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arXiv:2203.09989 · Quantum Science and Technology

Verification of colorable hypergraph states with stabilizer test

Hong Ge Tao, Xiaoqian Zhang, Lei Shao, Xiaoqing Tan

Published Nov 10, 2022• 3 citations
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Abstract

Abstract Many-body quantum states, as a matter of fact, are extremely essential to solve certain mathematical problems or simulate quantum systems in measurement-based quantum computation. However, how to verify large-scale quantum states, such as hypergraph states, is an exceedingly hard task for many-body quantum systems. Here, we propose a novel fault-tolerant solution for the verification of colorable hypergraph states by using the stabilizer test. Furthermore, our protocol is dramatically facilitated by making only Pauli-X and Pauli-Z measurements. For geometric structure hypergraph states, the computational complexity of our protocol is polynomial. As to appliance, it will be also applied to blind quantum computing based on the no-signaling principle.

DOI
10.1088/2058-9565/aca1d8

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Abstract Many-body quantum states, as a matter of fact, are extremely essential to solve certain mathematical problems or simulate quantum systems in measurement-based quantum computation. However, how to verify large-scale quantum states, such as hypergraph states, is an exceedingly hard task for many-body quantum systems. Here, we propose a novel fault-tolerant solution for the verification of colorable hypergraph states by using the stabilizer test. Furthermore, our protocol is dramatically facilitated by making only Pauli-X and Pauli-Z measurements. For geometric structure hypergraph states, the computational complexity of our protocol is polynomial. As to appliance, it will be also applied to blind quantum computing based on the no-signaling principle.
Hypergraph states are generalizations of graph states where controlled-ZZZ gates on edges are replaced with generalized controlled-ZZZ gates on hyperedges. Hypergraph states have several advantages over graph states. For example, certain hypergraph states, such as the Union Jack states, are universal resource states for measurement-based quantum computing with only Pauli measurements, while graph state measurement-based quantum computing needs non-Clifford basis measurements. Furthermore, it is impossible to classically efficiently sample measurement results on hypergraph states unless the polynomial hierarchy collapses to the third level. Although several protocols have been proposed to verify graph states with only sequential single-qubit Pauli measurements, there was no verification method for hypergraph states. In this paper, we propose a method for verifying a certain class of hypergraph states with only sequential single-qubit Pauli measurements. Importantly, no i.i.d. property of samples is assumed in our protocol: any artificial entanglement among samples cannot fool the verifier. As applications of our protocol, we consider verified blind quantum computing with hypergraph states, and quantum computational supremacy demonstrations with hypergraph states.
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.
Measurement-based quantum computing is one of the most promising quantum computing models. Although various universal resource states have been proposed so far, it was open whether only two Pauli bases are enough for both of universal measurement-based quantum computing and its verification. In this paper, we construct a universal hypergraph state that only requires X and Z-basis measurements for universal measurement-based quantum computing. We also show that universal measurement-based quantum computing on our hypergraph state can be verified in polynomial time using only X and Z-basis measurements. Furthermore, in order to demonstrate an advantage of our hypergraph state, we construct a verifiable blind quantum computing protocol that requires only X and Z-basis measurements for the client.
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.
Graph states known as the resource states for measurement-based quantum computation play an important role in quantum information processing. Verifying the correctness of graph states in an efficient way is crucial to scalable quantum computing. In this paper, we propose an efficient method for verifying two-colorable graph states with Pauli measurements, where the number of required measurement settings is a constant that is independent of the size of graph states. In addition, the number of required tests is less than the existing protocols. We also present several examples, such as the brickwork state and the two-dimensional square lattice state, to show how to get the specific verification strategy. Furthermore, we propose a robust verification of graph states for resisting the noise acting on the measurement device. Finally, based on the verification of two-colorable graph states in the adversarial scenario, we propose a verifiable blind quantum computing protocol, which can realize the less resource overhead.
Complex entangled states are the key resources for measurement-based quantum computations, which is realised by performing a sequence of measurements on initially entangled qubits. Executable quantum algorithms in the graph-state quantum computing model are determined by the entanglement structure and the connectivity of entangled qubits. By generalisation from graph-type entanglement in which only the nearest qubits interact to a new type of hypergraph entanglement in which any subset of qubits can be arbitrarily entangled via hyperedges, hypergraph states represent more general resource states that allow arbitrary quantum computation with Pauli universality. Here we report experimental preparation, certification and processing of complete categories of four-qubit hypergraph states under the principle of local unitary equivalence, on a fully reprogrammable silicon-photonic quantum chip. Genuine multipartite entanglement for hypergraph states is certificated by the characterisation of entanglement witness, and the observation of violations of Mermin inequalities without any closure of distance or detection loopholes. A basic measurement-based protocol and an efficient resource state verification by color-encoding stabilizers are implemented with local Pauli measurement to benchmark the building blocks for hypergraph-state quantum computation. Our work prototypes hypergraph entanglement as a general resource for quantum information processing.