Quantum Solutions for Training a Single Layer Binary Neural Network

Quantum computing is a relatively new field starting in the early 1980s when a physicist named Paul Benioff proposed a quantum mechanical model of the Turing machine, introducing quantum computers. Previously, the focus of most quantum computers was in the study of quantum applications instead of broad applications due to the fact that quantum technology is a newer field with many technology constraints, such as limited qubits and noisy environments. However, quantum computers are still capable of using quantum mechanics to solve specific algorithms with an exponential speed-up in comparison to their classical counterparts. One key algorithm is the HHL algorithm proposed by Harrow, Hassidim and Lloyd in 2009 [1]. This algorithm outlines a quantum approach to solve a linear systems of equations with a best case time complexity of O(poly(log N )), in comparison to the best case time complexity for classical algorithms of O(N^3 ). The HHL algorithm outlines a use for quantum circuits outside of quantum applications. One such application is in machine learning, as many networks use linear regression in their training algorithm. Currently it is not feasible to solve for weight vectors of floating point precision on a quantum computer, but if the weight vector is constrained to binary values 0 or 1 then the problem becomes small enough to implement even on current noisy quantum computers. This work outlines two different circuit designs to solve for 2 × 2 and 4 × 4 systems of equations, so long as the matrices follow the eigenvalue constraint of having eigenvalues be powers of 2. In addition, the problem of reading data from the quantum state to classical data is addressed through the use of a swap test between the solution state |x\u3e and an test state |test\u3e. By using a swap test vector of all 1s, it is shown it is possible to find how many ones lay in the solution vector; thus reducing the number of possible solution states without performing quantum tomography. While it is not possible to beat classical algorithms with the noise on current quantum circuits, this work shows it is possible to implement quantum algorithms for non-quantum applications establishing potential for future hybrid approaches