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Journal of Computational and Applied Mathematics

A sequential quadratic Hamiltonian scheme for solving non-smooth quantum control problems with sparsity

Tim Breitenbach, Alfio Borzı̀

Published Nov 5, 2019• 16 citations
Publisher
DOI
10.1016/j.cam.2019.112583

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We present a quantum algorithm for approximating the real time evolution e−iHt of an arbitrary d-sparse Hamiltonian to error є, given black-box access to the positions and b-bit values of its non-zero matrix entries. The query complexity of our algorithm is O((t√d||H||1 → 2)1+o(1)/єo(1)) with respect to the largest Euclidean row norm ||H||1 → 2, which is shown to be optimal up to subpolynomial factors through a matching lower bound, and it uses a factor O(b) more gates. This provides a polynomial speedup in sparsity for the common case where the spectral norm is known, and generalizes previous approaches which achieve optimal scaling, but with respect to more restrictive parameters. By exploiting knowledge of the spectral norm, our algorithm solves the black-box unitary implementation problem – O(d1/2+o(1)) queries suffice to approximate any d-sparse unitary in the black-box setting, which matches the quantum search lower bound of Ω(√d) queries and improves upon prior art [Berry and Childs, QIP 2010] of Õ(d2/3) queries. Combined with known techniques, we also solve systems of sparse linear equations with condition number κ using O((κ √d)1+o(1)/єo(1)) queries, which is a quadratic improvement in sparsity.
This work proposes an end-to-end framework for the learning-enabled control of closed quantum systems. The proposed learning technique is the first of its kind to utilize a hierarchical design which layers probing control, quantum state tomography, quantum process tomography, and Hamiltonian learning to identify both the internal and control Hamiltonians. Within this context, a novel quantum process tomography algorithm is presented which involves optimization on the unitary group, i.e., the space of unitary operators, to ensure physically meaningful predictions. Our scalable Hamiltonian learning algorithms have low memory requirements and tunable computational complexity. Once the Hamiltonians are learned, we formalize data-driven model-predictive quantum control (MPQC). This technique utilizes the learned model to compute quantum control parameters in a closed-loop simulation. Then, the optimized control input is given to a physical quantum system in an open-loop fashion. Simulations show model-predictive quantum control to be more efficient than the current state-of-the-art, quantum optimal control, when sequential quadratic programming (SQP) is used to solve each control problem.