Narayanan Rengaswamy, Duke University
In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal T and T^{-1} gates preserve the codespace. Our Heisenberg perspective reduces this question to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries of this result:
(a) For any non-degenerate [[n, k, d]] stabilizer code supporting a physical transversal T, there exists an [[n, k, d]] CSS code with the same property;
(b) Triorthogonal codes form the most general family of CSS codes that realize logical transversal T via physical transversal T;
(c) Triorthogonality is necessary for physical transversal T on a CSS code to realize the logical identity.
The main tool we use is a recent characterization of a particular family of diagonal gates in the Clifford hierarchy that are efficiently described by symmetric matrices over rings of integers [N. Rengaswamy et al., Phys. Rev. A 100, 022304]. We refer to these operations as Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing code constructions that realize logical gates via transversal T. We provide several examples of codes and briefly discuss connections to decreasing monomial codes, pin codes, generalized triorthogonality and quasitransversality. We partially extend these results towards characterizing all stabilizer codes that support transversal \pi/2^{\ell} Z-rotations. In particular, using Ax’s theorem on residue weights of polynomials, we provide an alternate characterization of logical gates induced by transversal \pi/2^{\ell} Z-rotations on a family of quantum Reed-Muller codes. We also briefly discuss a general approach to analyze QFD gates that might lead to a characterization of all stabilizer codes that support any given physical transversal 1- or 2-local diagonal gate.
Biography:
Narayanan Rengaswamy is a PhD candidate in the Department of Electrical and Computer Engineering at Duke University, Durham, USA. He works under the supervision of Prof. Henry Pfister and Prof. Robert Calderbank, and is affiliated to the Rhodes Information Initiative at Duke (iiD). His training is primarily in classical information and coding theory, compressed sensing and wireless communications. He received the Master of Science degree in Electrical Engineering from Texas A&M University in Dec. 2015, where he worked on Cyclic Polar Codes with Prof. Henry Pfister. He also conducted research on Spatially-Coupled LDPC on Burst Erasure Channels under the supervision of Dr. Laurent Schmalen and Dr. Vahid Aref while he was a research intern at Alcatel-Lucent Bell Labs, Stuttgart, Germany. Over the past few years, he has been actively learning about quantum information and quantum computation, and this is his primary area of research now. In particular, his recent research topics are characterizing desirable code properties for fault-tolerant quantum computation, and generalizing belief-propagation algorithms to the quantum setting where the messages are now qubit states and not probabilities. More information is available on his webpage: https://nrenga.github.io