Measurement-efficient quantum Krylov subspace diagonalisation
Graduate School of China Academy of Engineering Physics, Beijing 100193, China
| Published: | 2024-08-13, volume 8, page 1438 |
| Eprint: | arXiv:2301.13353v3 |
| Doi: | https://doi.org/10.22331/q-2024-08-13-1438 |
| Citation: | Quantum 8, 1438 (2024). |
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Abstract
The Krylov subspace methods, being one category of the most important classical numerical methods for linear algebra problems, can be much more powerful when generalised to quantum computing. However, quantum Krylov subspace algorithms are prone to errors due to inevitable statistical fluctuations in quantum measurements. To address this problem, we develop a general theoretical framework to analyse the statistical error and measurement cost. Based on the framework, we propose a quantum algorithm to construct the Hamiltonian-power Krylov subspace that can minimise the measurement cost. In our algorithm, the product of power and Gaussian functions of the Hamiltonian is expressed as an integral of the real-time evolution, such that it can be evaluated on a quantum computer. We compare our algorithm with other established quantum Krylov subspace algorithms in solving two prominent examples. To achieve an error comparable to that of the classical Lanczos algorithm at the same subspace dimension, our algorithm typically requires orders of magnitude fewer measurements than others. Such an improvement can be attributed to the reduced cost of composing projectors onto the ground state. These results show that our algorithm is exceptionally robust to statistical fluctuations and promising for practical applications.
Featured image: Comparison of the measurement costs across various quantum Krylov subspace diagonalisation algorithms. Here, we take a 10-spin Heisenberg chain as an example, and set the subspace dimension to 5. $\epsilon_K$ indicates the error of the classical Lanczos algorithm. $\gamma$ represents a measurement cost factor that depends on the basis of Krylov subspace. It can be observed that, to achieve the precision of the classical counterpart, the Gaussian-power (GP) basis we designed requires significantly fewer measurements than others.
Popular summary
Quantum Krylov subspace diagonalisation (QKSD) aims to achieve quantum advantage by transferring the measurement of subspace matrices to quantum computers. In contrast, quantum phase estimation requires substantial quantum computing resources, while the variational quantum eigensolver is limited by the ansatz and classical optimisation problems. For quantum computers, statistical errors due to finite measurement number are unavoidable. By random matrix theory, we analyse the statistical errors in measuring subspace matrices and propose a regularisation method to overcome the ill-conditioned problem. This ensures the energy estimation is variational and bounded with a controllable failure probability, unlike truncation strategies. We rigorously analyse the measurement cost of QKSD algorithms and find a factor that depends on the basis of Krylov subspace.
We design a quantum algorithm to construct the Gaussian-power basis, which is the product of Hamiltonian powers and Gaussian functions. It can be transformed into an integral over real-time evolution via Fourier transform and implemented on quantum computers using Monte Carlo sampling. Benchmarking strongly correlated systems shows that our algorithm typically requires significantly fewer measurements than others to match the precision of the classical Lanczos algorithm. By composing Chebyshev projectors, we prove the near-optimal measurement efficiency of our algorithm. Our theoretical framework serves as a universal platform for evaluating the measurement costs of QKSD algorithms, and the robustness of our algorithm to statistical fluctuations makes it less susceptible to noise in quantum circuits.
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