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Solving systems of linear equations on a quantum computer

Systems of linear equations are used to model a wide array of problems in all fields of science and engineering. Recently, it has been shown that quantum computers could solve linear systems exponentially faster than classical computers, making for one of the most promising applications of quantum computation. Here, we demonstrate this quantum algorithm by implementing various instances on a photonic quantum computing architecture. Our implementation involves the application of two consecutive entangling gates on the same pair of polarisation-encoded qubits. We realize two separate controlled-NOT gates where the successful operation of the first gate is heralded by a measurement of two ancillary photons. Our work thus demonstrates the implementation of a quantum algorithm with high practical significance as well as an important technological advance which brings us closer to a comprehensive control of photonic quantum information.