Quantum Singular Value Transformation - A Unifying framework of quantum algorithms

Abstract

An n-qubit quantum circuit performs a unitary operation on an exponentially large, 2^n-dimensional, Hilbert space, which is a major source of quantum speed-ups. We show how Quantum Singular Value Transformation can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator. The transformations are realized by quantum circuits with a very simple structure – typically using only a constant number of ancilla qubits – leading to optimal algorithms with appealing constant factors. We show that this framework allows describing and unifying many quantum algorithms on a high level, and enables remarkably concise proofs for many prominent quantum algorithms, ranging from optimal Hamiltonian simulation to quantum linear equation solving (i.e., the HHL algorithm) and advanced amplitude amplification techniques. Finally, we also prove a quantum lower bound on spectral transformations.

About
The purpose of the Team-Net Quantum Computing Colloquium series is to expose Polish and international researchers, as well as interested peers, to the most important recent achievements and trends in the field of quantum computing. Seminars will be taking place on a monthly basis, on Wednesdays at 16:00 CET. Topics of the colloquium include, but are not limited to:

• Near-term quantum algorithms
• Quantum supremacy experiments
• Resource-theoretic approaches to quantum computing
• Quantum machine learning
• Practical quantum error correction and error mitigation
• Mathematical aspects of quantum computing and many-body physics

Webpage of the project: nisq.pl

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