Quantum Algorithm for Linear Differential Equations with Exponentially Improved Dependence on Precision
We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.
Unable to display preview. Download preview PDF.
- 2.Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proceedings of the 35th ACM Symposium on Theory of Computing, pp. 20–29 (2003). arXiv:quant-ph/0301023
- 5.Berry, D.W., Childs, A.M., Cleve, R., Kothari, R., Somma, R.D.: Exponential improvement in precision for simulating sparse Hamiltonians. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp. 283–292 (2014). arXiv:1312.1414
- 7.Berry, D.W., Childs, A.M., Kothari, R.: Hamiltonian simulation with nearly optimal dependence on all parameters. In: Proceedings of the 56th Annual Symposium on Foundations of Computer Science, pp. 792–809 (2015). arXiv:1501.01715
- 10.Childs, A.M.: Quantum information processing in continuous time. Ph.D. thesis, Massachusetts Institute of Technology (2004)Google Scholar
- 11.Childs, A.M., Kothari, R., Somma, R.D.: Quantum linear systems algorithm with exponentially improved dependence on precision (2015). arXiv:1511.02306
- 14.Low, G.H., Chuang, I.L.: Hamiltonian simulation by qubitization. arXiv:1610.06546
- 16.Novo, L., Berry, D.W.: Improved Hamiltonian simulation via a truncated Taylor series and corrections. arXiv:1611.10033