(250i) Mitigating the Impact of Nondeterminism Due to Process Control Implemented on a Quantum Computer

Quantum computers that leverage quantum phenomena to perform tasks that may be performed by a classical computer are garnering increasing focus within a myriad of fields including finance, healthcare, communication, and technology by driving innovation across optimization and machine learning and may also find applications within the chemical and the biochemical industry [1-5]. Process control systems are comprised of classical computers computing the control actions to be implemented on a process. Quantum computers have nondeterminism and noise that can make the control actions be different from the control actions computed by a classical computer. Thus, to ensure safety of a process operated under control implemented on a quantum computer, approaches to handle the nondeterminism and noise must be developed. Realizing this, prior work in our group has analyzed a number of considerations and implications for control implemented on a quantum computer, particularly with respect to implementation of the control laws and consideration of safety [3-5]. In [4], an analysis was presented that explored how nondeterministic control actions from control implemented on a quantum computer impact the cybersecurity of the process (i.e., the randomness of the control actions due to noise was not capable alone of making an attack challenging for an attacker to carry out). In [5], the impact of nondeterminism due to control implemented on a quantum computer on the stability of an illustrative process example was analyzed using a numerical study. Theoretical considerations (by integrating a lookup table search for control actions from a Lyapunov-based economic model predictive control law [6] with a quantum algorithm based on Grover's algorithm [7]) were also presented that demonstrated that the control law implementation could maintain the closed-loop state within a set of safe states for a sampling period with the probability given by the probability of obtaining the desired result from Grover's algorithm. While these studies have laid a preliminary foundation for the consideration of control theory integrated with the theory of quantum computation, many questions remain with regard to how to handle nondeterminism and noise for a variety of different control algorithms implemented on quantum devices.

Motivated by this need for deeper theoretical understanding of the manner in which control laws and quantum computation interact, in this work, the stability of a process modeled by discrete-time linear time-invariant dynamics with nondeterminism from control implemented on a quantum computer is analyzed theoretically. In this analysis, we treat the nondeterminism originating from noise within a quantum circuit as bounded process disturbances and measurement noise which are inherent in process control implemented on a classical computer. This enables us to consider how control strategies which reject disturbances and noise may be utilized to make up for undesired inputs, and in particular to guarantee practical stability of the process under the control actions developed from the quantum computer. However, due to the presence of quantum noise, operation of a process with control implemented on a quantum computer may not be profitable. To mitigate the impact of quantum noise on the profitability of the process, a linear quadratic gaussian (LQG)-based control law [8] that guarantees optimality (in the sense of minimizing the expected value of a quadratic cost function) in the presence of bounded process disturbances and noise is considered. A fundamental assumption in the synthesis of LQG-based control law is that the nondeterminism in the process fits a Gaussian distribution. Though the noise in quantum devices may not obey this assumption, we utilize this assumption to provide initial steps toward a theoretical understanding of the impacts of noise on closed-loop performance when the noise fulfills certain assumptions. Therefore, implementing the LQG-based control law on a quantum circuit may cause the process to be operated under suboptimal conditions. To analyze the extent to which non-Gaussian characteristics of noise may impact optimality, the expected value of the quadratic cost function is evaluated by implementing LQG-based control law over two different sets of simulations of an illustrative process example: (1) considering control implemented on a noisy classical computer subject to noise modeled as variables drawn from a gaussian distribution, and (2) considering control implemented on a noisy quantum computer [9]. The expected value of the quadratic cost is compared across the two sets of simulations.

References:

[1] Andersson, M.P., Jones, M.N., Mikkelsen, K.V., You, F. and Mansouri, S.S., “Quantum computing for chemical and biomolecular product design”, Current Opinion in Chemical Engineering, volume 36, p.100754, 2022.

[2] Nourbakhsh A, Jones M.N., Kristjuhan K., Carberry D., Karon J., Beenfeldt C., Shahriari K., Andersson M. P., Jadidi M. A., Mansouri S.S., “Quantum computing: Fundamentals, trends and perspectives for chemical and biochemical engineers”, arXiv preprint arXiv:2201.02823, 2022.

[3] Nieman, K., Durand, H., Patel, S., Koch, D. and Alsing, P.M., “Investigating an amplitude amplification-based optimization algorithm for model predictive control”, Digital Chemical Engineering, volume 10, p.100134, 2024.

[4] Rangan, K. K., Abou Halloun, J., Oyama, H., Cherney, S., Assoumani, I. A., Jairazbhoy, N., Durand, H, and Ng, S.K., “Quantum computing and resilient design perspectives for cybersecurity of feedback systems”, Proceedings of the 13th IFAC Symposium on Dynamics and Control of Process Systems, including Biosystems, volume 55(7), pp. 703-708, Busan, Republic of Korea, 14–17 June 2022.

[5] Nieman, K., Rangan, K. K., and Durand, H., “Control Implemented on Quantum Computers: Effects of Noise, Non-Determinism, and Entanglement”, Industrial & Engineering Chemistry Research, volume 61(28), pp. 10133-10155, 2022.

[6] Heidarinejad, Mohsen, Jinfeng Liu, and Panagiotis D. Christofides. "Economic model predictive control of nonlinear process systems using Lyapunov techniques." AIChE Journal 58, no. 3 (2012): 855-870.

[7] Grover, Lov K. "A fast quantum mechanical algorithm for database search." In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pp. 212-219. 1996.

[8] Datta B., “Numerical methods for linear control systems”, Academic Press, 2004.

[9] Norlén H., “Quantum Computing in Practice with Qiskit® and IBM Quantum Experience®: Practical recipes for quantum computer coding at the gate and algorithm level with Python”, Packt Publishing Ltd, 2020.