Quantum Amplitude Amplification and Estimation
Gilles Brassard, Peter Hoyer, Michele Mosca, Alain Tapp
TL;DR
The paper generalizes Grover search through amplitude amplification, enabling efficient discovery of a good x even when the initial success probability a is unknown or known. It introduces the amplitude amplification operator Q and shows how it rotates the state in a two-dimensional subspace to achieve a quadratic speedup, with special considerations when a is known versus unknown. It then presents amplitude estimation, combining Grover-like amplitude techniques with phase estimation to estimate the success probability a and count the number of solutions t with provable, near-optimal error bounds. Across heuristic-driven and black-box settings, the work provides tight quantum algorithms for searching and counting, significantly advancing quantum speedups beyond naive Grover search and enabling practical approximate counting with rigorous guarantees.
Abstract
Consider a Boolean function $χ: X \to \{0,1\}$ that partitions set $X$ between its good and bad elements, where $x$ is good if $χ(x)=1$ and bad otherwise. Consider also a quantum algorithm $\mathcal A$ such that $A |0\rangle= \sum_{x\in X} α_x |x\rangle$ is a quantum superposition of the elements of $X$, and let $a$ denote the probability that a good element is produced if $A |0\rangle$ is measured. If we repeat the process of running $A$, measuring the output, and using $χ$ to check the validity of the result, we shall expect to repeat $1/a$ times on the average before a solution is found. *Amplitude amplification* is a process that allows to find a good $x$ after an expected number of applications of $A$ and its inverse which is proportional to $1/\sqrt{a}$, assuming algorithm $A$ makes no measurements. This is a generalization of Grover's searching algorithm in which $A$ was restricted to producing an equal superposition of all members of $X$ and we had a promise that a single $x$ existed such that $χ(x)=1$. Our algorithm works whether or not the value of $a$ is known ahead of time. In case the value of $a$ is known, we can find a good $x$ after a number of applications of $A$ and its inverse which is proportional to $1/\sqrt{a}$ even in the worst case. We show that this quadratic speedup can also be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform amplitude estimation, a process that allows to estimate the value of $a$. We apply amplitude estimation to the problem of *approximate counting*, in which we wish to estimate the number of $x\in X$ such that $χ(x)=1$. We obtain optimal quantum algorithms in a variety of settings.