Towards Optimizing the Expected Performance of Sampling-Based Quantum-Inspired Algorithms

TL;DR

The paper tackles the bottlenecks of inner product estimation and sampling from a linear combination in sampling-based quantum-inspired algorithms by introducing $ ext{SQ}_p$ data structures that generalize the traditional $ ext{SQ}_2$ framework to arbitrary $p\ge 1$. It derives performance bounds for inner product estimation and sampling under $ ext{SQ}_p$, analyzes average-case behavior to argue that $ ext{SQ}_1$ often minimizes sample complexity, and validates these claims with numerical simulations on real data. The work further demonstrates practical gains by tightening the Direct Fidelity Estimation (DFE) measurement bounds, notably for the W state and general well-conditioned states. Overall, the results provide concrete guidance on choosing norm-based data structures to optimize high-dimensional linear-algebra operations in quantum-inspired ML and quantum-state verification tasks, with implications for recommendation systems and state tomography-like verification. The findings highlight the importance of norm selection in dequantized quantum algorithms and propose a pathway to more efficient implementations across high-dimensional tasks.

Abstract

Quantum-inspired classical algorithms has received much attention due to its exponential speedup compared to existing algorithms, under certain data storage assumptions. The improvements are noticeable in fundamental linear algebra tasks. In this work, we analyze two major subroutines in sampling-based quantum-inspired algorithms, specifically, inner product estimation and sampling from a linear combination of vectors, and discuss their possible improvements by generalizing the data structure. The idea is to consider the average behavior of the subroutines under certain assumptions regarding the data elements. This allows us to determine the optimal data structure, and the high-dimensional nature of data makes our assumptions reasonable. Experimental results from recommendation systems also highlight a consistent preference for our proposed data structure. Motivated by this observation, we tighten the upper bound on the number of required measurements for direct fidelity estimation. We expect our findings to suggest optimal implementations for various quantum and quantum-inspired machine learning algorithms that involve extremely high-dimensional operations, which has potential for many applications.

Figures (4)

• Figure 1: Input data $x$ is encoded as $\Phi(x)$, with the assumption that querying the elements of $\Phi(x)$ can be done efficiently (in sublinear time with respect to its length). The weight vector $w$ is stored as a BST structure supporting efficient sampling and updating operations. The goal is to efficiently estimate the inner product $f(x) = w \cdot \Phi(x)$ to a desired accuracy.
• Figure 2: BST data structures for (a) $x \in \mathbb{R}^4$ ($\mathrm{SQ}_p(x)$) and (b) $A \in \mathbb{R}^{4 \times 4}$ ($\mathrm{SQ}_p(A)$) based on $L^p$ norm.
• Figure 3: $\mathbb{E}[M(p)]$ vs. $p$ for $f = \mathcal{N}(0, 1)$, $m = 1024$, and $n \in \{ 2, 4, \cdots , 4096 \}$. Assuming $n \rightarrow \infty$ gives a positively biased estimate for small $n$.
• Figure 4: $\mathbb{E}[M(p)]$ vs. $p$ for $f = \mathcal{U}_{(-1, 1)}$, $m = 1024$, and $n \in \{ 2, 4, \cdots , 4096 \}$. Assuming $n \rightarrow \infty$ gives a negatively biased estimate for small $n$.

Theorems & Definitions (22)

• Definition 1
• Lemma 1: tang2019quantum, tang2019quantum
• Corollary 1: tang2019quantum, tang2019quantum
• Proposition 1: tang2019quantum, tang2019quantum
• Proposition 2: tang2019quantum, tang2019quantum
• Definition 2
• Remark 1
• Remark 2
• Remark 3
• Definition 3: Well-conditioned state flammia2011direct