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Quantum Origami: Fault-tolerant Transversal Gates for Quantum Computation and Measurement of Topological Order

January 29, 2018 - 11:00am
Guanyu Zhu
Topologically ordered states of matter arise both as ground states of strongly interacting many-body quantum Hamiltonians and also as code subspaces of quantum error correction codes.  A conventional way to perform logical operations in this subspace is via adiabatic braiding of anyons which has a linear overhead with the system size or equivalently the code distance. An alternative way of doing these operations is via transversal logical gates (TLG).  TLGs are inherently fault-tolerant due to the locality of error propagation, while their one-shot nature (constant circuit depth) dramatically speeds up the time to perform a logical operation.   However, the current sets of TLGs are quite limited and cannot form a universal set.  Moreover, there is a lack of deep understanding, on a fundamental level, of certain types of transversal gates, such as those in color codes.
In this talk, I discuss a wide class of TLGs using modular transformations, which are elements of the mapping class group (MCG) of a genus-g surface. In particular, by considering multiple layers of a topological state together with appropriate gapped boundaries or twist defects, it is possible to implement modular transformations such as S and T transversally by local SWAP gates between the layers. Our discovery also provides a simple geometric interpretation for a class of TLGs,  i.e., “manifold origami”, involving folding of the manifold and permutation of code patches.   This new scheme not only leads to a deep understanding of existing TLGs, but also greatly extends the variety of them, especially to the realm of non-abelian phases. In particular, TLGs in non-abelian systems, such as Fibonacci and Ising phases, can be used to perform a universal set of logical gates, without the requirement of state distillation.  From an even broader perspective, we also reveal a deep connection between TLGs and anyon symmetry transformation in the context of symmetry-enriched topological phases. We propose an experimental implementation of these ideas using superconducting qubits, and also methods to measure the modular matrices, which contain the braiding statistics of quasiparticles.
Reference:  arXiv:1711.05752
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