Qudit quantum computation on matrix product states with global symmetry

Abstract

Resource states that contain nontrivial symmetry-protected topological order are identified for universal single-qudit measurement-based quantum computation. Our resource states fall into two classes: one as the qudit generalizations of the 1D qubit cluster state, and the other as the higher-symmetry generalizations of the spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) state, namely, with unitary, orthogonal, or symplectic symmetry. The symmetry in cluster states protects information propagation (identity gate), while the higher symmetry in AKLT-type states enables nontrivial gate computation. This work demonstrates a close connection between measurement-based quantum computation and symmetry-protected topological order.

High-dimensional one-way quantum processing implemented on d-level cluster states

“…Of significant interest are cluster states 4 , which can enable the realization of universal quantum computers by means of a 'oneway' scheme 5 , where processing is performed through measurements 6 . The use of d-level cluster states can increase the quantum resources while keeping the number of parties constant 7 and enable novel algorithms 8 . Here, we achieve their experimental realization, characterization, and test their noise sensitivity.…”

Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter

“…A complementing approach to topological quantum computation uses Majorana fermions located at the edges of one-dimensional (1D) chains with symmetryprotected topological (SPT) order [4][5][6]. On the other hand, quantum states exhibiting SPT order can be used as resources for instances of measurement-based quantum computation [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] -an insight most relevant to the present work. Various further examples can be found [23][24][25]; indeed, every time a new type of quantum order is discovered, it is not long before its uses in notions of quantum computation are being investigated.…”

Classes of topological qubits from low-dimensional quantum spin systems

“…Due to TOP order, SET qubits can also be encoded into anyonic excitations (or holes, defects) [12], braiding of which can enable universal quantum computing. While SPT order, by definition, does not support anyons; however, these systems also enable universal quantum computing [37][38][39][40]. A promising approach is to use 'extrinsic' defects in SPT systems, which, so far, have not been fully understood [95] except for a few notable systems [96][97][98].…”

“…and its unique ground state is |C 0 = n F n n CZ n (|+ ) ⊗L for CZ n as qudit version of controlled-Z gate on sites n and n + 1, F n as a Fourier operator on site n [40]. Several other cluster states are |C = (X ) ⊗L |C 0 for = 1, 2, .…”