⚛️ quantum physics

CDJ-Pontryagin Optimal Control for General Continuously Monitored Quantum Systems

This paper generalizes the Chantasri-Dressel-Jordan stochastic path integral formalism to arbitrary continuously monitored quantum systems by introducing a costate operator and a quantum Pontryagin's maximum principle, enabling the derivation of optimal control protocols that significantly improve state preparation fidelities in bosonic quantum computing applications.

Original authors: Tathagata Karmakar, Andrew N. Jordan

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tathagata Karmakar, Andrew N. Jordan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to guide a very shy, jittery ghost (a quantum particle) from one room to another. The ghost is invisible, but you have a flashlight that lets you see it only in brief, fuzzy flashes. Every time you look, the ghost jumps a little bit because your light disturbs it. This is what happens in the quantum world: continuous monitoring.

The problem is that the ghost is also being pushed around by random winds (noise). You want to get it to a specific destination (a target state) as reliably as possible. But because of the random winds and your own "fuzzy" observations, sometimes the ghost ends up in the wrong room.

This paper introduces a new, super-smart navigation system to help guide these quantum ghosts. Here is how it works, broken down into simple concepts:

1. The Old Map vs. The New GPS

Previously, scientists had a map (called the CDJ formalism) that could predict the "most likely path" a ghost would take if it were a simple, smooth object (like a ball rolling on a hill). But this map broke down when the ghost got complicated, weird, or "non-Gaussian" (like a ghost that can be in two shapes at once). It was like trying to use a flat paper map to navigate a 3D maze with moving walls.

The authors created a new GPS system (the CDJ-Pontryagin method). Instead of just tracking the ghost's position, this system introduces a "shadow twin" called a costate.

The Ghost (State): This is the actual quantum system you are trying to control.
The Shadow Twin (Costate): Think of this as a "future-looking" version of the ghost. It doesn't exist in reality, but it helps calculate the best moves. It's like having a co-pilot who knows exactly where you want to be in the future and tells you how to steer now to get there, accounting for all the random winds.

2. The "Bang-Bang" Steering Wheel

One of the coolest discoveries in this paper is how the best steering looks. Imagine you are driving a car on a slippery road with a strong crosswind. To stay on the perfect path, do you gently nudge the wheel left and right? Or do you slam it hard to the left, hold it there, and then slam it hard to the right?

The math shows that for these quantum systems, the best strategy is often "Bang-Bang."

The Analogy: It's like a surfer catching a wave. You don't gently adjust your board; you commit fully to a direction (Bang!), ride it, and then switch fully to the other direction (Bang!) at the exact right moment.
The Result: The paper proves that the optimal way to control the quantum system is to switch the control settings to their maximum or minimum limits instantly, rather than using a smooth, gradual change.

3. The "Most Likely Path" vs. Reality

The new GPS calculates the Most Likely Path (MLP). This is the single, perfect route the ghost should take if everything goes according to the plan.

The Reality: In the real world, the ghost is jittery. It will wander off the perfect line.
The Test: The authors ran a simulation with 10,000 ghosts.
- Group A (Sample Control): They used a "good enough" steering method (like a human driver guessing).
- Group B (Optimal Control): They used the new "Bang-Bang" GPS.

The Result: The GPS group was a massive success. While the "good enough" group got the ghost to the target only about 6% of the time with high accuracy, the GPS group got it there 196% more often (nearly 20% of the time). They didn't just get a few ghosts there; they got a lot of them there with incredible precision (over 95% accuracy).

4. Why Does This Matter?

This isn't just about moving ghosts around in a lab. This is the blueprint for building Quantum Computers.

Error Correction: Quantum computers are fragile. They make mistakes easily. This method helps "steer" the computer back to the right answer even when noise tries to push it off course.
Cat States: The paper tested this on "Schrödinger's Cat" states (quantum objects that are both dead and alive). They successfully guided these complex states from one form to another, which is a huge step toward building powerful quantum machines.

The Big Picture

Think of this paper as inventing a self-driving algorithm for the quantum world.

Before: We were trying to drive a car on ice with a blindfold, guessing which way to turn.
Now: We have a system that calculates the "shadow twin" of the car, predicts the perfect "Bang-Bang" turns needed to counter the ice, and guides the car to the destination with much higher success rates.

It turns a chaotic, jittery quantum journey into a highly predictable, optimized trip, paving the way for the next generation of quantum technology.

1. Problem Statement

The paper addresses the challenge of finding optimal control protocols for quantum systems undergoing continuous monitoring (continuous measurement).

Context: Near-term quantum hardware requires high-fidelity state preparation and stabilization (e.g., for error correction). Dissipation engineering and continuous feedback are promising approaches, but finding the optimal control parameters (Hamiltonian drives and measurement settings) to steer a system from an initial state to a target state is computationally difficult.
Limitations of Existing Methods:
- The Chantasri-Dressel-Jordan (CDJ) stochastic path integral formalism characterizes the statistics of readouts and "most likely paths" (MLP) but relies on parametrizing the quantum state using a finite set of coordinates (e.g., Bloch vectors for qubits or quadrature expectations for Gaussian states). This limits its applicability to non-Gaussian states and many-body systems.
- Previous applications of the Pontryagin Maximum Principle (PMP) to quantum systems often assumed linear evolution or specific cost functions, failing to capture the nonlinear nature of conditional evolution in continuously monitored systems.
Goal: To generalize the CDJ formalism to arbitrary continuously monitored systems (including non-Gaussian states) and derive a rigorous framework for optimal control using the Pontryagin Maximum Principle.

2. Methodology

The authors develop a unified framework combining stochastic path integrals and optimal control theory.

A. Generalization of CDJ Formalism via Costate Operators

Costate Operator ( $\hat{\sigma}$ ): Instead of parametrizing the state with coordinates, the authors introduce a Hermitian costate operator $\hat{\sigma}$ . This acts as a Lagrange multiplier for the stochastic master equation (SME).
Stochastic Hamiltonian: They construct a stochastic Hamiltonian $H(\hat{\sigma}, \hat{\rho}, r)$ where $\hat{\rho}$ is the density matrix and $r$ is the measurement readout.
$H \sim \text{Tr}(\hat{\sigma} \hat{F}_{\hat{\rho}}(\hat{\rho}, r)) + \ln P(r|\hat{\rho})$
Optimal Path Equations: By extremizing the stochastic action, they derive coupled evolution equations for the state $\hat{\rho}$ and the costate $\hat{\sigma}$ . Crucially, they show that the costate evolution is generally the negative conjugate of the state evolution, a relationship that simplifies to the standard negative adjoint only under specific cost functions (like readout probabilities).

B. CDJ-Pontryagin (CDJP) Formulation

Cost Function: The cost function is defined as the negative log-probability of the measurement readouts (the CDJ action). Minimizing this cost maximizes the probability of a specific trajectory.
Pontryagin's Maximum Principle (PMP): The authors formulate the PMP for quantum systems with arbitrary nonlinear evolution.
- Condition I: The optimal control must maximize the Hamiltonian $H$ at every instant.
- Condition II: Boundary conditions on the costate at the final time.
Derivation: They prove that the CDJ most likely paths are a special case of the general PMP when the cost function corresponds to readout probabilities. This bridges the gap between stochastic path integrals and deterministic optimal control theory.

C. Application to Monitored Harmonic Oscillators

The framework is applied to a quantum harmonic oscillator with:

Parametric Control: A tunable quadratic potential $\lambda_1(t) \hat{X}^2$ .
Dissipative Control: Adaptive quadrature measurements defined by an angle $\theta(t)$ .

Dimensionality Reduction: For the harmonic oscillator, the infinite-dimensional operator equations are reduced to a closed set of 10 scalar differential equations involving expectation values of products of position/momentum operators and the costate. This makes numerical simulation feasible.
Analytical Solutions: The PMP conditions yield analytical forms for the optimal controls:
- Bang-Bang Control: The optimal parametric potential strength $\lambda_1(t)$ is shown to be of the "bang-bang" form (switching between maximum and minimum bounds).
- Optimal Quadrature: The optimal measurement angle $\theta(t)$ is derived analytically based on the phase of the costate variables.

3. Key Contributions

Generalized Formalism: Extended the CDJ formalism from Gaussian states/finite-dimensional parametrizations to arbitrary continuously monitored systems using a costate operator, removing the restriction to specific state parametrizations.
CDJ-Pontryagin Connection: Rigorously derived the CDJ most likely paths as a special case of the Pontryagin Maximum Principle, establishing a systematic prescription for quantum optimal control under continuous monitoring.
Analytical Control Laws: Proved analytically that for parametrically driven oscillators, the optimal potential strength follows a bang-bang protocol.
Numerical Framework: Developed a scalable numerical method (reducing operator dynamics to 10 scalars) to solve for optimal controls in non-Gaussian regimes.

4. Results and Validation

The authors validated their theory using three specific state preparation tasks relevant to Bosonic quantum computing, comparing the optimal control (derived via CDJP) against a sample control (a heuristic control designed to reach the target but not necessarily optimal).

Example 1: Binomial Code Word Preparation
- Task: Transform an error word $|0\rangle - |4\rangle$ to a logical code word $|0\rangle + |4\rangle$ .
- Result: The optimal control achieved a final state fidelity of 95.46%.
- Statistical Gain: Under optimal control, 57.08% of 10,000 simulated trajectories reached >90% fidelity, compared to 46.01% for the sample control. The increase in trajectories reaching >95% fidelity was 196%.
Example 2: Parametric Cooling (Cat State to Ground State)
- Task: Cool an even cat state to the ground state $|0\rangle$ .
- Result: Final fidelity of 97.49%.
- Statistical Gain: 79.51% of trajectories reached >95% fidelity under optimal control vs. 56.73% for sample control (a 40% increase in high-fidelity trajectories).
Example 3: Cat State to Cat State Evolution
- Task: Evolve one cat state to another with different parameters.
- Result: Final fidelity of 95.59%.
- Statistical Gain: 47.17% of trajectories reached >95% fidelity under optimal control vs. 18.89% for sample control (a 150% increase).

Key Observation: In all cases, the optimal control derived from the fixed-endpoint assumption significantly increased the probability of the system reaching the target state with high fidelity, even though the optimization was performed for a specific trajectory rather than maximizing the ensemble success rate directly.

5. Significance

Scalability: The costate-based approach overcomes the "parametrization bottleneck" of previous CDJ methods, making it applicable to many-body systems and non-Gaussian states (crucial for bosonic error correction codes).
Practical Utility: The results demonstrate that optimal control protocols derived from theoretical "most likely paths" are highly effective in practice, significantly outperforming heuristic controls in noisy, monitored environments.
Theoretical Unification: The work unifies stochastic path integral theory with deterministic optimal control (PMP), providing a robust mathematical foundation for designing feedback loops and state preparation schemes in quantum hardware.
Future Applications: The framework is directly applicable to continuous quantum error correction, autonomous stabilization, and the control of mechanical resonators and superconducting cavities, paving the way for more efficient bosonic quantum computing architectures.