Computational Power of Symmetry-Protected Topological Phases

Abstract

We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order as resources for measurement-based quantum computation (MBQC). We show that, for a wide range of SPT phases, the computational power of ground states is uniform throughout each phase. This computational power, defined as the Lie group of executable gates in MBQC, is determined by the same algebraic information that labels the SPT phase itself. We prove that these Lie groups always contain a full set of single-qubit gates, thereby affirming the long-standing conjecture that general SPT phases can serve as computationally useful phases of matter. Introduction. In many-body physics, the essential properties of a quantum state are determined by the phase of matter in which it resides. Recent years have witnessed tremendous progress in the discovery and classification of quantum phases [1][2][3][4][5][6][7][8][9][10], and it is thus pertinent to ask: what can a phase of matter be used for? A traditional example is the ubiquitous superconductor, while newly discovered phases such as topological insulators [11] and quantum spin liquids [12] have promising future applications. Quantum phases are useful in quantum information processing as well: certain topological phases allow for error-resilient topological quantum computation via the braiding and fusion of their anyonic excitations [13,14]. These applications all operate due to properties of a phase rather than a particular quantum state, hence they enjoy passive protection against certain sources of noise and error.

Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

“…However, this method does not generalize conveniently to arbitrary members of the surrounding SPTO phase since measurements away from the X basis violate the relevant symmetries. Fortunately MBQC can be performed in manner that minimizes symmetry violating operations [31,32]. We review and make use of this method in Sec.…”

“…A scheme for MBQC with 1D-SPT phases that leverages the symmetry to do universal MBQC at arbitrary points in the phase was proposed in Refs. [30][31][32]. Note, however, that this approach cannot be immediately applied to 2D SPTO because strict single-site locality of measurements is required for MBQC.…”

“…Recently, these results have been extended to 2D resource states lying in a quasi-1D SPTO phase, protected by so-called subsystem symmetries, giving rise to quantum phases of matter in which every state is universal resource for * AustinDaniel@unm.edu MBQC. Remarkably, the computational power of such phases is a direct consequence of the symmetries they possess [31]. In particular, the first example of a computationally universal phase, known as the 2D cluster phase, was constructed from the rigid line-like symmetries of the square lattice cluster state in Ref.…”

Harnessing symmetry-protected topological order for quantum memories

“…However, our procedure in Eq. (14) only allows us to evaluate the map on elements of the form |m m|, and so producing all the operator basis elements seems out of reach. Thankfully, there exists an algorithm for producing the off-diagonal elements, namely…”

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

“…Examples include the 1D cluster state and the 1D spin-1 AKLT state, which can be used for arbitrary single-qubit gates [33,34]. Such utility in quantum gates has been further established [35][36][37][38].…”