Quantum Origami: Fault-tolerant Transversal Gates for Quantum Computation and Measurement of Topological Order
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.
College Park, MD 20742