[Paper Review] Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements

Yuki Takeuchi, Tomoyuki Morimae+1 more|arXiv (Cornell University)|Sep 20, 2018

Quantum Computing Algorithms and Architecture33 references3 citations

TL;DR

This paper proposes a universal hypergraph state that enables deterministic, universal measurement-based quantum computing using only Pauli-X and Z basis measurements. It achieves both universality and efficient verification under the same measurement constraints, surpassing prior resource states that required additional measurement bases or lacked efficient verifiability.

ABSTRACT

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.

Motivation & Objective

• To construct a universal resource state for measurement-based quantum computing that requires only Pauli-X and Z basis measurements.
• To demonstrate that universal quantum computation can be verified in polynomial time using only X and Z basis measurements on the proposed hypergraph state.
• To enable a verifiable blind quantum computing protocol where the client performs only X and Z basis measurements.
• To show that a hypergraph state can achieve both universality and efficient verifiability with minimal measurement bases, improving upon prior graph and weighted graph states.

Proposed method

• The universal hypergraph state |Gₙ^d⟩ is constructed using a three-step process involving hypergraph states with CZ and CCZ entangling gates.
• The state is defined via a lexicographic ordering of triples (i,j,k) with 1≤i<j<k≤n, assigning a unique index t using a derived formula involving summations over (n−l) terms.
• The construction ensures that the state supports universal quantum computation via adaptive X and Z basis measurements alone.
• Verification is achieved by measuring each qubit in X or Z basis and checking stabilizer conditions, enabling polynomial-time verification.
• The protocol leverages the fact that stabilizer operators of the hypergraph state are real linear combinations of X and Z tensor products, enabling efficient verification with only these bases.
• A verifiable blind quantum computing protocol is designed where the client only needs to perform X and Z basis measurements, ensuring security and efficiency.

Experimental results

Research questions

• RQ1 Can a universal resource state be constructed that supports universal measurement-based quantum computing using only Pauli-X and Z basis measurements?
• RQ2 Is it possible to verify the correctness of such a universal resource state efficiently using only X and Z basis measurements?
• RQ3 Can a verifiable blind quantum computing protocol be realized where the client performs only X and Z basis measurements?
• RQ4 How does the complexity of the resource state relate to the number of required measurement bases for universality and verifiability?

Key findings

• The proposed hypergraph state |Gₙ^d⟩ enables universal quantum computation via adaptive X and Z basis measurements alone, achieving deterministic universality.
• The state is efficiently verifiable in polynomial time using only X and Z basis measurements, a property not known to hold for the Mølmer-Sørensen weighted graph state under the same constraints.
• The construction uses only CZ and CCZ entangling gates, and the resource state is a hypergraph state generalizing graph states.
• The method provides a verifiable blind quantum computing protocol where the client performs only X and Z basis measurements, enhancing practicality for cloud quantum computing.
• The hypergraph state achieves both universality and verifiability with only two measurement bases, outperforming prior universal resource states requiring three or more bases.
• The stabilizer structure of the state allows verification through real linear combinations of X and Z tensor products, which is essential for efficient verification with minimal measurement bases.