Quantum Linear Systems Solvers

Workshop Description

Full-day workshop on the HHL algorithm and quantum linear solvers for logistics applications. Covers Kalman filtering, state-space inference, and network flow computation with honest assessment of circuit depth requirements and the fault-tolerant timeline.

Many logistics computations reduce to solving systems of linear equations. Kalman filters for demand state estimation, state-space models for supply chain dynamics, and multi-commodity network flow problems all spend most of their runtime solving Ax=b. The HHL algorithm (Harrow, Hassidim, and Lloyd 2009) promises exponential speedup for sparse linear systems: O(log N) versus O(N) for classical methods. The fine print is substantial. Quantum state preparation and readout constraints (Aaronson 2015) limit the practical contexts where HHL delivers advantage. Circuit depth requirements exceed what NISQ hardware can execute reliably, meaning HHL is a fault-tolerant era algorithm requiring error-corrected qubits. This workshop teaches participants the full HHL algorithm, maps it to three specific logistics applications (Kalman filtering, state-space inference, network flow), and provides an honest comparison against classical alternatives that work today. Participants implement an HHL circuit for a small state-space model and measure how circuit depth scales with system dimension. The workshop also covers VQLS (Bravo-Prieto et al. 2020) as a NISQ-compatible alternative with shallower circuits but weaker theoretical guarantees. The goal is preparation: understanding when quantum linear solvers will become practical so organisations can be ready when hardware catches up.

What participants cover

• HHL algorithm: quantum phase estimation, controlled rotation, amplitude amplification, and the O(log N) speedup claim with its caveats (Aaronson 2015)
• Kalman filtering for logistics: demand state estimation, fleet position tracking, and inventory inference as linear system problems
• State-space models: hidden Markov structure in supply chain dynamics where transition matrix solves dominate computation
• Network flow: multi-commodity flow on logistics networks where linear system solves are the runtime bottleneck
• Circuit depth reality: HHL requires O(log^2 N) depth with controlled rotations that exceed NISQ coherence, plus error correction overhead
• NISQ alternatives: variational quantum linear solvers (VQLS, Bravo-Prieto et al. 2020) with shallower circuits and their accuracy trade-offs