# Low depth algorithms for quantum amplitude estimation

Tudor Giurgica-Tiron2,3, Iordanis Kerenidis1,5, Farrokh Labib2,4, Anupam Prakash1, and William Zeng2

1QC Ware Corp., Palo Alto, USA and Paris, France.
2Goldman, Sachs & Co., New York, USA.
3Stanford University, Palo Alto, USA.
4CWI Amsterdam, Netherlands.
5CNRS, Université Paris, France.

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### 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 $\beta \in (0,1]$, our algorithms require $N= \tilde{O}( \frac{1}{ \epsilon^{1+\beta}})$ oracle calls and require the oracle to be called sequentially $D= O( \frac{1}{ \epsilon^{1-\beta}})$ times to perform amplitude estimation within additive error $\epsilon$. These algorithms interpolate between the classical algorithm $(\beta=1)$ and the standard quantum algorithm ($\beta=0$) and achieve a tradeoff $ND= O(1/\epsilon^{2})$. 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 [24]. The algorithm works for $\beta \in (0,1]$ 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 $\beta =q/k$ where $k \geq 2$ is the number of distinct coprime moduli used by the algorithm and $1 \leq q \leq k-1$, 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.

Amplitude estimation (AE) is one of the fundamental quantum algorithms that enables quantum computers to achieve a quadratic speedup over classical algorithms for several statistical estimation tasks. Amplitude estimation also underlies quantum speedups for quantum Monte Carlo methods.

The standard AE algorithm requires $O(1/\epsilon)$ queries to an oracle circuit that is sequentially run $O(1/\epsilon)$ times in order to obtain an $O(\epsilon)$ accuracy. In this work, we address the question of the speedups in the near term setting where the quantum computer is limited in the depth for queries made to the oracle. We a provide provably correct algorithm that achieves a tradeoff of the form $O(ND) = O(1/\epsilon^{2})$ where $N$ is the total number of oracle queries and $D$ the maximum depth of the query.

The results demonstrate the possibility of a $D$-fold speedup over classical algorithms when the quantum computer is able to query the oracle at depth $D$. The new algorithmic idea is to leverage the Chinese remainder theorem to boost the precision of the AE estimates.

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[3] Bo Yang, Rudy Raymond, and Shumpei Uno, "Efficient quantum readout-error mitigation for sparse measurement outcomes of near-term quantum devices", Physical Review A 106 1, 012423 (2022).

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[12] M. C. Braun, T. Decker, N. Hegemann, and S. F. Kerstan, "Error Resilient Quantum Amplitude Estimation from Parallel Quantum Phase Estimation", arXiv:2204.01337.

[13] Zhicheng Zhang, Qisheng Wang, and Mingsheng Ying, "Parallel Quantum Algorithm for Hamiltonian Simulation", arXiv:2105.11889.

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[17] Tomoki Tanaka, Shumpei Uno, Tamiya Onodera, Naoki Yamamoto, and Yohichi Suzuki, "Noisy quantum amplitude estimation without noise estimation", Physical Review A 105 1, 012411 (2022).

[18] Koichi Miyamoto, "Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation", arXiv:2108.09014.

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