(374p) Enhancing Quantum Algorithms through Integration with Model Predictive Control Concepts

Under ideal conditions, quantum algorithms may allow for some classes of complex computational problems potentially intractable on even the most powerful classical computers to be solved using quantum computational resources. However, today's quantum computers are far from ideal, offering a limited number of qubits which are quickly overwhelmed by quantum noise after a small number of operations, limiting the depth of quantum circuits in which these operations (known as "gates") are implemented. A class of quantum algorithms known as variational quantum algorithms (VQAs) [6],[7] have been developed to function within the limitations of today's quantum computers. VQAs use classical optimization to develop parameters for quantum circuits aimed at achieving optimization tasks including solving discrete optimization problems. The goals of VQAs are conceptually similar to the goals of quantum control [8].

Building on this quantum control connection, the Feedback-based ALgorithm for Quantum OptimizatioN (FALQON) [9] borrows the concept of quantum Lyapunov control [10] to implement a feedback algorithm capable of monotonically decreasing an objective function with respect to circuit depth in the ideal case. When implemented on real quantum hardware, however, errors propagate after several gates, and the algorithm is unable to obtain high-quality solutions. In particular, FALQON's Lyapunov-based approach works by iteratively appending gates to a quantum circuit to decrease the expectation value of a “problem” Hamiltonian, which acts akin to a control Lyapunov function. Compared with VQAs, FALQON circumvents the need for optimization over circuit parameters by instead feeding back measurement information to determine parameter values. At the beginning of each iteration, qubits are initialized in a desired state, and gates corresponding to evolutions under the problem and driver Hamiltonians (i.e., where the latter is a controlled dynamic term used to drive qubits towards a minimum of the control Lyapunov function), respectively, are appended to the quantum circuit. Then, an expectation value associated with the derivative of the expectation value of the problem Hamiltonian is estimated and used to modify the driver Hamiltonian such that the expectation value of the problem Hamiltonian decreases monotonically with circuit depth. Iterations of the algorithm are performed to develop a quantum circuit intended to drive the system towards a small neighborhood of the minimum. It is worth emphasizing that although the expectation value of the problem Hamiltonian decreases monotonically, FALQON may not develop quantum circuits which optimally decrease the problem Hamiltonian.

In this work, we investigate whether an approach based on model predictive control (MPC) [11] for determining circuit parameters may offer advantages in regards to optimal circuit design. Moving towards an MPC-based strategy, however, offers new challenges. Importantly, model-based control designs employ optimization algorithms requiring the solution of a dynamic process model, which can be expensive to integrate as the dimension of the quantum system (i.e., number of qubits) grows. Additionally, obtaining an initial quantum state estimate to seed the algorithm can be costly (measurements probabilistically collapse the state of the qubits, thus requiring many copies of the quantum state to be measured to provide this estimate), and design heuristics for selecting parameters such as prediction horizon length need to be developed.

In this talk, we present an MPC-based quantum algorithm for decreasing the expectation value of a problem Hamiltonian. A reduced-order modeling strategy is developed and paired with the concept of classical shadows [12] to estimate initial quantum states used in integrating the dynamic model using less computational resources than current strategies (such as quantum state tomography). We discuss the benefits and limitations of the MPC-based approach with FALQON's Lyapunov-based approach, evaluate the reduced-order modeling effort in terms of accuracy and efficiency, and investigate how the selection of design parameters such as prediction horizon length impact the algorithm. This work demonstrates how process systems engineering concepts including optimization and control can be used to develop new quantum algorithms.

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Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. SAND No. SAND2024-04226A.