Block encoding is a key ingredient in the recently developed quantum singular value transformation (QSVT) framework, which provides a unifying description for many quantum algorithms. Initially introduced to simplify and optimize resource utilization in various problems, such as searching, amplitude estimation, and Hamiltonian simulation, it is reasonable to expect that the capabilities of QSVT extend beyond these applications and offer untapped potential for designing new quantum algorithms. In this article, we affirm this perspective by leveraging block encoding to substantially enhance two previously proposed quantum algorithms: largest eigenvalue estimation and quantum gradient descent. Unlike previous works that rely on sophisticated approaches, our findings demonstrate that even just elementary operations within the unitary block encoding framework can eliminate major scaling factors present in their original counterparts. This results in significantly more efficient quantum algorithms capable of tackling target computational problems with remarkable efficiency. Furthermore, we illustrate how our proposed method can be extended to other contexts, including matrix inversion and multiple eigenvalue estimation.Block encoding is a key ingredient in the recently developed quantum singular value transformation (QSVT) framework, which provides a unifying description for many quantum algorithms. Initially introduced to simplify and optimize resource utilization in various problems, such as searching, amplitude estimation, and Hamiltonian simulation, it is reasonable to expect that the capabilities of QSVT extend beyond these applications and offer untapped potential for designing new quantum algorithms. In this article, we affirm this perspective by leveraging block encoding to substantially enhance two previously proposed quantum algorithms: largest eigenvalue estimation and quantum gradient descent. Unlike previous works that rely on sophisticated approaches, our findings demonstrate that even just elementary operations within the unitary block encoding framework can eliminate major scaling factors present in their original counterparts. This results in significantly more efficient quantum algorithms capable of tackling target computational problems with remarkable efficiency. Furthermore, we illustrate how our proposed method can be extended to other contexts, including matrix inversion and multiple eigenvalue estimation.

arXiv:2312.14786Dec 22, 20230 citations

Quantum techniques for eigenvalue problems

Dean Lee

arXiv:2307.03889The European Physical Journal ANov 20, 20234 citations

Improving quantum linear system solvers via a gradient descent perspective

Sander Gribling, Iordanis Kerenidis, Dániel Szilágyi

Solving systems of linear equations is one of the most important primitives in quantum computing that has the potential to provide a practical quantum advantage in many different areas, including in optimization, simulation, and machine learning. In this work, we revisit quantum linear system solvers from the perspective of convex optimization, and in particular gradient descent-type algorithms. This leads to a considerable constant-factor improvement in the runtime (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth). More precisely, we first show how the asymptotically optimal quantum linear system solver of Childs, Kothari, and Somma is related to the gradient descent algorithm on the convex function $\|A\vec x - \vec b\|_2^2$ : their linear system solver is based on a truncation in the Chebyshev basis of the degree- $(t-1)$ polynomial (in $A$ ) that maps the initial solution $\vec{x}_1 := \vec{b}$ to the $t$ -th iterate $\vec{x}_t$ in the basic gradient descent algorithm. Then, instead of starting from the basic gradient descent algorithm, we use the optimal Chebyshev iteration method (which can be viewed as an accelerated gradient descent algorithm) and show that this leads to considerable improvements in the quantum solver.Solving systems of linear equations is one of the most important primitives in quantum computing that has the potential to provide a practical quantum advantage in many different areas, including in optimization, simulation, and machine learning. In this work, we revisit quantum linear system solvers from the perspective of convex optimization, and in particular gradient descent-type algorithms. This leads to a considerable constant-factor improvement in the runtime (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth). More precisely, we first show how the asymptotically optimal quantum linear system solver of Childs, Kothari, and Somma is related to the gradient descent algorithm on the convex function $\|A\vec x - \vec b\|_2^2$: their linear system solver is based on a truncation in the Chebyshev basis of the degree-$(t-1)$ polynomial (in $A$) that maps the initial solution $\vec{x}_1 := \vec{b}$ to the $t$-th iterate $\vec{x}_t$ in the basic gradient descent algorithm. Then, instead of starting from the basic gradient descent algorithm, we use the optimal Chebyshev iteration method (which can be viewed as an accelerated gradient descent algorithm) and show that this leads to considerable improvements in the quantum solver.

arXiv:2109.04248Sep 9, 20211 citation

An improved noisy gradient descent bit-flipping decoding algorithm for LDPC codes

Jun Asatani, Yuya Kondo, Kengo Katayama +2 more

Gradient descent bit-flipping (GDBF) algorithm achieves good error performance for decoding low density parity check codes. In the GDBF algorithm, a search point is apt to be trapped in a local optimum, which causes performance degradation of the GDBF algorithm. To overcome this issue, the noisy GDBF algorithm introduces random perturbation into the inversion function of each symbol, which decreases the probability of being trapped in a local optimum. In this paper, we develop the noisy GDBF algorithm by enhancing the threshold adaptation in multi-bit flipping procedure to improve the BER performance without increasing the time complexity.Gradient descent bit-flipping (GDBF) algorithm achieves good error performance for decoding low density parity check codes. In the GDBF algorithm, a search point is apt to be trapped in a local optimum, which causes performance degradation of the GDBF algorithm. To overcome this issue, the noisy GDBF algorithm introduces random perturbation into the inversion function of each symbol, which decreases the probability of being trapped in a local optimum. In this paper, we develop the noisy GDBF algorithm by enhancing the threshold adaptation in multi-bit flipping procedure to improve the BER performance without increasing the time complexity.

International Symposium on Information Theory and its ApplicationsOct 1, 20162 citations

Calculating unknown eigenvalues with a quantum algorithm

Xiaoqi Zhou, Pruet Kalasuwan, Timothy C. Ralph +1 more

arXiv:1110.4276Nature PhotonicsFeb 22, 201358 citations

\|A\vec x - \vec b\|_2^2

: their linear system solver is based on a truncation in the Chebyshev basis of the degree-

(t-1)

polynomial (in

A

) that maps the initial solution

\vec{x}_1 := \vec{b}

to the

t

-th iterate

\vec{x}_t

in the basic gradient descent algorithm. Then, instead of starting from the basic gradient descent algorithm, we use the optimal Chebyshev iteration method (which can be viewed as an accelerated gradient descent algorithm) and show that this leads to considerable improvements in the quantum solver.