Hamiltonian singular value transformation and inverse block encoding

Lloyd, Seth; Kiani, Bobak T.; Arvidsson-Shukur, David R. M.; Bosch, Samuel; Palma, Giacomo De; Kaminsky, William M.; Liu, Ziwen; Marvian, Milad

doi:10.48550/arxiv.2104.01410

Cited by 4 publications

(6 citation statements)

References 14 publications

(23 reference statements)

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“…The curious ability to locate invariant qubit-like subspaces in larger Hilbert spaces and perform QSP simultaneously within them, obliviously to the eigenbases or singular vector bases of these subsystems as well as the eigenvalues or singular values, led to the far expanded QSVT [1], whose uses, robustness, and applications [4,1] have been recently explored. Finally, rephrasing QSVT in terms of Hamiltonian simulation [5] has both simplified the presentation and in some ways brought this algorithmic story full circle. Ongoing work continues to simplify the presentation of these algorithms.…”

Section: Prior Workmentioning

confidence: 99%

“…Much of the interest in QSP-like algorithms stems from their use at the core of algorithms for manipulating the eigenvalues or singular values of larger linear systems embedded in unitary matrices [1,5,13]. QSP can be thought of as the special case in which this linear operator is just the single scalar value in the top left of a representation of an SU(2) operator.…”

Section: Lifting M-qsp To M-qsvtmentioning

confidence: 99%

“…In addition to improving the performance of many known quantum algorithms [2,3], QSVT has great explanatory utility: unifying the presentation of most major known quantum algorithms [4]. This includes Hamiltonian simulation [5,3], search [1], phase estimation [2], quantum walks [1], fidelity estimation [6], sophisticated techniques for measurement [7], and channel discrimination [8]. QSVT has found purchase in surprisingly disparate subfields, from undergirding a general theory of quantum-inspired classical algorithms for low-rank machine learning [9], to quantum cryptographic protocols with zero-knowledge properties [10].…”

Section: Introductionmentioning

confidence: 99%

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Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Rossi

Chuang

2022

Quantum

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Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, saying nothing about computing joint properties of two or more oracles; these can be far cheaper to determine given an ability to pit oracles against one another coherently. This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons, and remain numerically stable and efficient. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms, and provides a bridge from quantum algorithms to algebraic geometry.

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Section: Prior Workmentioning

confidence: 99%

Section: Lifting M-qsp To M-qsvtmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Rossi

Chuang

2022

Quantum

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“…Alternatively, the methods discussed in Ref. [43] or quantum singular value transformation [18,19,44] can be applied. The techniques to realize f (H) on a quantum system are also summarized in Ref.…”

Section: Construction Of Appropriate Transformations Of Hamiltonianmentioning

confidence: 99%

A Gauss-Newton based Quantum Algorithm for Combinatorial Optimization

Takeori¹,

Yamamoto²,

Ohira³

et al. 2022

Preprint

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In this work, we present a Gauss-Newton based quantum algorithm (GNQA) for combinatorial optimization problems that rapidly converges to one of the optimal solutions without being trapped in local minima or plateaus. Quantum optimization algorithms have been explored for decades, but more recent investigations have been on variational quantum algorithms, which often suffer from the aforementioned problems. Our approach mitigates those by employing a tensor product state that accurately represents the optimal solution, and an appropriate function for the Hamiltonian, containing all the combinations of binary variables. Numerical experiments presented here demonstrate the effectiveness of our approach, and they show that GNQA outperforms other optimization methods in both convergence properties and accuracy for all problems considered here. Finally, we briefly discuss the potential impact of the approach to other problems, including those in quantum chemistry and higher order binary optimization.

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“…Quantum computing has been applied in many important tasks, including breaking encryption [1], searching databases [2], and simulating quantum evolution [3]. Recent advances in quantum computing show that quantum singular value transformation (QSVT) introduced by Gilyén et al [4] has led to a unified framework of the most known quantum algorithms [5], including amplitude amplification [4], quantum walks [4], phase estimation [5,6], and Hamiltonian simulations [7][8][9][10]. This framework can further be used to develop new quantum algorithms such as quantum entropies estimation [11][12][13], fidelity estimation [14], ground state preparation and ground energy estimation [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Quantum Phase Processing: Transform and Extract Eigen-Information of Quantum Systems

Wang¹,

Wang²,

Yu³

et al. 2022

Preprint

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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 a new algorithmic framework "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 eigen-information 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 new quantum phase estimation algorithm without quantum Fourier transform, which requires the least ancilla qubits and matches the best performance so far. We further exploit the power of our QPP framework by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy, and quantum entropies estimation, demonstrating improvements or optimality for almost all cases.

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Hamiltonian singular value transformation and inverse block encoding

Cited by 4 publications

References 14 publications

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

A Gauss-Newton based Quantum Algorithm for Combinatorial Optimization

Quantum Phase Processing: Transform and Extract Eigen-Information of Quantum Systems

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