While quantum computers can achieve dramatic speedups over the classical computers familiar to us, identifying the origin of this quantum advantage in physical systems remains a major goal of quantum information science. A useful tool here is measurement-based quantum computation (MQC), a computational framework utilizing the quantum entanglement found in many-body resource states. Not all resource states are useful for quantum computation however, so an important question is what properties of many-body entanglement characterize universal resource states, which can implement any quantum computation. Many-body states are also studied in condensed matter physics, where the collective behavior of quantum many-body systems sometimes define topological phases of matter. These phases are defined by nonlocal many-body entanglement, making topologically-ordered states natural candidates for MQC. We might wonder if these topological phases could be organized as phases of quantum computation, so that every state within the phase is universal for MQC. While phases of symmetry-protected topological order (SPTO) have arisen as natural candidates, previous attempts to demonstrate an MQC-SPTO correspondence were mostly limited to nonuniversal 1D spin chains, leaving the important 2D setting wide open. In this dissertation, we explore the wide and varied connections between MQC and SPTO, and obtain new results for 1D and 2D systems. After identifying a new MQC-SPTO correspondence within 1D spin chains, we move up and explore the operational use of 2D states with two complementary forms of SPTO. We create a new Union Jack resource state, whose different form of SPTO than previous 2D resource states permits a hierarchical notion of MQC universality. This state leads us to consider an idealized model of 2D SPTO, where we show that an additional symmetry condition makes these model states form universal resources for MQC only when they have nontrivial SPTO. We finally study the intrinsic complexity of SPTO-inspired states for classically intractable sampling, and identify inherent advantages of MQC for this purpose. Our work highlights the rich complexity available in states of entangled quantum matter, providing new evidence which sharpens our understanding of the diverse connections between MQC and SPTO.
What kinds of symmetry-protected topologically ordered (SPTO) ground states can be used for universal measurement-based quantum computation in a similar fashion to the 2D cluster state? 2D SPTO states are classified not only by global on-site symmetries but also by subsystem symmetries, which are fine-grained symmetries dependent on the lattice geometry. Recently, all states within so-called SPTO cluster phases on the square and hexagonal lattices have been shown to be universal, based on the presence of subsystem symmetries and associated structures of quantum cellular automata. Motivated by this observation, we analyze the computational capability of SPTO cluster phases on all vertex-translative 2D Archimedean lattices. There are four subsystem symmetries here called ribbon, cone, fractal, and 1-form symmetries, and the former three are fundamentally in one-to-one correspondence with three classes of Clifford quantum cellular automata. We conclude that nine out of the eleven Archimedean lattices support universal cluster phases protected by one of the former three symmetries, while the remaining lattices possess 1-form symmetries and have a different capability related to error correction.
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.
Measurement-based quantum computation is a model for quantum information processing utilizing local measurements on suitably entangled resource states for the implementation of quantum gates. A complete characterization for universal resource states is still missing. It has been shown that symmetry-protected topological order in one dimension can be exploited for the protection of certain quantum gates in measurement-based quantum computation. In this paper we show that the two-dimensional (2D) plaquette states on arbitrary lattices exhibit nontrivial symmetry-protected topological order in terms of symmetry fractionalization and that they are universal resource states for quantum computation. Our results of the nontrivial symmetry-protected topological order on arbitrary 2D lattices are based on an extension of the recent construction by Chen et al. [Phys. Rev. B 87, 155114 (2013)] on the square lattice.
Symmetry-protected topological (SPT) phases with gapless edge excitations have been shown to exist in strongly interacting bosonic/fermionic systems and it is highly desirable to identify practical systems realizing such phases through numerical simulation. A central question to be addressed is how to determine the SPT order in the system given the simulation result, while no local order parameter can be measured to distinguish the phases from a trivial one. In the tensor network approach to simulate strongly interacting systems, the quantum state renormalization algorithm has been shown to be effective in identifying intrinsic topological orders. Here, we show that a modified algorithm can identify SPT orders by extracting the symmetry-protected fixed point entanglement pattern in the ground-state wave function, which is essential for the existence of SPT order. The key to this approach is to add proper symmetry protection to the renormalization process. We demonstrate the effectiveness of this algorithm with examples of nontrivial SPT phases with internal symmetry in 1D and internal and translation symmetry in 2D.
Symmetry-protected topological (SPT) phases exhibit nontrivial short-ranged entanglement protected by symmetry and cannot be adiabatically connected to trivial product states while preserving the symmetry. In contrast, intrinsic topological phases do not need ordinary symmetry to stabilize them and their ground states exhibit long-range entanglement. It is known that for a given symmetry group
G, the 2D SPT phase protected by
G is dual to the 2D topological phase exemplified by the twisted quantum double model ${D}^{\ensuremath{\omega}}(G)$ via gauging the global symmetry
G. Recently it was realized that such a general gauging map can be implemented by some local unitaries and local measurements when
G is a finite, solvable group. Here, we review the general approach to gauging a $G\text{\ensuremath{-}}\mathrm{SPT}$ starting from a fixed-point ground-state wave function and applying a
N-step gauging procedure. We provide an in-depth analysis of the intermediate states emerging during the
N-step gauging and provide tools to measure and identify the emerging symmetry-enriched topological order (SET) of these states. We construct the generic lattice parent Hamiltonians for these intermediate states and show that they form an entangled superposition of a twisted quantum double (TQD) with an SPT-ordered state. Notably, we show that they can be connected to the TQD through a finite-depth, local quantum circuit which does not respect the global symmetry of the SET order. We introduce the so-called symmetry branch line operators and show that they can be used to extract the symmetry fractionalization classes (SFC) and symmetry defectification classes (SDC) of the SET phases with the input data
G and $[\ensuremath{\omega}]\ensuremath{\in}{H}^{3}(G,U(1))$ of the pregauged SPT-ordered state. We illustrate the procedure of preparing and characterizing the emerging SET-ordered states for some Abelian and non-Abelian examples such as dihedral groups
Dn and the quaternion group
Q8.