Low depth algorithms for quantum amplitude estimation (2012.03348v2)

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 \cite{S20}. 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.

• The paper introduces two novel algorithms, Power Law AE and QoPrime AE, designed to balance quantum speedup with reduced circuit depth for Quantum Amplitude Estimation on noisy near-term quantum devices.
• Power Law AE uses a tunable parameter $eta$ to smoothly interpolate between classical and standard quantum approaches, achieving a balance where total oracle calls times maximum depth equals $O(1/ obreakspace ext{epsilon}^2)$.
• QoPrime AE employs number-theoretic techniques and the Chinese Remainder Theorem with coprime moduli to significantly reduce circuit depth to $O(1/ obreakspace ext{epsilon}^{1-q/k})$.

Analysis of Low Depth Algorithms for Quantum Amplitude Estimation

The paper examines the development and analysis of two innovative algorithms tailored for Quantum Amplitude Estimation (QAE), highlighting their ability to balance quantum speedup with reduced circuit depth. These algorithms are framed as solutions to the limitations posed by current noisy quantum devices, known as NISQ devices, aiming to optimize the performance of quantum algorithms under practical constraints.

Overview of Proposed Algorithms

The authors introduce two distinct algorithms: Power Law Amplitude Estimation (AE) and QoPrime AE. Each is designed to provide quantum speedups while minimizing the number of sequential calls to the quantum oracle—a critical factor given the noise thresholds of NISQ devices.

1. Power Law AE: This algorithm leverages power-law schedules, modifying the depth of circuit executions in a manner sensitive to a tunable parameter, $\beta$. By adjusting $\beta$, users can interpolate between classical ($\beta=1$) and standard quantum ($\beta=0$) amplitude estimation methods. The algorithm achieves a balance, where $N \cdot D = O(1/\epsilon^2)$, with $N$ being the total oracle calls and $D$ the maximum circuit depth.
2. QoPrime AE: Utilizing number-theoretic techniques, this algorithm combines low-depth estimates through the Chinese Remainder Theorem to achieve greater accuracy. It further adapts the standard AE approach by corralling estimates with coprime moduli. This effectively reduces circuit depth to $O(1/\epsilon^{1-q/k})$, using $O(1/\epsilon^{1+q/k})$ oracle calls in total.

Numerical and Theoretical Insights

The algorithms are examined for their performance under depolarizing noise conditions, typical of NISQ environments. Theoretically, both Power Law and QoPrime AE are constructed to transition smoothly from quantum to classical regimes as errors $\epsilon$ cross a threshold relative to the noise rate. This adaptive scaling is confirmed by benchmarks against existing algorithms like IQAE, showing a significant improvement in terms of oracle call efficiency under noise.

Implications for Quantum Computing

By creating algorithms that adapt to both quantum advantages and classical limitations, the authors address critical requirements for making quantum computational techniques more applicable to near-term devices. The approaches could fundamentally enhance quantum Monte Carlo methods and other applications in machine learning and quantum finance. The reduced depth requirements mean real-world applications of quantum speedup in computational tasks could occur sooner than expected.

Future Directions

The research opens pathways for further refinement of QAE algorithms, particularly in optimizing $\beta$ dynamically based on noise levels or refining the coprime selection for QoPrime AE. Exploring these areas will enhance algorithm robustness to noise and other real-world computational challenges. Moreover, the integration of such advanced QAE techniques into broader computational models may yield new insights into complex problem-solving across various quantum computing fields.

In conclusion, the paper illustrates a substantial theoretical and practical advance in making quantum algorithms more accessible without sacrificing their advantages, reinforcing the potential for immediate applications of quantum computing technologies.