Data Science Part 26: Quantum Machine Learning

 This will be a very short article discussing the explorations of quantum machine learning. Google's sycamore 53-qubit processor takes about 200 seconds to sample a complex probability distribution while it would take summit, the world's second largest computer in the world 2.5 days. IMB announced a 433 qubit chip and we will have way more qubits in the near future. A qubit is a basic unit of quantum information regarding the basic split of an electron. In qubits, we can superimpose states simultaneously.

We can use Quantum Computers in finding new catalysts for nitrogen fixation, or extremely energy intensive processes, with equations where we cannot solve using classical computers, such as designing room temperature superconductors. We can solve age of universe problems in a matter of a few days. Classical computing relies on a bitstream while quantum relies on a Qubit or a complex unit vector. The gates for Classical are Boolean Gates while they are Unitary Matrices for Quantum Computers. We output Bitstream in classical computing whereas we output distribution over bitstreams for quantum computing. We have transistors for classical computing whereas there are many candidates for hardware for Quantum Computing.  The superposition for N Qubits are defined as follows:

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Here is a quantum circuit. 

Born's rule indicates that the probability of getting outcome |x> is proportional to the square magnitude of the coefficient. 

Grover's algorithm asks us to find an element W in a list of N elements. We increase the amplitude of getting W while decreasing the amplitude of the rest of the items of the circuit. We repeat (N)^(1/2) times and you will get W with a high probability. The items of the circuit are denoted as 

with the chart

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We want to incorporate quantum effects into data-driven models to speed up learning and understand more complex datasets. Running Quantum Machine Learning (QML) algorithms on classical data requires extra overhead. However, we might suffer from some Gradient Decay. 

A Unitary Quantum Neural Network is defined as 

Where

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The network looks as follows: 

and the qubits we measure at the end are visible units and the rest are called the hidden units. 
ρ(θ) is the density matrix and ρv is the reduced density. 

Let's talk about the Variational Quantum Eigen solver (VQE) now. Consider a Hermitian Matrix with a minimum eigenvalue λ0. We can show that for any arbitrary state ψ(θ) that  (ψ(θ)| H |ψ(θ)> ≥ λ0. We want to minimize θ to get close to λ0 (argmin ( ψ(θ) | H |  ψ(θ))).

We can find the upper bound of an eigenvalue using this quantum neural network: 

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Here is the comparison between the classical and the Quantum Circuit: 

. The probability that a gradient is nonzero is exponentially small as a function of a number of qubits. 

Barren Plateaus are related to how the loss function is evaluated, and arise from excess of entanglement between the visible and hidden units. Highly expressible models are hard to train using linear (Bounded) objective functions. To train Quantum Neural Networks, we train a loss function without bounds, an unbounded loss function. 

Thermal states, or ϕdata, are used in many applications denoted by equation 

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