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Recent results by Harrow et. al. and by Ta-Shma, suggest that quantum computers may have an exponential advantage in solving a wealth of linear algebraic problems, over classical algorithms. Building on the quantum intuition of these results, we step back into the classical domain, and explore its usefulness in designing classical algorithms. We achieve an algorithm for solving the major linear-algebraic problems in time $O(n^{omega+ u})$ for any $ u>0$, where $omega$ is the optimal matrix-product constant. Thus our algorithm is optimal w.r.t. matrix multiplication, and comparable to the state-of-the-art algorithm for these problems due to Demmel et. al. Being derived from quantum intuition, our proposed algorithm is completely disjoint from all previous classical algorithms, and builds on a combination of low-discrepancy sequences and perturbation analysis. As such, we hope it motivates further exploration of quantum techniques in this respect, hopefully leading to improvements in our understanding of space complexity and numerical stability of these problems.
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Quantum algorithms have been developed for efficiently solving linear algebra tasks. However, they generally require deep circuits and hence universal fault-tolerant quantum computers. In this work, we propose variational algorithms for linear algebra tasks that are compatible with noisy intermediate-scale quantum devices. We show that the solutions of linear systems of equations and matrix-vector multiplications can be translated as the ground states of the constructed Hamiltonians. Based on the variational quantum algorithms, we introduce Hamiltonian morphing together with an adaptive ansatz for efficiently finding the ground state, and show the solution verification. Our algorithms are especially suitable for linear algebra problems with sparse matrices, and have wide applications in machine learning and optimisation problems. The algorithm for matrix multiplications can be also used for Hamiltonian simulation and open system simulation. We evaluate the cost and effectiveness of our algorithm through numerical simulations for solving linear systems of equations. We implement the algorithm on the IBM quantum cloud device with a high solution fidelity of 99.95%.
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We use our Clifford algebra technique, that is nilpotents and projectors which are binomials of the Clifford algebra objects $gamma^a$ with the property ${gamma^a,gamma^b}_+ = 2 eta^{ab}$, for representing quantum gates and quantum algorithms needed in quantum computers in an elegant way. We identify $n$-qubits with spinor representations of the group SO(1,3) for a system of $n$ spinors. Representations are expressed in terms of products of projectors and nilpotents. An algorithm for extracting a particular information out of a general superposition of $2^n$ qubit states is presented. It reproduces for a particular choice of the initial state the Grovers algorithm.
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