Efficient verification of pure quantum states with applications to hypergraph states

Huangjun Zhu, Masahito Hayashi

Graph and hypergraph states are of wide interest in quantum information processing as well as fundamental physics, and efficient verification of these states is key to various applications. The authors propose a simple recipe for verifying hypergraph states that requires only two distinct Pauli measurements for each party, and is dramatically more efficient than conventional protocols based on local measurements. This approach enables verification of hypergraph states and genuine multipartite entanglement of thousands of qubits, even in an adversarial scenario.

Ankur Raina

Huangjun Zhu, Masahito Hayashi

Graph states and hypergraph states are of wide interest in quantum information processing and foundational studies. Efficient verification of these states is a key to various applications. Here we propose a simple method for verifying hypergraph states, which requires only two distinct Pauli measurements for each party, yet its efficiency is comparable to the best strategy based on entangling measurements. For a given state, the overhead is bounded by the chromatic number and degree of the underlying hypergraph. Our protocol is dramatically more efficient than all previous protocols based on local measurements, including tomography and direct-fidelity estimation. It enables the verification of hypergraph states and genuine multipartite entanglement of thousands of qubits. The protocol can also be generalized to the adversarial scenario, while achieving almost the same efficiency. This merit is particularly appealing to demonstrating blind measurement-based quantum computation and quantum supremacy.

Matthias P. Kl�y

Damian Markham

Huangjun Zhu, Masahito Hayashi

Bipartite and multipartite entangled states are of central interest in quantum information processing and foundational studies. Efficient verification of these states, especially in the adversarial scenario, is a key to various applications, including quantum computation, quantum simulation, and quantum networks. However, little is known about this topic in the adversarial scenario. Here we initiate a systematic study of pure-state verification in the adversarial scenario. In particular, we introduce a general method for determining the minimal number of tests required by a given strategy to achieve a given precision. In the case of homogeneous strategies, we can even derive an analytical formula. Furthermore, we propose a general recipe to verifying pure quantum states in the adversarial scenario by virtue of protocols for the nonadversarial scenario. Thanks to this recipe, the resource cost for verifying an arbitrary pure state in the adversarial scenario is comparable to the counterpart for the nonadversarial scenario, and the overhead is at most three times for high-precision verification. Our recipe can readily be applied to efficiently verify bipartite pure states, stabilizer states, hypergraph states, weighted graph states, and Dicke states in the adversarial scenario, even if only local projective measurements are accessible. This paper is an extended version of the companion paper Zhu and Hayashi, Phys. Rev. Lett. 123, 260504 (2019).