotheses proposing that classical mechanics alone cannot explain consciousness, positing instead that quantum-mechanical phenomena, such as entanglement and superposition, may play an imp ...
, which posits that quantum effects play a role in cognitive function. However, typical research in quantum neural networks involves combining classical
artificial neural network
Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains.
An ANN is based on a collection of connected unit ...
models (which are widely used in machine learning for the important task of pattern recognition) with the advantages of
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
in order to develop more efficient algorithms. One important motivation for these investigations is the difficulty to train classical neural networks, especially in
big data applications. The hope is that features of
quantum computing such as
quantum parallelism or the effects of
interference
Interference is the act of interfering, invading, or poaching. Interference may also refer to:
Communications
* Interference (communication), anything which alters, modifies, or disrupts a message
* Adjacent-channel interference, caused by extr ...
and
entanglement can be used as resources. Since the technological implementation of a quantum computer is still in a premature stage, such quantum neural network models are mostly theoretical proposals that await their full implementation in physical experiments.
Most Quantum neural networks are developed as
feed-forward
Feedforward is the provision of context of what one wants to communicate prior to that communication. In purposeful activity, feedforward creates an expectation which the actor anticipates. When expected experience occurs, this provides confirmato ...
networks. Similar to their classical counterparts, this structure intakes input from one layer of qubits, and passes that input onto another layer of qubits. This layer of qubits evaluates this information and passes on the output to the next layer. Eventually the path leads to the final layer of qubits.
The layers do not have to be of the same width, meaning they don't have to have the same number of qubits as the layer before or after it. This structure is trained on which path to take similar to classical
artificial neural network
Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains.
An ANN is based on a collection of connected unit ...
s. This is discussed in a lower section. Quantum neural networks refer to three different categories: Quantum computer with classical data, classical computer with quantum data, and quantum computer with quantum data.
Examples
Quantum neural network research is still in its infancy, and a conglomeration of proposals and ideas of varying scope and mathematical rigor have been put forward. Most of them are based on the idea of replacing classical binary or
McCulloch-Pitts neurons with a
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
(which can be called a “quron”), resulting in neural units that can be in a
superposition of the state ‘firing’ and ‘resting’.
Quantum perceptrons
A lot of proposals attempt to find a quantum equivalent for the
perceptron
In machine learning, the perceptron (or McCulloch-Pitts neuron) is an algorithm for supervised learning of binary classifiers. A binary classifier is a function which can decide whether or not an input, represented by a vector of numbers, belon ...
unit from which neural nets are constructed. A problem is that nonlinear activation functions do not immediately correspond to the mathematical structure of quantum theory, since a quantum evolution is described by linear operations and leads to probabilistic observation. Ideas to imitate the perceptron activation function with a quantum mechanical formalism reach from special measurements to postulating non-linear quantum operators (a mathematical framework that is disputed). A direct implementation of the activation function using the
circuit-based model of quantum computation has recently been proposed by Schuld, Sinayskiy and Petruccione based on the
quantum phase estimation algorithm
In quantum computing, the quantum phase estimation algorithm (also referred to as quantum eigenvalue estimation algorithm), is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. More precisely, given ...
.
Quantum networks
At a larger scale, researchers have attempted to generalize neural networks to the quantum setting. One way of constructing a quantum neuron is to first generalise classical neurons and then generalising them further to make unitary gates. Interactions between neurons can be controlled quantumly, with
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
gates
Gates is the plural of gate, a point of entry to a space which is enclosed by walls. It may also refer to:
People
* Gates (surname), various people with the last name
* Gates Brown (1939-2013), American Major League Baseball player
* Gates McFadde ...
, or classically, via
measurement of the network states. This high-level theoretical technique can be applied broadly, by taking different types of networks and different implementations of quantum neurons, such as
photonically implemented neurons
and
quantum reservoir processor (quantum version of
reservoir computing
Reservoir computing is a framework for computation derived from recurrent neural network theory that maps input signals into higher dimensional computational spaces through the dynamics of a fixed, non-linear system called a reservoir. After the in ...
). Most learning algorithms follow the classical model of training an artificial neural network to learn the input-output function of a given
training set
In machine learning, a common task is the study and construction of algorithms that can learn from and make predictions on data. Such algorithms function by making data-driven predictions or decisions, through building a mathematical model from ...
and use classical feedback loops to update parameters of the quantum system until they converge to an optimal configuration. Learning as a parameter optimisation problem has also been approached by adiabatic models of quantum computing.
Quantum neural networks can be applied to algorithmic design: given
qubits
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
with tunable mutual interactions, one can attempt to learn interactions following the classical
backpropagation
In machine learning, backpropagation (backprop, BP) is a widely used algorithm for training feedforward artificial neural networks. Generalizations of backpropagation exist for other artificial neural networks (ANNs), and for functions gener ...
rule from a
training set
In machine learning, a common task is the study and construction of algorithms that can learn from and make predictions on data. Such algorithms function by making data-driven predictions or decisions, through building a mathematical model from ...
of desired input-output relations, taken to be the desired output algorithm's behavior. The quantum network thus ‘learns’ an algorithm.
Quantum associative memory
The quantum associative memory algorithm was introduced by Dan Ventura and Tony Martinez in 1999. The authors do not attempt to translate the structure of artificial neural network models into quantum theory, but propose an algorithm for a
circuit-based quantum computer that simulates
associative memory. The memory states (in
Hopfield neural networks saved in the weights of the neural connections) are written into a superposition, and a
Grover-like quantum search algorithm retrieves the memory state closest to a given input. An advantage lies in the exponential storage capacity of memory states, however the question remains whether the model has significance regarding the initial purpose of Hopfield models as a demonstration of how simplified artificial neural networks can simulate features of the brain.
Classical neural networks inspired by quantum theory
A substantial amount of interest has been given to a “quantum-inspired” model that uses ideas from quantum theory to implement a neural network based on
fuzzy logic.
Training
Quantum Neural Networks can be theoretically trained similarly to training classical/
artificial neural network
Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains.
An ANN is based on a collection of connected unit ...
s. A key difference lies in communication between the layers of a neural networks. For classical neural networks, at the end of a given operation, the current
perceptron
In machine learning, the perceptron (or McCulloch-Pitts neuron) is an algorithm for supervised learning of binary classifiers. A binary classifier is a function which can decide whether or not an input, represented by a vector of numbers, belon ...
copies its output to the next layer of perceptron(s) in the network. However, in a quantum neural network, where each perceptron is a qubit, this would violate the
no-cloning theorem In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theore ...
.
A proposed generalized solution to this is to replace the classical
fan-out
In digital electronics, the fan-out is the number of gate inputs driven by the output of another single logic gate.
In most designs, logic gates are connected to form more complex circuits. While no logic gate input can be fed by more than one ...
method with an arbitrary
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
that spreads out, but does not copy, the output of one qubit to the next layer of qubits. Using this fan-out Unitary (
) with a dummy state qubit in a known state (Ex.
in the
computational basis), also known as an
Ancilla bit
Ancilla bits are some extra bits being used to achieve some specific goals in computation (e.g. reversible computation). In classical computation, any memory bit can be turned on or off at will, requiring no prior knowledge or extra complexity. ...
, the information from the qubit can be transferred to the next layer of qubits.
This process adheres to the quantum operation requirement of
reversibility.
Using this quantum feed-forward network, deep neural networks can be executed and trained efficiently. A deep neural network is essentially a network with many hidden-layers, as seen in the sample model neural network above. Since the Quantum neural network being discussed utilizes fan-out Unitary operators, and each operator only acts on its respective input, only two layers are used at any given time.
In other words, no Unitary operator is acting on the entire network at any given time, meaning the number of qubits required for a given step depends on the number of inputs in a given layer. Since Quantum Computers are notorious for their ability to run multiple iterations in a short period of time, the efficiency of a quantum neural network is solely dependent on the number of qubits in any given layer, and not on the depth of the network.
Cost functions
To determine the effectiveness of a neural network, a cost function is used, which essentially measures the proximity of the network’s output to the expected or desired output. In a Classical Neural Network, the weights (
) and biases (
) at each step determine the outcome of the cost function
.
When training a Classical Neural network, the weights and biases are adjusted after each iteration, and given equation 1 below, where
is the desired output and
is the actual output, the cost function is optimized when
= 0. For a quantum neural network, the cost function is determined by measuring the fidelity of the outcome state (
) with the desired outcome state (
), seen in Equation 2 below. In this case, the Unitary operators are adjusted after each iteration, and the cost function is optimized when C = 1.
Equation 1
Equation 2
See also
*
Differentiable programming
Differentiable programming is a programming paradigm in which a numeric computer program can be differentiated throughout via automatic differentiation.
This allows for gradient-based optimization of parameters in the program, often via grad ...
*
Optical neural network
An optical neural network is a physical implementation of an artificial neural network with photonics, optical components. Early optical neural networks used a photorefractive Volume hologram to interconnect arrays of input neurons to arrays of o ...
*
Holographic associative memory
*
Quantum cognition
*
Quantum machine learning
Quantum machine learning is the integration of quantum algorithms within machine learning programs. The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a quantum computer, i.e. qu ...
References
External links
Recent review of quantum neural networks by M. Schuld, I. Sinayskiy and F. PetruccioneArticle by P. Gralewicz on the plausibility of quantum computing in biological neural networks
{{emerging technologies, quantum=yes, other=yes
Artificial neural networks
Neural circuits
Quantum information science
Quantum programming