Variational algorithms for linear algebra (1909.03898v3)

Abstract: 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%.

• The paper introduces variational quantum algorithms that frame linear algebra problems as ground state searches, enabling solutions on noisy intermediate-scale quantum (NISQ) devices.
• Numerical simulations and tests on IBM hardware show these algorithms have polynomial scaling and achieve high solution fidelities, demonstrating their practical viability.
• These algorithms expand the potential for near-term quantum applications in areas like optimization and machine learning by operating within current hardware constraints without requiring error correction.

Variational Algorithms for Linear Algebra on NISQ Devices

This paper presents variational quantum algorithms aimed at solving linear algebra problems like linear systems of equations and matrix-vector multiplications using noisy intermediate-scale quantum (NISQ) devices. The proposed algorithms circumvent the need for fault-tolerant quantum computers, which require deep circuits that are not currently feasible given present technological limitations.

Quantum computing holds the potential to provide significant advantages in solving certain linear algebra tasks. Notably, the HHL algorithm provides an exponential speed-up over classical algorithms for solving linear systems with complexity that grows polynomially with the logarithm of the matrix size and its condition number. However, algorithms such as HHL necessitate quantum error correction and deep quantum circuits. The current work addresses these constraints by leveraging variational quantum algorithms that can operate within the constraints of NISQ technology.

Methodology

The core idea of the approach is to frame the solutions to linear algebra problems as ground states of meticulously constructed Hamiltonians. The algorithms employ variational methods like the variational quantum eigensolver (VQE), imaginary time evolution (ITE), and adaptive variational techniques to locate these ground states. Specifically, the ground state of a Hamiltonian representing a linear system or matrix-vector multiplication task yields the desired linear algebraic solution. The algorithms are particularly effective for sparse matrices, which is advantageous given the challenges of large-scale quantum simulations.

Cost and Fidelity Assessment

The practical viability of these algorithms is validated via numerical simulations that demonstrate polynomial scaling of circuit depth and computation time with matrix size and condition number. Importantly, performance tests on IBM's quantum cloud yielded solution fidelities as high as 99.95%, underscoring the algorithms' effectiveness and compatibility with current quantum hardware.

Implications and Future Directions

From a theoretical standpoint, these variational algorithms potentially expand the scope of quantum computing applications, particularly in domains requiring linear algebra operations such as optimization problems and machine learning. Practically, their ability to operate on NISQ devices without error correction augurs well for near-term applications of quantum computing in solving commercially relevant problems.

Looking ahead, further development could involve refining ansätze to enhance algorithm performance or applying the algorithmic framework to other linear algebra problems like singular value decomposition. Additionally, future research may explore more compact circuit designs and sophisticated error mitigation techniques to improve efficiency and scalability. The application of these methods to simulate real and imaginary time dynamics in quantum systems could also prove beneficial, offering alternative approaches to quantum simulation challenges.

Conclusion

The paper introduces a promising framework for solving linear algebra problems using quantum computing, tailored specifically for the constraints and capabilities of NISQ hardware. As quantum technology progresses, these algorithms could become crucial computational tools across various scientific and industrial applications.