Co-designed Quantum Discrete Adiabatic Linear System Solver Via Dynamic Circuits

Existing quantum discrete adiabatic approaches are hindered by circuit depth that increases linearly with the number of evolution steps, a significant challenge for current quantum hardware with limited coherence times. To address this, we propose a co-designed framework that synergistically integrates dynamic circuit capabilities with real-time classical processing. This framework reformulates the quantum adiabatic evolution into discrete, dynamically adjustable segments. The unitary operator for each segment is optimized on-the-fly using classical computation, and circuit multiplexing techniques are leveraged to reduce the overall circuit depth scaling from $O(\text{steps}\times\text{depth}(U))$ to $O(\text{depth}(U))$. We implement and benchmark a quantum discrete adiabatic linear solver based on this framework for linear systems of $W \in \{2,4,8,16\}$ dimensions with condition numbers $\kappa \in \{10,20,30,40,50\}$. Our solver successfully overcomes previous depth limitations, maintaining over 80% solution fidelity even under realistic noise models. Key algorithmic optimizations contributing to this performance include a first-order approximation of the discrete evolution operator, a tailored dynamic circuit design exploiting real-imaginary component separation, and noise-resilient post-processing techniques.