Hierarchy of universal entanglement in 2D measurement-based quantum computation

Abstract

Measurement-based quantum computation (MQC) is a paradigm for studying quantum computation using many-body entanglement and single-qubit measurements. Although MQC has inspired wide-ranging discoveries throughout quantum information, our understanding of the general principles underlying MQC seems to be biased by its historical reliance upon the archetypal 2D cluster state. Here we utilise recent advances in the subject of symmetry-protected topological order (SPTO) to introduce a novel MQC resource state, whose physical and computational behaviour differs fundamentally from that of the cluster state. We show that, in sharp contrast to the cluster state, our state enables universal quantum computation using only measurements of single-qubit Pauli X, Y, and Z operators. This novel computational feature is related to the 'genuine' 2D SPTO possessed by our state, and which is absent in the cluster state. Our concrete connection between the latent computational complexity of many-body systems and macroscopic quantum orders may find applications in quantum many-body simulation for benchmarking classically intractable complexity. INTRODUCTIONThe idea of measurement-based quantum computation (MQC), where computation is carried out solely through single-qubit measurements on a fixed many-body resource state and classical feed-forward of measurement outcomes, 1-3 is quite surprising. This is because it highlights the origins of quantum advantage in terms of entanglement and non-commutative measurements, uniquely quantum effects without counterparts in classical mechanics. In particular, so-called universal resource states, the states that are capable of efficiently implementing universal MQC, represent a class of maximal entanglement in the classification of many-body entanglement, 4 so that the structure and complexity of their entanglement is of great interest for advancing the understanding of quantum computation. Following the canonical example of the two-dimensional (2D) cluster state, 5 many other universal resource states have been found, including cluster states defined on various lattices, 4 some tensor network states, 4,6-10 and model ground states in condensed matter physics such as 2D Affleck-Kennedy-Lieb-Tasaki (AKLT) states. [10][11][12][13][14][15] Given the existence of these various known universal resource states, a natural question is whether we might be able to find any common key feature, so as to explore more their variety in fundamental structures as well as in practical applications. Although the earliest resource states for MQC were found in short-range correlated states described as somewhat artificial tensor network states, 4,6-10 a new insight has been that a class of short-ranged entangled states structured by symmetry, endowed with so-called symmetry-protected topological order (SPTO), [16][17][18][19][20][21][22][23][24] make excellent candidate resource states systematically. Indeed, in the setting of 1D spin chains, the ground states of several SPTO phases have already been shown to poss...

Contextuality as a Resource for Models of Quantum Computation with Qubits

“…2 for illustration.) Note that an alternative resource state for one-qubit Pauli measurements is the so-called "union-jack" hypergraph state of Ref. [14].…”

Verification of hypergraph states

“…(For the definition of hypergraph states and their properties, see below.) First, certain hypergraph states, such as the Union Jack states, are universal resource states for measurement-based quantum computing with only Pauli measurements [39]. This result solves the first problem, namely, the requirement of non-Clifford basis measurements for the user.…”

“…Ref. [39] also pointed out that hypergraph states are important in the study of symmetry-protected topological orders. Second, it was shown in Ref.…”

Universal measurement-based quantum computation in two-dimensional symmetry-protected topological phases

“…Miller and Miyake also provide examples of Z 3 2 SPTsymmetric fixed-point wave function on the union-jack lattice [40,41]. But an important question of whether there exists an extended region (if not the whole phase) in an SPT phase that supports universal MBQC.…”

“…However, the development in two dimensions and higher is far limited, with only a handful of fixed-point wave functions providing universal resources [39][40][41], and the usefulness in some cases may depend on the underlying lattices. An important question remains open is whether the quantum computational universality can be extended beyond the fixed points in two or highdimensional SPT phases.…”