2026-08-12 –, Room 6
This talk explores algebraic methods for quantum control polynomial optimization problems. We show how degeneracy, numerical stability and scalability are central issues in quantum control and how ConicSolve.jl has been extended to use face reduction and symmetry reduction (via the Wedderburn decomposition).
We close with open questions and further work so such tools may become a practical reality. We hope that this work acts as a foundation for further research, new tools and methods for realizing quantum control systems where certification and high precision are paramount.
Many quantum control problems are posed as polynomial optimization problems. Sum of Squares programming is effective at utilizing symmetric self-dual solvers due to such solvers exhibiting polynomial-time convergence and achieving solutions to high precision. Despite this solving Semidefinite programs and moreover Sum of Squares programs remain challenging. First is the computational blowup in memory and runtime cost as the problem size increases, second is the ill conditioning of solving polynomial optimization problems using the monomial basis, third is the numerical struggle that solvers must deal with.
Sum of squares programming is widely used in control (Lyapunov stability), robotics (trajectory optimization) and the validation and verification of safety critical systems. Global convergence and solution quality are becoming increasingly relevant in quantum control and quantum information sciences. Scaling quantum systems without compromising solution quality remains an ongoing challenge. This motivates the question; how can we adapt existing well established algebraic and computational tools in the quantum realm.
In this talk we’ll focus on the fixed-time control problem of finding the control that achieves as close as possible a given target unitary at the end of a given evolution time. The QCPOP method popularized by Bondar et al., 2025 uses TSSOS.jl for solving the sum of squares programming problem. We take a different approach by proposing a computational pipeline based on the ideas discussed in Permenter, 2017 to apply face reduction on top of symmetry reduction as a preprocessing step before solving the SOS problem. This is attractive for several theoretical/practical reasons, it exploits problem structure without relaxing the original problem, these mathematical methods can act as additional layers to extend existing solvers, packages such as SymbolicWedderburn.jl (which we use) already exists for exploiting algebraic structure.
An ongoing challenge with adopting these tools is a robust solver architecture. We’ll show the changes in ConicSolve.jl made to further reduce degeneracy and numerical ill-conditioning, particularly the use of matrix decompositions, equilibration and regularization methods. We hope this talk stimulates discussion and interest in further advancing Julia’s ecosystem to realize and explore new algebraic and computational methods. It is without many Julia packages like OperatorScaling.jl and SymbolicWedderburn.jl that this talk and exploration of these ideas would remain just that, a theoretical ideal.
I'm a software engineer based in Brisbane, Australia. My interests are in mathematical optimization and numerical methods for engineering and science. I code only in Julia in my spare time.