Block Encoding and QSVT for solving differential equations

Our approach combines block encoding with Quantum Singular Value Transformation (QSVT) to construct a full pipeline for solving differential equations on quantum hardware. The central methodological contribution lies in developing a hardware-efficient framework for block encoding sparse matrices with explicit gate-level realizations.

We begin by encoding the sparse matrix into a unitary operator using a structured block encoding scheme. To reduce overhead, we introduce a combinatorial optimization strategy that assigns control qubits in a way that satisfies nearest-neighbor connectivity constraints typical of superconducting architectures. This significantly reduces the need for costly multi-controlled X (MCX) gates. Additionally, we design coherent permutation operators that enable amplitude reordering while preserving quantum superposition, avoiding measurement-based or classical preprocessing steps.

Once the block encoding is constructed, QSVT is applied to implement polynomial transformations of the encoded matrix, enabling the solution of linear systems derived from discretized PDEs. The quantum register is prepared using basis encoding of the right-hand side vector, and polynomial sequences are implemented via phase-controlled rotations.

Finally, the solution is extracted through measurement of observables or overlap estimation, with post-selection used where necessary. We explicitly account for success probabilities and their dependence on system parameters. The methodology is demonstrated on tridiagonal systems, the heat equation with mixed boundary conditions, and Carleman-linearized Burgers' equation, providing a complete workflow from problem formulation to quantum circuit execution.

We demonstrate the correctness and feasibility of our approach through a combination of simulation and hardware-aware circuit construction. First, all proposed quantum circuits-including block encoding and QSVT layers-are fully specified at the gate level, enabling direct implementation on existing quantum platforms.

The algorithms are implemented and tested using quantum simulators to validate numerical correctness against classical solutions. For benchmark problems such as tridiagonal systems and discretized heat equations, we compare quantum outputs (obtained via measurement statistics) with exact classical solutions, verifying convergence and accuracy within expected error bounds.

In addition, we map the circuits to realistic hardware topologies, specifically IBM superconducting processors with heavy-hex and square lattice connectivity. This includes explicit routing of qubits, decomposition of multi-qubit gates into native gate sets, and estimation of two-qubit gate depth. These hardware-aware implementations allow us to quantify the practical cost of our approach in terms of circuit depth and noise sensitivity.

We also analyze post-selection success probabilities and their impact on runtime, providing a realistic assessment of execution feasibility. While full-scale execution on current devices is limited by noise and circuit depth, smaller instances of the algorithm can be executed or emulated, demonstrating the validity of the approach and highlighting the gap between theoretical algorithms and practical implementations.

The proposed framework provides a pathway toward scalable quantum solutions for differential equations, but its practical advantage depends on careful hybridization and hardware improvements. A key aspect of scalability lies in combining classical preprocessing-such as discretization, Carleman linearization, and sparsity exploitation-with quantum subroutines for solving the resulting linear systems.

Our complexity analysis shows that, in principle, QSVT-based solvers can achieve polylogarithmic scaling in system size under favorable conditions, particularly for well-conditioned sparse matrices. However, the actual circuit depth required for block encoding and polynomial transformations remains a limiting factor on near-term devices. Our hardware-aware optimizations reduce this overhead, making larger problem instances more accessible as quantum hardware improves.

We identify regimes-such as moderately sized, structured sparse systems-where quantum methods may begin to outperform classical solvers, especially when high precision or repeated solves are required. At the same time, our results highlight critical bottlenecks, including circuit depth, post-selection overhead, and noise accumulation, which currently prevent large-scale advantage.

This work therefore contributes both a practical implementation strategy and a realistic roadmap for future progress. It clarifies where quantum advantage may emerge and where classical methods remain dominant, aligning with the "Insights from Failure" perspective by explicitly identifying the limitations that must be overcome to achieve scalable quantum computational fluid dynamics and PDE solvers.