This paper addresses the task of cross-platform verification, which involves comparing the output states of different quantum platforms using only local quantum operations and classical communication. The authors propose a novel protocol based on *Pauli sampling*, a subroutine that generates Pauli strings according to their weight in the expansion of a quantum state in the Pauli basis. They show that their protocols for both Pauli sampling and cross-platform verification are efficient for quantum states with low magic and entanglement (order \(O(\log n)\)). Conversely, they demonstrate super-polynomial lower bounds on the complexity of these tasks for states with \(\omega(\log(n))\) magic and entanglement. Interestingly, when considering states with real amplitudes, the requirements for cross-platform verification can be significantly relaxed. The paper begins by introducing the problem of distributed quantum inner product estimation (IP), where two parties, Alice and Bob, aim to estimate the overlap \(\text{tr}(\rho \sigma)\) of two unknown states \(\rho\) and \(\sigma\) using local quantum operations and classical communication. The authors provide both positive and negative results, including a no-go theorem for efficient Pauli sampling and distributed IP for a wide class of states, and efficient algorithms for these tasks for states with low magic and entanglement. The main contributions of the paper include: 1. **Pauli Sampling**: A novel algorithm for approximate Pauli sampling that is efficient for states with low magic and entanglement. 2. **Distributed Inner Product Estimation**: Efficient algorithms for distributed IP using Pauli sampling, particularly for states with low magic and entanglement. The authors also discuss the relationship between Pauli sampling and other quantum tasks, such as quantum learning and matrix product state (MPS) tomography, and highlight the advantages of their approach over existing methods. Finally, they outline future directions for research, including numerical performance studies and experimental implementation of their protocols.This paper addresses the task of cross-platform verification, which involves comparing the output states of different quantum platforms using only local quantum operations and classical communication. The authors propose a novel protocol based on *Pauli sampling*, a subroutine that generates Pauli strings according to their weight in the expansion of a quantum state in the Pauli basis. They show that their protocols for both Pauli sampling and cross-platform verification are efficient for quantum states with low magic and entanglement (order \(O(\log n)\)). Conversely, they demonstrate super-polynomial lower bounds on the complexity of these tasks for states with \(\omega(\log(n))\) magic and entanglement. Interestingly, when considering states with real amplitudes, the requirements for cross-platform verification can be significantly relaxed. The paper begins by introducing the problem of distributed quantum inner product estimation (IP), where two parties, Alice and Bob, aim to estimate the overlap \(\text{tr}(\rho \sigma)\) of two unknown states \(\rho\) and \(\sigma\) using local quantum operations and classical communication. The authors provide both positive and negative results, including a no-go theorem for efficient Pauli sampling and distributed IP for a wide class of states, and efficient algorithms for these tasks for states with low magic and entanglement. The main contributions of the paper include: 1. **Pauli Sampling**: A novel algorithm for approximate Pauli sampling that is efficient for states with low magic and entanglement. 2. **Distributed Inner Product Estimation**: Efficient algorithms for distributed IP using Pauli sampling, particularly for states with low magic and entanglement. The authors also discuss the relationship between Pauli sampling and other quantum tasks, such as quantum learning and matrix product state (MPS) tomography, and highlight the advantages of their approach over existing methods. Finally, they outline future directions for research, including numerical performance studies and experimental implementation of their protocols.