Quantum Speed-ups Approximate Semidefinite Relaxations with Runtime, Achieving Accuracy for Polynomial Optimization in the -ball

Solving complex optimisation problems represents a major challenge in mathematics and computer science, and researchers continually seek faster methods to tackle them. Daniel Stilck França from University of Copenhagen and Ngoc Hoang Anh Mai from Institute of Mathematics, Vietnam Academy of Science and Technology, have now developed a quantum algorithm that significantly accelerates the computation of Lasserre relaxations, a technique used to approximate solutions to polynomial optimisation problems. Their work demonstrates a super-quadratic speedup over existing classical methods, particularly when dealing with problems where solutions lie within a specific range or within a simplex, and offers a substantial advance in the field. This improvement stems from a novel application of matrix multiplicative weights within a quantum framework, potentially revolutionising areas such as portfolio optimisation and other complex modelling tasks.

radius 1/2, or when the problem lies within a simplex. After appropriately rescaling coefficients, the researchers developed a quantum algorithm based on matrix multiplicative weights that approximates a solution to accuracy ε with runtime, for fixed k. This represents a significant improvement over classical matrix multiplicative-weights methods, which scale even in the unconstrained case. As an example, the team achieved an algorithm for portfolio optimization, improving over the classical bound.

Block Encoding for Matrix Representation and Manipulation

This research details a method for efficiently encoding and manipulating matrices within quantum algorithms, aiming to reduce the computational cost of solving probabilistic optimization problems. The core technique involves block encoding, a method for representing a matrix using a smaller quantum circuit. This allows quantum operations on the matrix to be performed with fewer qubits than traditional methods. Probabilistic optimization problems arise in fields like machine learning and finance, and often require solving complex mathematical formulations. Block encoding is crucial because it allows quantum computers to work with large matrices efficiently.

The team utilizes unitary operators, the building blocks of quantum algorithms, and leverages concepts from semidefinite programming and duality to formulate and solve these problems. The method involves representing a matrix B(γ) using a unitary operator Uγ. This approach is efficient because it exploits the matrix’s structure and requires a relatively small number of quantum gates. The researchers extended this technique to handle constrained problems by using a block-diagonal matrix and a linear combination of unitaries. They provide theoretical guarantees about the accuracy and efficiency of their block encoding method, demonstrating its potential for practical implementation.

This work contributes to the development of quantum algorithms that could outperform classical algorithms for solving these complex problems. The proposed block encoding method is efficient in terms of the number of quantum gates required, which is crucial for implementing quantum algorithms on real quantum hardware. The techniques presented could potentially be scaled to handle larger and more complex problems, with applications in machine learning, finance, and logistics. In essence, this research provides a way to represent complex puzzles in a quantum computer so they can be solved much faster. The block encoding is a special code that allows the quantum computer to work with the puzzle efficiently, and the techniques they describe help to make this code as compact and fast as possible.

Quantum Speedup for Lasserre Hierarchy Approximations

This work presents a new quantum algorithm for approximating solutions to Lasserre’s hierarchy, a technique used to solve polynomial optimization problems. The researchers achieved a significant speedup by leveraging quantum computation to tackle the complex calculations involved in these hierarchies, which are frequently used in diverse fields including combinatorial optimization and quantum information theory. The algorithm focuses on efficiently approximating solutions to semidefinite programs, a core component of Lasserre’s hierarchy, and builds upon the matrix multiplicative weights method. The team developed a quantum approach that, under specific conditions, delivers a super-quadratic speedup in problem dimension when computing Lasserre relaxations.

Specifically, the algorithm achieves this speedup when dealing with polynomial optimization problems where the optimum is attained within a ball of radius, or when the problem lies within a simplex. A key aspect of the method involves rescaling coefficients to ensure efficient computation, and the algorithm utilizes block encodings to represent the problem’s matrices without requiring a quantum random access memory. For a polynomial optimization problem with a sparsity bound, the quantum algorithm approximates solutions to an accuracy with a runtime. This represents a substantial improvement over classical matrix multiplicative-weights methods, which scale even in the unconstrained case. As a practical demonstration, the researchers applied their algorithm to portfolio optimization, achieving an improvement over the classical bound. The work also details how to implement the necessary block encodings without requiring specific computational steps, further enhancing the algorithm’s practicality and potential for real-world applications.

Quantum Speedup for Polynomial Optimization Problems

This work presents new quantum algorithms for approximating solutions to Lasserre’s hierarchy, a technique used to solve polynomial optimization problems. The researchers developed a quantum method, based on matrix multiplicative weights, that achieves a speedup over classical approaches when certain conditions are met regarding the problem’s structure and the sparsity of its constraints. Specifically, the algorithm demonstrates improved performance for problems where the optimum lies within a defined ball or simplex, and offers a super-quadratic speedup in problem dimension for computing these relaxations. The team successfully applied this approach to portfolio optimization, a practical application of polynomial optimization, achieving a demonstrable improvement over existing classical bounds.

Importantly, the researchers also detail how to implement the necessary components of the algorithm without requiring specific, potentially limiting, computational steps. While acknowledging that the speedup relies on the problem meeting certain structural requirements, this research significantly advances the potential for quantum computers to efficiently tackle complex optimization challenges. The authors note that the performance of the algorithm is linked to the sparsity of the coefficient-matching matrices associated with the constraints, suggesting that further research could focus on techniques to exploit or reduce this sparsity. Future work could also explore the applicability of this method to a wider range of polynomial optimization problems and investigate potential extensions to handle more complex constraints or larger problem instances.