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arXiv:2004.11086 · Physical Review A

Quantum gradient algorithm for general polynomials

Pan Gao, Keren Li, Shijie Wei, Jiancun Gao, Guilu Long

Published Apr 1, 2021• 14 citations
arXivHTMLPDF

Abstract

Gradient-based algorithms provide the fundamental tools for the optimizations and are the essential components of many modern technologies. Theoretically, the extreme points of a cost function can be found by iterative descent(ascent) of the variable along the direction of the gradient. However, the time complexity to calculate a gradient is at a level of $O(poly(d))$ for the $d$-dimension problems. For processing the high-dimension case, especially modern machine-learning engineering with the number of optimized parameters being in billions, quantum techniques are introduced. Via the dressed amplitude encoding and nonunitary module, a gradient-based quantum optimization algorithm for general polynomials is proposed, aiming at solving the fast-convergence problems with both time and memory complexity in $O(poly(logd))$. Furthermore, numerical simulations are carried out to inspect the performance of the newly born protocol by considering the noises or perturbations during the initialization, operations, and truncations. For the potential values in high-dimension optimizations, this gradient-based quantum optimization algorithm is supposed to facilitate the polynomial optimizations, being a subroutine for future practical quantum computers.

DOI
10.1103/physreva.103.042403

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Gradient-based algorithms provide the fundamental tools for the optimizations and are the essential components of many modern technologies. Theoretically, the extreme points of a cost function can be found by iterative descent(ascent) of the variable along the direction of the gradient. However, the time complexity to calculate a gradient is at a level of O(poly(d))O(poly(d))O(poly(d)) for the ddd-dimension problems. For processing the high-dimension case, especially modern machine-learning engineering with the number of optimized parameters being in billions, quantum techniques are introduced. Via the dressed amplitude encoding and nonunitary module, a gradient-based quantum optimization algorithm for general polynomials is proposed, aiming at solving the fast-convergence problems with both time and memory complexity in O(poly(logd))O(poly(logd))O(poly(logd)). Furthermore, numerical simulations are carried out to inspect the performance of the newly born protocol by considering the noises or perturbations during the initialization, operations, and truncations. For the potential values in high-dimension optimizations, this gradient-based quantum optimization algorithm is supposed to facilitate the polynomial optimizations, being a subroutine for future practical quantum computers.
Abstract The gradient descent method is central to numerical optimization and is the key ingredient in many machine learning algorithms. It promises to find a local minimum of a function by iteratively moving along the direction of the steepest descent. Since for high-dimensional problems the required computational resources can be prohibitive, it is desirable to investigate quantum versions of the gradient descent, such as the recently proposed (Rebentrost et al. 1 ). Here, we develop this protocol and implement it on a quantum processor with limited resources. A prototypical experiment is shown with a four-qubit nuclear magnetic resonance quantum processor, which demonstrates the iterative optimization process. Experimentally, the final point converged to the local minimum with a fidelity >94%, quantified via full-state tomography. Moreover, our method can be employed to a multidimensional scaling problem, showing the potential to outperform its classical counterparts. Considering the ongoing efforts in quantum information and data science, our work may provide a faster approach to solving high-dimensional optimization problems and a subroutine for future practical quantum computers.
Abstract Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton’s method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.
Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.
We consider a generic framework of optimization algorithms based on gradient descent. We develop a quantum algorithm that computes the gradient of a multi-variate real-valued function f : Rd → R by evaluating it at only a logarithmic number of times in superposition. Our algorithm is an improved version of Jordan's gradient computation algorithm [28], providing an approximation of the gradient ▽f with quadratically better dependence on the evaluation accuracy of f, for an important class of smooth functions. Furthermore, we show that objective functions arising from variational quantum circuits usually satisfy the necessary smoothness conditions, hence our algorithm provides a quadratic improvement in the complexity of computing their gradient. We also show that in a continuous phase-query model, our gradient computation algorithm has optimal query complexity up to poly-logarithmic factors, for a particular class of smooth functions. Moreover, we show that for low-degree multivariate polynomials our algorithm can provide exponential speedups compared to Jordan's algorithm in terms of the dimension d.One of the technical challenges in applying our gradient computation procedure for quantum optimization problems is the need to convert between a probability oracle (which is common in quantum optimization procedures) and a phase oracle (which is common in quantum algorithms) of the objective function f. We provide efficient subroutines to perform this delicate interconversion between the two types of oracles incurring only a logarithmic overhead, which might be of independent interest. Finally, using these tools we improve the runtime of prior approaches for training quantum auto-encoders, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA).
We consider a generic framework of optimization algorithms based on gradient descent. We develop a quantum algorithm that computes the gradient of a multi-variate realvalued function f : ℝd → ℝ by evaluating it at only a logarithmic number of times in superposition. Our algorithm is an improved version of Jordan's gradient computation algorithm [28], providing an approximation of the gradient ▽f with quadratically better dependence on the evaluation accuracy of f, for an important class of smooth functions. Furthermore, we show that objective functions arising from variational quantum circuits usually satisfy the necessary smoothness conditions, hence our algorithm provides a quadratic improvement in the complexity of computing their gradient. We also show that in a continuous phase-query model, our gradient computation algorithm has optimal query complexity up to poly-logarithmic factors, for a particular class of smooth functions. Moreover, we show that for low-degree multivariate polynomials our algorithm can provide exponential speedups compared to Jordan's algorithm in terms of the dimension d. One of the technical challenges in applying our gradient computation procedure for quantum optimization problems is the need to convert between a probability oracle (which is common in quantum optimization procedures) and a phase oracle (which is common in quantum algorithms) of the objective function f. We provide efficient subroutines to perform this delicate interconversion between the two types of oracles incurring only a logarithmic overhead, which might be of independent interest. Finally, using these tools we improve the runtime of prior approaches for training quantum auto-encoders, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA).
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 ∥Ax⃗−b⃗∥22\|A\vec x - \vec b\|_2^2∥Ax−b∥22​: their linear system solver is based on a truncation in the Chebyshev basis of the degree-(t−1)(t-1)(t−1) polynomial (in AAA) that maps the initial solution x⃗1:=b⃗\vec{x}_1 := \vec{b}x1​:=b to the ttt-th iterate x⃗t\vec{x}_txt​ 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.