From Gates to Polynomials: High-Level Quantum Programming Abstractions for Physicists
Quantum computing is approaching the threshold of practical utility for simulating strongly correlated systems and complex physical phenomena. However, the prevailing paradigm of gate-by-gate circuit design remains a fundamental bottleneck. To probe non-equilibrium dynamics or quantum chaos, theoretical physicists require robust, high-level programming abstractions that map physical logic directly to quantum hardware.
This seminar explores a paradigm shift in quantum algorithm design, inspired by the seminal "Grand Unification of Quantum Algorithms" (Martyn, Chuang, et al.). We will lay down the foundations of (Generalized) Quantum Signal Processing ((G)QSP), (Generalized) Quantum Singular Value Transformation ((G)QSVT), and (Generalized) Quantum Eigenvalue Transformation ((G)QET), which elegantly reframe quantum computation as polynomial transformations of block-encoded matrices.
We will demonstrate how the Eclipse Qrisp framework (v0.8) utilizes these advanced techniques via its new BlockEncoding class. By abstracting away arithmetic overhead and qubit management, Qrisp allows researchers to effortlessly construct block encodings from physical Hamiltonians. The core of the talk will focus on manipulating operator spectra using optimal Chebyshev polynomial approximations. As it turns out, Chebyshev is one's best friend in that endeavor.
Specifically, we will showcase concrete applications tailored for advanced physics research:
- Eigenstate Filtering & Ground State Preparation: Applying Gaussian spectral filters via GQSP to efficiently isolate ground states in frustrated magnetism and spin-liquid models, bypassing deep time-evolution circuits via Quantum Lanczos methods.
- Hamiltonian Simulation: Evolving complex many-body systems (e.g., the 1D Heisenberg model) and exploring algebraic manipulations for non-equilibrium dynamics.
- Solving Quantum Linear Systems: Tackling matrix inversion ($Ax = b$) with exponential speedups to invert ill-conditioned discrete Laplacians, opening pathways for solving partial differential equations in soft matter and fluid dynamics.
Finally, we will bridge theory and execution by showing how to compile these fault-tolerant protocols using deterministic "Repeat-Until-Success" logic, complete with rigorous, on-the-fly quantum resource estimation (gate counts, circuit depth, and qubit scaling).
The future of Quantum Linear Algebra is here, and it is Qrisp.
The lecture is organized as part of the UL VIP project KTTK21 (ARIS SN-ZRD/22-27/510).
