Efficient encoding of the Schrodinger equation on quantum computers
- December 5 (Mon) at 14:00 - 15:30, 2022 (JST)
- Ermal Rrapaj (Postdoctoral Researcher, iTHEMS)
- Tetsuo Hatsuda
The continuous space Schrödinger equation is reformulated in terms of spin Hamiltonians. For the kinetic energy operator, the critical concept facilitating the reduction in model complexity is the idea of position encoding. A binary encoding of position produces a spin-1/2 Heisenberg-like model and yields exponential improvement in space complexity when compared to classical computing. Encoding with a binary reflected Gray code (BRGC), and a Hamming distance 2 Gray code (H2GC) reduces the model complexity down to the XZ and transverse Ising model respectively. Any real potential is mapped to a series of k-local Ising models through the fast Walsh transform. As a first step, the encoded Hamiltonian is simulated for quantum adiabatic evolution. As a second step, the time evolution is discretized, resulting in a quantum circuit with a gate cost that is better than the Quantum Fourier transform. Finally, a simple application on an ion-based quantum computer is provided as proof of concept.
- Improving Schrödinger Equation Implementations with Gray Code for Adiabatic Quantum Computers, Chia Cheng Chang, Kenneth S. McElvain, Ermal Rrapaj, and Yantao Wu, PRX Quantum 3, 020356, doi: 10.1103/PRXQuantum.3.020356
- Ermal Rrapaj, Kenneth S. McElvain, Chia Cheng Chang, Yantao Wu, André Walker-Loud, Gate Based Implementation of the Laplacian with BRGC Code for Universal Quantum Computers, doi: 10.48550/arXiv.2207.11647
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