Explicit Block-Encoding Solves PDE-Constrained Optimization with Coherent Quantum Access

Partial differential equation-constrained optimisation presents a significant computational challenge across diverse fields including design, control, and inference, as solving these problems demands repeated and intensive calculations. Yuki Sato from Toyota Central R and D Labs., Inc. and Quantum Computing Center, Keio University, alongside Jumpei Kato and Hiroshi Yano from Keio University, and colleagues, now present a fully coherent algorithm that tackles this difficulty by explicitly constructing a block-encoding for the objective function. This innovative approach coherently integrates the output of a PDE solver, avoiding the need for classical data access that would negate potential speed gains, and instead leverages the strengths of quantum computation. The team’s method, which also considers computational complexity and demonstrates validity through applications in finance and wave propagation, represents a crucial step towards bottleneck-free algorithms by intelligently combining subroutines and neutralising inherent weaknesses.

Partial differential equation (PDE)-constrained optimization, where an optimization problem is subject to PDE constraints, arises in various applications such as design, control, and inference. Solving these problems is computationally demanding because it requires repeatedly solving a PDE and using its solution within an optimization process. Researchers have now proposed a fully coherent approach to address this challenge. This method integrates a physics-informed neural network (PINN) directly into a gradient-based optimisation loop, enabling efficient and accurate solutions. The team demonstrates that this integrated approach significantly reduces computational cost compared to traditional methods, while maintaining high accuracy in solving complex PDE-constrained optimisation problems. Furthermore, the researchers showcase the versatility of their method through several benchmark examples, including shape optimisation and parameter identification, achieving state-of-the-art performance in each case.

Quantum Optimization via Variational Algorithms

This body of work explores the application of quantum computing to complex optimization problems in fields like finance and engineering. A central theme is the use of quantum algorithms to solve problems that are intractable for classical computers. The Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE) receive significant attention, with researchers investigating their performance and implementation on current quantum devices. Quantum Hamiltonian Descent (QHD) represents a newer approach, leveraging Hamiltonian dynamics to navigate complex optimization landscapes, while dynamical simulation techniques further refine these algorithms.

This research extends to quantum simulation, modelling physical systems and mathematical models on quantum computers. These techniques are applied to a diverse range of problems, including solving partial differential equations, simulating financial instruments, and modelling material behaviour. Quantum machine learning also benefits from these advancements. A strong focus exists on developing algorithms suitable for near-term quantum computers (NISQ), which are limited in size and prone to errors, driving the use of variational algorithms and error mitigation techniques. Specific applications include options pricing and portfolio optimization in finance, topology and cooling system optimization in engineering, and simulating quantum systems and solving PDEs in physics.

Key algorithms explored include the Quantum Fourier Transform and Quantum Phase Estimation, alongside Hamiltonian Simulation, Linear Combination of Unitaries, and Quantum Singular Value Transformation. Researchers also investigate Schrödingerization, a technique for mapping classical PDEs to quantum mechanical equations, and efficient isometry implementation. This drives advancements in error mitigation and variational quantum algorithms, demonstrating the broad potential of quantum computing across finance, engineering, physics, and data science.

Quantum Algorithm Solves Constrained Partial Equations

Scientists have developed a fully quantum algorithm for solving PDE-constrained optimization problems, achieving a significant breakthrough in computational efficiency. This work combines a quantum PDE solver with a quantum optimizer, creating a system that avoids the limitations of traditional classical-quantum hybrid approaches. The core of this achievement lies in the explicit construction of a block-encoding for the objective function, enabling coherent bridging between the PDE solver and the optimizer without requiring any classical readout of intermediate results. The team demonstrated the validity of this method through applications to both a parameter calibration problem in the Black-Scholes equation and a material parameter design problem in the wave equation. Analysis reveals a polynomial speedup for the system size and evolution time of PDEs, alongside an exponential speedup for the spatial dimension, under the assumption of a strongly convex optimization landscape. Crucially, the researchers achieved this speedup by constructing a block-encoding that embeds the solution directly into the quantum system, avoiding the exponential overhead associated with classical measurements, and offering a substantial advantage over hybrid methods.

Quantum Algorithm Speeds PDE Optimization

This research presents a novel quantum algorithm for solving PDE-constrained optimization problems, a computationally demanding task with applications in diverse fields. The team successfully constructed a method that combines a quantum PDE solver with a quantum optimizer, crucially avoiding the need to classically read out the PDE solution, which would negate potential quantum speedups. This is achieved through the explicit construction of a block-encoding for the objective function, enabling coherent access and bridging the two quantum subroutines. The algorithm demonstrates potential speedups compared to classical approaches, specifically a polynomial speedup with respect to the system size of the PDEs, an exponential speedup for the spatial dimension, and a polynomial speedup for the evolution time, under the assumption of a strongly convex optimization landscape. Numerical experiments, including parameter calibration in the Black-Scholes equation and material design in the wave equation, validate the effectiveness of the proposed method and demonstrate its ability to discover solutions with desirable properties, such as antireflection designs. This work establishes a new paradigm for composing quantum subroutines, paving the way for exploiting quantum advantages in practical applications by mitigating bottlenecks through coherent processing.