Low depth algorithms for quantum amplitude estimation
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by
Tudor Giurgica-Tiron, Iordanis Kerenidis, Farrokh Labib, Anupam Prakash, William Zeng
2022
Abstract
We design and analyze two new low depth algorithms for amplitude estimation
(AE) achieving an optimal tradeoff between the quantum speedup and circuit
depth. For β∈ (0,1], our algorithms require N= Õ( 1/ϵ^1+β) oracle calls and require the oracle to be called
sequentially D= O( 1/ϵ^1-β) times to perform amplitude
estimation within additive error ϵ. These algorithms interpolate
between the classical algorithm (β=1) and the standard quantum algorithm
(β=0) and achieve a tradeoff ND= O(1/ϵ^2). These algorithms
bring quantum speedups for Monte Carlo methods closer to realization, as they
can provide speedups with shallower circuits.
The first algorithm (Power law AE) uses power law schedules in the framework
introduced by Suzuki et al <cit.>. The algorithm works for β∈
(0,1] and has provable correctness guarantees when the log-likelihood function
satisfies regularity conditions required for the Bernstein Von-Mises theorem.
The second algorithm (QoPrime AE) uses the Chinese remainder theorem for
combining lower depth estimates to achieve higher accuracy. The algorithm works
for discrete β =q/k where k ≥ 2 is the number of distinct coprime
moduli used by the algorithm and 1 ≤ q ≤ k-1, and has a fully rigorous
correctness proof. We analyze both algorithms in the presence of depolarizing
noise and provide numerical comparisons with the state of the art amplitude
estimation algorithms.
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