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.
We investigate the usefulness of ground states of quantum spin chains with symmetry-protected topological order (SPTO) for measurement-based quantum computation. We show that, in spatial dimension one, if an SPTO phase supports quantum wire, then, subject to an additional symmetry condition that is satisfied in all cases so far investigated, it can also be used for quantum computation.
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