A Shapley Value Estimation Speedup for Efficient Explainable Quantum AI

TL;DR

The paper addresses the computational bottleneck of Shapley-value explanations in AI, including quantum models, by introducing quantum algorithms that estimate Shapley values with a quadratic speedup over classical Monte Carlo methods (up to polylog factors). It develops a quantum framework that encodes coalitions and value functions, leveraging amplitude estimation and partition-based sampling to obtain $\Phi(i)$ efficiently, with rigorous error and complexity analysis. The authors demonstrate concrete examples on weighted voting games and quantum binary classifiers, and propose an improved method using Quantum CORDIC to avoid expensive partitions, broadening applicability to quantum explainability. Overall, the work provides a principled, scalable approach to additive explanations in quantum AI and related cooperative games, with potential impact on both theoretical and applied explainability tasks.

Abstract

This work focuses on developing efficient post-hoc explanations for quantum AI algorithms. In classical contexts, the cooperative game theory concept of the Shapley value adapts naturally to post-hoc explanations, where it can be used to identify which factors are important in an AI's decision-making process. An interesting question is how to translate Shapley values to the quantum setting and whether quantum effects could be used to accelerate their calculation. We propose quantum algorithms that can extract Shapley values within some confidence interval. Our method is capable of quadratically outperforming classical Monte Carlo approaches to approximating Shapley values up to polylogarithmic factors in various circumstances. We demonstrate the validity of our approach empirically with specific voting games and provide rigorous proofs of performance for general cooperative games.

Figures (4)

• Figure 1: This circuit $R_j$ is a controlled rotation of the $j$th player qubit, where $R_y(\theta) = (\cos(\theta/2), -\sin(\theta/2);\sin(\theta/2), \cos(\theta/2))$. (Note: Library used for visualizing circuits can be found here in Ref. kay_2023).
• Figure 2: Visual representation of $\beta_{n,m}$ being approximated using Riemann sums of function $b_{n,m}(x)=x^m(1-x)^{n-m}$ over partition $P_\ell$, $t \in [0,1]$, $n=4$, $m=1$. The $k^\text{th}$ rectangle's height is $b_{n,m}(t'_\ell(k))=(t'_\ell(k))^m(1-~t'_\ell(k))^{n-m}$, and its width is $w_\ell(k)$.
• Figure 3: Circuit of $U_V^\pm$ for a weighted voting game. This circuit takes an basis state input $\ket {h_n, \dots, h_{i+1}, x, h_{i-1}, \dots, h_0}$ and outputs $\ket{V^-(S_h)}$ when $x=0$ or $\ket{V^+(S_h)}$ when $x=1$ to the utility register (Recall, $S_h$ is defined in Definition \ref{['def:S_h']}). The auxiliary register contains the total vote count. Just before the $\geq q$ gate, the $Aux$ register is in a basis state corresponding to the vote count of $S_h$, including or excluding player $i$s vote depending on $x$. The $\geq q$ gate uses the auxiliary register as an input and outputs whether the vote count exceeds threshold $q$ in the Ut register. After this gate, it is trivial to clear the $Aux$ register by subtracting each player's contribution. Results and simulation code are available in a https://github.com/iain-burge/QuantumShapleyValueAlgorithmgithubEntry.
• Figure 4: This figure demonstrates the exponentially small error introduced in Step 1 with respect to $\ell$. We generated $64$ random weighted voting games for each condition, e.g., for each combination of $\ell$ and number of players. Random weights $w_j\in \mathbb{N}$ were assigned for each case such that $q\leq \sum_j w_j < 2q$. There were four primary scenarios: (1) Four players, voting threshold $q=8$; (2) Six players, voting threshold $q=16$; (3) Eight players, voting threshold $q=32$; and (4) Ten players, voting threshold $q=32$. We approximated every player's Shapley value for each scenario with our quantum algorithm using various $\ell$'s, taking only Step 1's error into account. Next, we found the absolute error of our approximation by comparing each approximated Shapley value to its true value. The graphs show the mean and maximum absolute errors of each condition. Results and simulation code are available in a https://github.com/iain-burge/QuantumShapleyValueAlgorithmgithubEntry.

Theorems & Definitions (40)

• Definition 1: Coalitional game
• Definition 2: Payoff vector
• Definition 3: Shapley value shapley1952value
• Remark 1
• Lemma 1
• Definition 4: Monotonic game
• Definition 5
• Theorem 1
• Theorem 2
• Definition 6: Modified Beta Function