Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states

Abstract

While the circuit model of quantum computation defines its logical depth or "computational time" in terms of temporal gate sequences, the measurement-based model could allow totally different temporal ordering and parallelization of logical gates. By developing techniques to analyze Pauli measurements on multi-qubit hypergraph states generated by the Controlled-Controlled-Z (CCZ) gates, we introduce a deterministic scheme of universal measurement-based computation. In contrast to the cluster-state scheme where the Clifford gates are parallelizable, our scheme enjoys massive parallelization of CCZ and SWAP gates, so that the computational depth grows with the number of global applications of Hadamard gates, or, in other words, with the number of changing computational bases. A logarithmic-depth implementation of an N -times Controlled-Z gate illustrates a novel trade-off between space and time complexity.

General framework for verifying pure quantum states in the adversarial scenario

“…The four figures of merit defined in Eq. (20) are tied to the two-dimensional region composed of all the points (p ρ , f ρ ) for density matrices ρ, that is,…”

“…In this section we develop a general method for computing the figures of merit defined in Eq. (20), which characterize the verification precision in the adversarial scenario. We also clarify the properties of these figures of merit in preparation for latter study.…”

“…The same conclusion also applies to the figures of merit F (N, δ, Ω), F (N, f, Ω), ζ(N, δ, Ω), and η(N, f, Ω) defined in Eq. (20) given that they are completely determined by the region R N,Ω . For example, F (N, δ, Ω) corresponds to the lower boundary of the intersection of R N,Ω and the vertical line p ρ = δ as long as δ ≥ τ N .…”

“…Next, we introduce alternative definitions of the figures of merit defined in Eq. (20), which are easier to analyze and compute. Definẽ…”

“…For example, bipartite entangled states, especially maximally entangled states, are crucial to quantum teleportation, dense coding, and quantum cryptography [2,3]. Multipartite entangled states, such as graph states [4] and hypergraph states [5][6][7][8][9] are especially useful in (blind) measurement-based quantum computation (MBQC) [10][11][12][13][14][15][16][17][18][19][20], quantum error correction [21,22], quantum networks [23][24][25], and foundation studies [26][27][28][29]. Another important class of multipartite states, including Dicke states [30,31], are useful in quantum metrology [32].…”