Amplitude amplification is a central tool used in Grover's quantum search algorithm and has been used in various forms in numerous quantum algorithms since then. It has been shown to completely eliminate one-sided error of quantum search algorithms where input is accessed in the form of black-box queries. We generalize amplitude amplification for two-sided error quantum algorithm for decision problems in the familiar form where input is accessed in the form of initial states of quantum circuits and where arbitrary projective measurements may be used to ascertain success or failure. This generalization allows us to derive interesting applications of amplitude amplification for distinguishing between two given distributions based on their samples, detection of faults in quantum circuits and eliminating error of one and two-sided quantum algorithms with exact errors.
Amplitude amplification is a central tool used in Grover's quantum search algorithm and has been used in various forms in numerous quantum algorithms since then. It has been shown to completely eliminate one-sided error of quantum search algorithms where input is accessed in the form of black-box queries. We generalize amplitude amplification for two-sided error quantum algorithm for decision problems in the familiar form where input is accessed in the form of initial states of quantum circuits and where arbitrary projective measurements may be used to ascertain success or failure. This generalization allows us to derive interesting applications of amplitude amplification for distinguishing between two given distributions based on their samples, detection of faults in quantum circuits and eliminating error of one and two-sided quantum algorithms with exact errors.
Grover's quantum search and its generalization, quantum amplitude amplification, provide quadratic advantage over classical algorithms for a diverse set of tasks, but are tricky to use without knowing beforehand what fraction λ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction, but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability, and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of λ.
Grover's quantum search and its generalization, quantum amplitude amplification, provide a quadratic advantage over classical algorithms for a diverse set of tasks but are tricky to use without knowing beforehand what fraction λ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of λ.
We present a technique to reduce the error probability of quantum algorithms that determine whether its input has a specified property of interest. The standard process of reducing this error is statistical processing of the results of multiple independent executions of an algorithm. Denoting by $\ensuremath{\rho}$ an upper bound of this probability (wlog; assume $\ensuremath{\rho}\ensuremath{\le}\frac{1}{2}$), classical techniques require $O(\frac{\ensuremath{\rho}}{{[(1\ensuremath{-}\ensuremath{\rho})\ensuremath{-}\ensuremath{\rho}]}^{2}})$ executions to reduce the error to a negligible constant. We investigate when and how quantum algorithmic techniques like amplitude amplification and estimation may reduce the number of executions. On one hand, the former idea does not directly benefit algorithms that can err on both yes and no answers and the number of executions in the latter approach is $O(\frac{1}{(1\ensuremath{-}\ensuremath{\rho})\ensuremath{-}\ensuremath{\rho}})$. We propose a quantum approach called amplitude separation that combines both these approaches and achieves $O(\frac{1}{\sqrt{1\ensuremath{-}\ensuremath{\rho}}\ensuremath{-}\sqrt{\ensuremath{\rho}}})$ executions, which betters existing approaches when the errors are high. In the multiple-weight decision problem, the input is an n-bit Boolean function f() given as a black box and the objective is to determine the number of x for which f(x)=1, denoted wt(f), given some possible values w1,...,wk for wt(f). When our technique is applied to this problem, we obtain the correct answer, maybe with a negligible error, using O(log2k2n) calls to f(), which shows a quadratic speedup over classical approaches and currently known quantum algorithms.
We present a family of quantum amplitude amplification algorithms that interleave ordinary Grover rotations with tunable weak measurements in loops. A successful measurement yields the target state and stops the algorithm, while after a failure, the loop resumes from the post-measurement state without restarting. We express the leading costs in expected Grover iterations as p→0. For a known target probability p, an exact weak measurement-conditioned loop for states near the angle π/4 uses (π/8+1/4+o(1))/p iterations, improving on the standard Grover constant π/4 and on optimised restart Grover, and matching the corresponding infimum in our continuous-angle analysis. When only a lower bound 0<p0≤p<1/2 is available, a discrete Lyapunov equation gives the exact expected oracle cost for every fixed measurement strength and a closed-form optimal strength. Using the strength selected from p0, the expected number of Grover iterations is at most 221(p1+p01), where the expected cost decreases as the actual p increases, and gives the leading constant 1/2 at the promise boundary p=p0. For completely unknown p, we identify measurement strength κ(t)=Θ(1/t) as the critical scale within the regular schedules considered here and analyse κb(t)=min{1/2,b/t}. For each fixed b>2, we rigorously derive a closed-form Gamma-function expression C(b) giving (C(b)/2+o(1))/p expected iterations. Numerical minimisation of this explicit expression gives b≈5.2 and C(b)/2≈1.01. The results show that weak measurements preserve the Θ(1/p) search scale while adding an explicit fixed-point control mechanism and provable oracle savings.