Low-depth amplitude estimation via statistical eigengap estimation

TL;DR

This work makes the key observation that amplitude estimation is equivalent to estimating the energy gap of an effective Hamiltonian, whereby discrete time evolution is generated by amplitude amplification, and develops two amplitude estimation algorithms for both Heisenberg-limited and low-depth circuit regimes.

Abstract

Amplitude estimation, in its original form, is formulated as phase estimation upon the Grover walk operator. Since its introduction, subsequent improvements to the algorithm have removed the use of phase estimation and introduced low-depth variants that trade speedup factors for lower circuit depth. We make the key observation that amplitude estimation is equivalent to estimating the energy gap of an effective Hamiltonian, whereby discrete time evolution is generated by amplitude amplification. This enables us to develop two amplitude estimation algorithms for both Heisenberg-limited and low-depth circuit regimes, inspired by statistical phase estimation techniques developed for seemingly unrelated early fault-tolerant ground-state energy estimation. Our approach has significant technical and practical benefits, and uses simplified classical post-processing compared to prior techniques -- our theoretical and numerical results indicate that we achieve state-of-the-art performance. Furthermore, while our approach achieves Heisenberg-limited scaling, we also establish optimal query-depth tradeoffs up to polylogarithmic factors in the low-depth regime with provable theoretical guarantees. Due to its flexibility, generality, and robustness, we expect our approach to be a key enabler for a broad range of early fault-tolerant applications.

Figures (11)

• Figure 1: Overview. We provide two algorithms for both Heisenberg-limited and low-depth amplitude estimation, utilizing techniques from early fault-tolerant statistical phase estimation algorithms. Both algorithms follow a simple three-step protocol. First, a series of $\{m\}$ is sampled from a truncated discrete Gaussian. Second, we execute amplitude amplification circuits with $m$ applications of $U_\psi$. Lastly, we construct a target function based on the measurement result for which the peak lies on an $\epsilon$-close estimate $\tilde{a}$ of the target amplitude $a= \lVert P\ket\psi\rVert^2$.
• Figure 2: Comparison of the circuit depth of near-Heisenberg-limited amplitude estimation algorithms (left) and low-depth amplitude estimation (right). GLSAE and GDMAE achieve state-of-the-art performance, comparable to ChebAE and CSAE. In the low-depth regime (right), we implement the case with circuit depth and sample complexity equivalent to uniform time sampling where $M, N \in \tilde{\mathcal{O}}(\epsilon^{-2/3})$. Our algorithm improves over prior art like Power law AE.
• Figure 3: Depth--query invariance of low-depth amplitude estimation. We show depth--query invariance of running GLSAE on low-depth amplitude amplification circuits. Varying the depth $\mathcal{D}$ and query $\mathcal{N}$ counts yields estimates of approximately identical precision as long as the product $\mathcal{D}\mathcal{N}$ is fixed.
• Figure 4: Regional functional properties of the periodic Gaussian. In the periodic Gaussian with standard derivation $T\ge2/\pi$, up to periodicity $2\pi$, the region $[\frac{1}{T}, 2\pi -\frac{1}{T}]$ (red) is convex (\ref{['lemCvxGauss']}), the region $[-\frac{1}{2T}, \frac{1}{2T}]$ (blue) is $\frac{T^2}{2}$-strongly concave (\ref{['lemStrongCvx']}), and the entire function is $T^2$-smooth (\ref{['lemSmoothGauss']}).
• Figure 5: Failure regions for GLSAE. We draft the ideal loss function for GLSAE without the measurement errors to show why GLSAE fails in regions close to 0 and $\pi/2$. In panel (a), we see that when plotted, the ideal loss function (black solid line) has symmetric minima at $\theta = \pm \lambda$ (blue dashed lines) formed from overlapping Gaussian peaks from the periodic Gaussians. In low depth situations, when the standard deviation $T$ of the discrete Gaussian is small, the minimum can still be found when $\lambda$ is far from either 0 or $\pi/2$ as shown in panel (b). However, when $\lambda$ is close to either 0 or $\pi/2$ (in the red shaded area), the minimum is not easy to discern, and worse so under noise. When the standard deviation $T$ of the discrete Gaussian is high enough, then the overlap can be reduced, and the invalid region (red shaded area) shrinks, such that we can still discern the peak even under noise, as known in panel (d).

Theorems & Definitions (37)

• proposition 0: Effective Hamiltonian of amplitude amplification
• proposition 0: Effective Hamiltonian eigengap estimation
• theorem 1: GLSAE
• corollary 1
• theorem 2: GDMAE
• proposition B.0: Effective Hamiltonian of amplitude amplification
• proof
• proposition B.0: Effective Hamiltonian eigengap estimation
• proof
• lemma C.1: Range of convexity of the periodic Gaussian