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Quantum signal processing and the quantum singular value transformation

Steven Herbert

Published Dec 8, 2025• 0 citations
Publisher

Abstract

Abstract The quantum algorithms presented in the previous chapters are underpinned by three fundamental features of quantum computation. First, it is possible to construct Grover iterates that amplify amplitude and concentrate probability mass on desired outcomes quadratically faster than their classical counterparts; second, quantum phase estimation is a subroutine that uses the quantum Fourier transform to interfere a quantum state such that wrong answers cancel out, leaving a state from which the solution to the problem can be extracted; and finally, using a digital quantum computer, Hamiltonian evolution can be simulated. Remarkably, these three phenomena can be seen to be instances of the same underlying computational effect: the quantum singular value transformation – a breakthrough that has been dubbed ‘the grand unification of quantum algorithms’, and which is introduced and analysed in this chapter.

DOI
10.1093/9780191964381.003.0014

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Abstract The quantum algorithms presented in the previous chapters are underpinned by three fundamental features of quantum computation. First, it is possible to construct Grover iterates that amplify amplitude and concentrate probability mass on desired outcomes quadratically faster than their classical counterparts; second, quantum phase estimation is a subroutine that uses the quantum Fourier transform to interfere a quantum state such that wrong answers cancel out, leaving a state from which the solution to the problem can be extracted; and finally, using a digital quantum computer, Hamiltonian evolution can be simulated. Remarkably, these three phenomena can be seen to be instances of the same underlying computational effect: the quantum singular value transformation – a breakthrough that has been dubbed ‘the grand unification of quantum algorithms’, and which is introduced and analysed in this chapter.
Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many prominent quantum algorithms. We propose a framework of quantum signal processing and quantum singular value transformation on U ( N ) , which realizes multiple polynomials simultaneously from a block-encoded input, as a generalization of those on U ( 2 ) in the original frameworks. We provide a comprehensive characterization of achievable polynomial matrices and give recursive algorithms to construct the quantum circuits that realize desired polynomial transformations. As three example applications, we propose a framework to realize bi-variate polynomial functions, demonstrate N -interval decision achieving O ( d ) query complexity with a log 2 ⁡ N improvement over iterative U ( 2 ) -QSP requiring O ( d log 2 ⁡ N ) queries, and present a quantum amplitude estimation algorithm achieving the Heisenberg limit without adaptive measurements.
Abstract The discovery of physical systems that exhibit what we now call quantum mechanical behaviour was unquestionably one of the most astonishing events of modern physics. No less remarkable was the realisation that quantum mechanics presents a much more powerful paradigm for computation and information processing than its classical counterpart. But this power is somewhat subtle, and only by building up a rigorous model of quantum computation from the fundamental physics can it be fully understood and analysed. Developing and then using the quantum circuit model as a suitable representation of quantum mechanics, this textbook presents a canon of the most important quantum algorithms. Starting with historically important query algorithms, ‘classic’ results such as Grover’s and Shor’s algorithm are then covered, followed by more recent breakthroughs including the quantum algorithm for solving linear systems (HHL) and the quantum singular value transformation. The thread that runs through the book is the hunt for quantum advantage, in all of its manifestations, and to this end the final chapter moves from the idealised setting of closed quantum systems, to address real ‘noisy’ open quantum systems. Quantum error correction is introduced, culminating in the seminal result, the threshold theorem, which roughly says that the idealised performance predicted by the analysis of quantum mechanics as a model of computation can actually be realised on real-world quantum computers.
Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop an algorithmic toolbox, quantum phase processing, that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigeninformation of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase related. As a notable application, we propose a quantum phase estimation algorithm without quantum Fourier transform, which requires the fewest ancilla qubits and matches the best performance so far. We further exploit the power of our method by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy, and quantum entropy estimation, demonstrating improvements or optimality for almost all cases.
Abstract This chapter introduces and analyses two of the most important quantum algorithmic primitives that are used in myriad applications, namely quantum phase estimation and quantum amplitude estimation (QAE). Quantum phase estimation (in the presentation covered in this chapter) uses the quantum Fourier transform, itself an extremely important quantum subroutine. In its original presentation, QAE combines quantum amplitude amplification (a generalisation of the circuit used in Grover’s algorithm) and quantum phase estimation to estimate the amplitude of a quantum state. However, more recent approaches achieve amplitude estimation without phase estimation, which is potentially practically advantageous. This chapter covers both, as well as some important applications of QAE such as quantum Monte Carlo integration and quantum counting.
Quantum signal processing (QSP) provides a representation of scalar polynomials of degree ddd as products of matrices in SU(2)\mathrm{SU}(2)SU(2), parameterized by (d+1)(d+1)(d+1) real numbers known as phase factors. QSP is the mathematical foundation of quantum singular value transformation (QSVT), which is often regarded as one of the most important quantum algorithms of the past decade, with a wide range of applications in scientific computing, from Hamiltonian simulation to solving linear systems of equations and eigenvalue problems. In this article we survey recent advances in the mathematical and numerical analysis of QSP. In particular, we focus on its generalization beyond polynomials, the computational complexity of algorithms for phase factor evaluation, and the numerical stability of such algorithms. The resolution to some of these problems relies on an unexpected interplay between QSP, nonlinear Fourier analysis on SU(2)\mathrm{SU}(2)SU(2), fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.