Large deviation-based noise mitigation in coupled quantum robotic systems

Classical robotics relies on control strategies that track specific trajectories, ensuring that robots follow defined translational and angular paths to perform tasks. However, quantum systems introduce inherent complexities due to principles like the Heisenberg uncertainty principle, which makes exact trajectory tracking impossible. Instead, quantum systems require the tracking of desired wave functions, which correspond to the quantum moments of observables, representing the average values of physical quantities in a quantum state.

This shift from classical trajectory tracking to quantum wave function tracking carries significant implications. For example, in molecular chemistry, adjusting bond angles affects a molecule’s chemical properties, which are governed by the wave functions of its constituent particles [6]. Effective wave function tracking is essential for achieving desired outcomes in drug design and molecular engineering applications.

Beyond simply tracking average observables, such as the average position or momentum of a particle, quantum control must also manage the dispersion of observables within a quantum state. This dispersion, which can be thought of as the spread or uncertainty in the measurement of these observables, is particularly important in precision-sensitive applications like drug manufacturing, where controlling molecular dynamics is critical to success. Maintaining control over both the average state and its dispersion ensures system stability and reliability in environments prone to disturbances.

However, quantum systems are highly vulnerable to environmental noise, including Gaussian and Poisson noise, which can cause significant deviations from the intended quantum state. This susceptibility underscores the need for robust control techniques to mitigate noise and ensure reliable quantum robotic operations, even in the face of unpredictable disturbances.

The LDP offers a promising framework to tackle these challenges. By estimating the probabilities of rare events, LDP allows for precise calculations of the likelihood that a quantum system will deviate from its desired trajectory due to noise. This enables the development of control strategies to minimize such deviations, thereby improving the robustness and reliability of quantum robotic systems.

This research applies LDP-based control methods to address the quantum robot tracking problem and minimize its impact on system dynamics. While noise analysis in quantum systems often involves open quantum systems theory, our approach focuses on classical noise perturbations, making it well suited for analysis through stochastic Schrödinger equations. This work provides foundational insights for developing highly reliable quantum robotic systems with potential applications in molecular chemistry, quantum computing, and other fields.

The key contributions of this research are:These contributions provide a significant advancement in noise control for quantum robotics, offering both theoretical insights and practical solutions to improve the reliability and performance of quantum systems under stochastic noise conditions. This research is particularly significant in the field of quantum robotics, where the management of noise is a critical factor in the development of robust and efficient systems.

This subsection derives the Hamiltonian governing the dynamics of coupled quantum robots, starting from a classical Lagrangian and extending to a master–slave interaction model with stochastic control terms. The Hamiltonian of a quantum robot is given bywhere \(\tau (t)\) is the applied classical torque and M(q) is the robot’s mass moment of inertia matrix. Here, \(q \in \mathbb {R}^n\) represents the canonical position vector of the robot, encompassing both translational and rotational degrees of freedom. The canonical momentum operator vector is \(p = -i \nabla _q\). This Hamiltonian is derived by applying the Legendre transformation to the LagrangianWe consider a system of two robots operating in teleoperation with each other via a control interaction Hamiltonian, the coefficients of which are designed by minimizing an appropriate cost functional. The total Hamiltonian of the system iswhere \(\theta \) is the control parameter vector for the interaction Hamiltonian. The torque \(\tau (t)\) acts only on the master robot, while an environmental force acts on the slave robot performing the surgery. This environmental interaction Hamiltonian with the slave is represented by the time-varying term \(G(t, q_s, p_s)\), which plays a significant role in the system’s dynamics.

The environmental force G has a classical component of randomness, and similarly, the master torque \(\tau (t)\) incorporates a classical component of randomness due to tremors in the hand operator. This randomness is modelled as a superposition of a white Gaussian noise process and a white Poisson differential noise process.

This subsection formulates a control problem for quantum robots using the Schrödinger wave function and large deviation principles. The goal is to design control parameters \(\theta \) that minimize rare event deviations in the slave robot trajectory by leveraging the rate functional of the perturbed wave function.

Let \(\psi (t,q_m,q_s)\) denote the joint wave function of the master and slave robots in position space. Schrödinger’s equation governing its evolution iswhere \(H(t,q_m,q_s,p_m,p_s|\theta )\) is the total Hamiltonian of the system.

We assume prior knowledge of the surgical environment, specifically the trajectory \(q_d(t)\) that the slave robot is expected to follow. The control vector \(\theta \) must be chosen such that the average slave position tracks \(q_d(t)\), while minimizing the expected deviation

Here, the expectation \(\mathbb {E}\) reflects classical randomness in the wave field \(\psi \), stemming from noise in the torque \(\tau \) and environmental interaction \(G\). Assuming this randomness is small, we use large deviation principles (LDP) to quantify the probability of extreme deviations.

Letting \(\epsilon \) denote the small parameter governing the noise strength, we approximate the probability of a perturbation \(\delta \psi \) as \(\exp (-I(\delta \psi )/\epsilon )\), where \(I\) is the associated rate functional. The integration \(D\delta \psi \) in the subsequent expressions denotes a formal functional measure over the space of admissible wave function perturbations—i.e. a path integral over fluctuations \(\delta \psi \) at each spatial point \((q_m, q_s)\).

To compute \(I\), we first solve (4) for the deterministic wave function \(\psi _0(\cdot |\theta )\) at \(\epsilon = 0\). Linear perturbation theory yields a PDE for the fluctuation \(\delta \psi (\cdot |\theta )\), and the Dawson–Gärtner theorem [22, 23] is used to construct \(I(\delta \psi |\theta )\). The statistical moments become:The optimal control strategy minimizes the following LDP-informed objective functional:More generally, for an observable \(X(q,p)\), we define \(\langle X \rangle (t) = \int \psi ^* X \psi \, dq\) and compute the rate function \(I_T(X|\theta )\). The control \(\theta \) should maximize the probability that the error stays below a threshold \(\epsilon \), by solving:A full matrix–vector stochastic differential equation (SDE) form of (4), capturing both Gaussian and Poissonian noise, is derived in Appendix A.1.

The goal of this subsection is to derive an approximate large deviation rate function \( I(X|\theta ) \) associated with the stochastic quantum state vector \( X(t) \), governed by Itô–Poisson dynamics. This rate function is essential for evaluating the probability that observable deviations from desired quantum trajectories exceed a given threshold.

We use first-order perturbation theory to expand the solution \( X(t) = X_0(t) + \delta X(t) \), where \( \delta X(t) \) is the fluctuation caused by Gaussian and Poisson noise sources. The moment-generating functional of \( \delta X(t) \) is then used to obtain the Gartner–Ellis limiting log-moment-generating function \( \Lambda _T(f|\theta ) \), from which the rate function is derived by a Legendre–Fenchel transform.

The resulting rate function has the form:which leads to the equation for the maximizing \( f \):This implicit equation is solved perturbatively by expanding \( f = f_0 + \delta f_1 + \delta ^2 f_2 + \dots \), yielding an approximation to the rate function \( I(X|\theta ) \) up to \( O(\delta ^2) \).

The full derivation of this result, including the discretization, perturbation expansion, and Gaussian–Poisson decomposition of the log-moment-generating functional, is provided in Appendix A.2.

Our objective in this subsection is to derive an approximate rate function for the wave function under noisy and controlled quantum evolution. To this end, we apply a combination of time-independent and time-dependent perturbation theory to model the system’s evolution in the continuous position domain. The process is outlined as follows.

The Hamiltonian is given bywhere \(V(t|\theta )\) includes both the random and control parameter terms and \(\delta \) is assumed to be a small parameter. The unperturbed time-independent Hamiltonian is:whereandHere, \(M_0\) and \(N_0\) are the parts of the mass moment of inertia matrices that are not functions of the robot’s angular or linear coordinates. This perturbative expansion leverages the matrix inversion lemma—commonly referred to as the Woodbury identity—to efficiently compute updates to the inverse inertia matrix, assuming the linear and angular displacements remain small [24]. Then,represents a multivariate free particle Hamiltonian, which has a continuous spectrum. The evolution operator \(U_{00}(t) = \exp (-it H_{00})\) can be computed easily in position space.

Time-dependent perturbation theory, using the Dyson series, gives the correction to the evolution up to \(O(\epsilon )\) as:Another application of the Dyson series expansion (15) provides the unitary evolution up to \(O(\delta )\), accounting for both the random and control terms:The formula (16) can be used to calculate the approximate rate function of the wave function. Specifically, if \(\psi _0(q_m, q_s)\) is the initial wave function, then the approximate wave function at time t is given by:whererepresents the deviation in the wave function due to the random and control terms. This formula will later be used to estimate the deviation probability of wavefunction observables under noisy dynamics.

We now design the control parameters to minimize the observable deviations induced by noise, using the Large Deviations Principle (LDP).

Let X be any observable on the system space. For instance, X may be specified by its position representation Hermitian kernel:The average value of X in the evolving state is given, up to \(O(\delta )\), by:where \(\langle X \rangle _0(t) = \langle \psi _{0t} | X | \psi _{0t} \rangle \) represents the average of X in the absence of random terms (i.e. noise) and, without any control terms, tracks the desired classical trajectory \(X_d(t)\). The question then is how to design the control parameters \(\theta \) so that the average trajectory \(\langle X \rangle (t)\) deviates by only a tiny amount from the desired trajectory \(\langle X \rangle _0(t)\) over the time interval [0, T]. This involves minimizing:We observe that:Assume that \(V(t|\theta )\) is a Gaussian process that takes operator values in the Hilbert space \(L^2(\mathbb {R}^{2d})\), where both \(q_m\) and \(q_s\) are \(\mathbb {R}^d\)-valued. We can write:where \(M(t|\theta ) = \mathbb {E}(V(t|\theta ))\) is the classical statistical mean of \(V(t|\theta )\) and \(V_k(t|\theta )\) are non-random Hermitian operator-valued processes acting in the Hilbert space \(L^2(\mathbb {R}^{2d})\). The \(y_k(t)\) terms are zero-mean Gaussian random processes with:Thus, we can express \(\delta X(t)\) as:where

For \(\delta \rightarrow 0\), the scaled logarithmic moment-generating functional of the process \(\delta \sum _k y_k(t) v_k(t|\theta ), t \in [0, T]\) is given by:It follows that the rate function of \(\delta X(t), t \in [0, T]\) is given by:where \(Q_v(t, s|\theta )\) is the inverse kernel of \(R_v(t, s|\theta ) = \sum _{k,m} v_k(t|\theta ) v_m(s|\theta )\)\( R_{km}(t, s|\theta )\).

The large deviation principle (LDP) then gives:whereThe control design can thus be specified by maximizing the quantity:with respect to \(\theta \). Alternatively, if we define:and aim to minimize \(P\left( \int _0^T |\delta X(t)|^2 dt \ge \epsilon \right) \), then we would maximize:with respect to \(\theta \). This would amount to maximizing the minimum eigenvalue of the kernel \(Q_v(.,.|\theta )\) or, equivalently, minimizing the maximum eigenvalue of \(R(.,.|\theta )\). The maximum eigenvalue of the kernel R can be calculated approximately using the Rayleigh variational principle.

For computational purposes, we now work directly in the wave function domain to derive a large deviation-based algorithm for fine-tuning the parameters \(\theta _1, \theta _2\) such that the effects of noise are minimized. Additionally, we derive a maximum likelihood algorithm for estimating these parameters from measurements of wave function deviations, which can be obtained from measurements of the average values of a class of observables.

We begin with the Schrödinger equation for the desired wave function \(\psi _d\) and the noisy wave function \(\psi \), incorporating control terms. The desired wave function satisfies:whose solution iswhere \(U_0(t) = \exp (-it H_0)\) is the unperturbed evolution operator. The noisy wave function satisfies the stochastic differential equation (SDE):where \(H_0, H_1, H_2, V\) are \(2 \times 2\) real symmetric matrices and \(P = V^2/2\) is the Ito correction term. The control parameters \(\theta _1\) and \(\theta _2\) are introduced to mitigate the effects of noise. Here, B(t) denotes standard Brownian motion, and \(dB(t)/dt = W(t)\) represents a white noise process. The perturbed wave function dynamics are then given by:whereis the wave function perturbation. We assume that the initial perturbation is zero, i.e. \(\delta \psi (0) = 0\), which eliminates the homogeneous solution term from the mild formulation. The solution to this equation is:This assumption of zero initial perturbation is maintained in the discrete-time formulation as well, allowing us to omit any homogeneous evolution term in \(\delta \psi [0]\). In the discrete time domain, using a time discretization step size \(\delta \), Eq. (50) is expressed as:where \(W[t], t=0, 1, 2, \dots \) is an iid N(0, 1) process. Collecting all K time samples, Eq. (51) is expressed in vector notation as:where \(f_1\) is a 2M vector with components \(-i \delta \sum _{k=0}^t U_0[t-k] H_1 \psi _d[k]\) and \(f_2\) is a 2M vector with components \(-i \delta \sum _{k=0}^t U_0[t-k] H_2 \psi _d[k]\). The matrix G is a \(2M \times 2M\) matrix with \(2 \times 2\) block components \(g(t,k) = U_0[t-k] V \psi _d[k]\).

By measuring the wave function deviation, we can estimate the control parameters using the maximum likelihood method:where

This gives the parameter estimate solution as follows:Here, the superscript \({}^*\) denotes the Hermitian conjugate (complex conjugate transpose), consistent with earlier usage.

Alternatively, the control problem can be posed as determining the parameters \(\theta \) such that for a given threshold \(\epsilon \), the probability that the time-averaged mean square wave function deviation \(\sum _{t=1}^T \parallel \delta \psi (t) \parallel ^2 = \parallel \delta \psi \parallel ^2 = \delta \psi ^* \delta \psi \) exceeds \(\epsilon \) is minimized. From the large deviation principle, this probability for small \(\delta \) is approximately \( \exp (-Q(\theta )/\delta )\), whereThe optimal \(\theta \) is obtained by maximizing \(Q(\theta )\). This optimization problem can be solved using Lagrange multipliers by setting \(\delta \psi = a \eta \), where \(a = \sqrt{\epsilon }\) and \(\eta ^* \eta = 1\). We then minimize \(\parallel G^{-1} (a \eta - F \theta ) \parallel ^2\) subject to the constraint \(\eta ^* \eta = 1\).

The function to be optimized is given by:where \(M = (G G^*)^{-1}\). Setting the gradient of S with respect to \(\psi ^*\) to zero gives:leading to the solution:The Lagrange multiplier \(\lambda \) is obtained from the equation:This gives \(\lambda \) as a function of \(\theta \) and \(\epsilon \), denoted as \(\lambda (\theta , \epsilon )\). The optimal value of \(\delta \psi \) is then:Finally, the optimal value of \(\theta \) is obtained by maximizing:Rather than solving this problem in its entirety, we can directly formulate a joint optimization problem to optimize \(S(\delta \psi , \lambda | \theta )\) concerning \(\delta \psi \), \(\theta \), and \(\lambda \):The simultaneous equations to be solved are:which yield, after noting that \(\theta \) is a real parameter vector, the following system of equations:

The second equation can be rewritten as:This system allows us to solve for the optimal control parameters \(\theta \) that minimize the effect of noise on the wave function deviation while maximizing the likelihood of tracking the desired trajectory under noise perturbations.

By solving this joint optimization problem, we obtain both the control parameters \(\theta _1, \theta _2\) and the wave function deviation \(\delta \psi \) that minimize the probability of exceeding the threshold \(\epsilon \), thereby ensuring robust performance in noisy environments.

The simulation studies demonstrate that the large deviation-based control design effectively mitigates the effects of noise in the quantum robotic system. By discretizing the time and solving the stochastic differential equations for the wave function perturbations, we can calculate the optimal control parameters \(\theta _1, \theta _2\) that minimize the wave function deviation. Furthermore, the maximum likelihood estimation method provides an efficient way to estimate these parameters from observable measurements.

The study confirms that the proposed control framework can reduce the impact of noise, as evidenced by the reduction in the maximum eigenvalue of the autocorrelation matrix \(R_x(\theta _1, \theta _2)\) across different simulation scenarios. Optimization ensures that the control parameters are fine-tuned to achieve robust performance, even in significant noise disturbances.