# Fast quantum algorithm for numerical gradient estimation.

@article{Jordan2005FastQA,
title={Fast quantum algorithm for numerical gradient estimation.},
author={Stephen P. Jordan},
journal={Physical review letters},
year={2005},
volume={95 5},
pages={
050501
}
}
• S. Jordan
• Published 25 May 2004
• Computer Science
• Physical review letters
Given a black box for f, a smooth real scalar function of d real variables, one wants to estimate [symbol: see text]f at a given point with n bits of precision. On a classical computer this requires a minimum of d + 1 black box queries, whereas on a quantum computer it requires only one query regardless of d. The number of bits of precision to which f must be evaluated matches the classical requirement in the limit of large n.
84 Citations
Quantum algorithms and lower bounds for convex optimization
• Computer Science
Quantum
• 2020
A quantum algorithm is presented that can optimize a convex function over an n-dimensional convex body using O~(n) queries to oracles that evaluate the objective function and determine membership in the conveX body.
A quantum algorithm for solving systems of nonlinear algebraic equations
• Computer Science
• 2019
A quantum algorithm to solve systems of nonlinear algebraic equations when the set of equations has many variables and not so high orders is proposed.
Simulated quantum computation of global minima
• Physics
• 2009
It is demonstrated that a modified Grover's quantum algorithm can be applied to real problems of finding a global minimum using modest numbers of quantum bits and combined with the classical Pivot method for global optimisation to treat larger systems.
Optimizing quantum optimization algorithms via faster quantum gradient computation
• Computer Science
SODA
• 2019
A quantum algorithm that computes the gradient of a multi-variate real-valued function by evaluating it at only a logarithmic number of points in superposition is developed, and it is shown that for low-degree multivariate polynomials the algorithm can provide exponential speedups compared to Jordan's algorithm in terms of the dimension $d. Near-optimal Quantum algorithms for multivariate mean estimation • Mathematics, Computer Science STOC • 2022 It is shown that, outside this low-precision regime, there does exist a quantum estimator that outperforms any classical estimator, and it is proved that the approximation error can be decreased by a factor of about the square root of the ratio between the dimension and the sample complexity. Optimizing a polynomial function on a quantum processor • Computer Science, Physics • 2021 This work develops this protocol and implements it on a quantum processor with limited resources to provide a faster approach to solving high-dimensional optimization problems and a subroutine for future practical quantum computers. Quantum gradient algorithm for general polynomials • Computer Science • 2020 This paper proposes a quantum gradient algorithm for optimizing general polynomials with the dressed amplitude encoding, aiming at solving fast-convergence polynmials problems within both time and memory consumption in$\mathcal{O}(poly (\log{d}))\$.
Quantum algorithmic differentiation
• Computer Science, Physics
Quantum Inf. Comput.
• 2021
This work presents an algorithm to perform algorithmic differentiation in the context of quantum computing, and presents two versions, one which is fully quantum and one which employees a classical step (hybrid approach).
Quantum algorithms for multivariate Monte Carlo estimation
• Computer Science, Mathematics
• 2021
Interestingly, this work designs quantum algorithms that provide a quadratic speed-up in query complexity with respect to the precision of the resulting estimates and shows that these complexities are asymptotically optimal in each of the authors' scenarios.
Quantum Probability Oracles & Multidimensional Amplitude Estimation
It is shown that all coordinates of p can be estimated up to error ε per coordinate using Õ ( 1 ε ) applications of Up and its inverse, which generalizes the normal amplitude estimation algorithm, which solves the problem for n = 2.

## References

Quantum Computation and Quantum Information
• Physics, Computer Science
• 2000
This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.