Introduction to Qubitization

Introduction to Qubitization

This PennyLane tutorial introduces qubitization, a powerful and asymptotically optimal technique for Hamiltonian simulation and for encoding the spectrum of a Hamiltonian into a quantum circuit, central to modern fault-tolerant quantum algorithms. The core idea builds on block encoding, embedding a (suitably normalized) Hamiltonian as a sub-block of a larger unitary, and on a clever construction, the qubitization walk operator, whose eigenvalues are related to those of the Hamiltonian by a simple cosine, so that applying powers of this walk operator effectively implements functions of the Hamiltonian with optimal query complexity. The tutorial explains how a block encoding is constructed from the Hamiltonian's decomposition into a linear combination of unitaries (using prepare and select operations), how the qubitization operator is built from that block encoding plus a reflection, and how its spectrum relates to the Hamiltonian's. It implements these constructions in PennyLane and shows how qubitization is used for phase estimation of the Hamiltonian's eigenvalues. By making this advanced but foundational technique concrete, the tutorial connects block encodings to optimal Hamiltonian simulation and provides a stepping stone to fault-tolerant algorithm design in PennyLane.

Run it

pip install -r requirements.txt
python demo.py

Source and license

Imported from demonstrations_v2/tutorial_qubitization/demo.py in PennyLaneAI/demos at c52c0abeb5122218aa96b38eea848864cce7323f, under the Apache License 2.0. Original authors: Xanadu and the PennyLane community. The upstream LICENSE is included alongside this example.