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A Grand Unification of Quantum Algorithms
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Quantum algorithms offer significant speedups over their classical counterparts for a variety of problems. The strongest arguments for this advantage are borne by algorithms for quantum search, quantum phase estimation, and Hamiltonian simulation, which appear as subroutines for large families of composite quantum algorithms. A number of these quantum algorithms were recently tied together by a novel technique known as the quantum singular value transformation (QSVT), which enables one to perform a polynomial transformation of the singular values of a linear operator embedded in a unitary matrix. In the seminal GSLW'19 paper on QSVT [Gily\'en, Su, Low, and Wiebe, ACM STOC 2019], many algorithms are encompassed, including amplitude amplification, methods for the quantum linear systems problem, and quantum simulation. Here, we provide a pedagogical tutorial through these developments, first illustrating how quantum signal processing may be generalized to the quantum eigenvalue transform, from which QSVT naturally emerges. Paralleling GSLW'19, we then employ QSVT to construct intuitive quantum algorithms for search, phase estimation, and Hamiltonian simulation, and also showcase algorithms for the eigenvalue threshold problem and matrix inversion. This overview illustrates how QSVT is a single framework comprising the three major quantum algorithms, thus suggesting a grand unification of quantum algorithms.
Forward citations
Cited by 5 Pith papers
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Faster quantum linear system solver beyond the condition number
Two quantum linear system solvers are presented with query complexity independent of the condition number, scaling instead with an effective condition number or a solution-norm ratio.
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Singular value transformation for unknown quantum channels
An algorithm block-encodes the Liouville representation of an unknown quantum channel from black-box access, enabling polynomial transformations of its singular values via QSVT.
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Fixing Divergence in Carleman Linearization via Analytical Continuation
A regularized function inserted into Carleman linearization, derived from a Möbius conformal map, removes the long-time divergence for logistic, KPP-Fisher, and phase-field models and supports an LCU quantum implementation.
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A quantum algorithm for modular flow
A QSVT-based algorithm implements modular flow of an operator with respect to a density matrix in O~(κ²|t|log(κ²/ε)) queries to a block encoding of the state.
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Krein space quantization and New Quantum Algorithms
A proposed Krein-space block-matrix regularization for singular linear systems reduces to a parameter-dependent normal-equation solve and is not demonstrated as a quantum algorithm.
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