Universal quantum computation with group surface codes

TL;DR

It is shown that group surface codes can be leveraged to perform non-Clifford gates in $\mathbb{Z}_2$ surface codes, thus enabling universal computation with well-established means of performing logical Clifford gates.

Abstract

We introduce group surface codes, which are a natural generalization of the $\mathbb{Z}_2$ surface code, and equivalent to quantum double models of finite groups with specific boundary conditions. We show that group surface codes can be leveraged to perform non-Clifford gates in $\mathbb{Z}_2$ surface codes, thus enabling universal computation with well-established means of performing logical Clifford gates. Moreover, for suitably chosen groups, we demonstrate that arbitrary reversible classical gates can be implemented transversally in the group surface code. We present the logical operations in terms of a set of elementary logical operations, which include transversal logical gates, a means of transferring encoded information into and out of group surface codes, and preparation and readout. By composing these elementary operations, we implement a wide variety of logical gates and provide a unified perspective on recent constructions in the literature for sliding group surface codes and preparing magic states. We furthermore use tensor networks inspired by ZX-calculus to construct spacetime implementations of the elementary operations. This spacetime perspective also allows us to establish explicit correspondences with topological gauge theories. Our work extends recent efforts in performing universal quantum computation in topological orders without the braiding of anyons, and shows how certain group surface codes allow us to bypass the restrictions set by the Bravyi-K{ö}nig theorem, which limits the computational power of topological Pauli stabilizer models.

Figures (6)

• Figure 1: Lattice of the GSC. The GSC is defined on a square lattice with rough boundaries on the left and right and smooth boundaries on the top and bottom. The edges are oriented either to the right or upwards.
• Figure 2: Mapping a flux-free state to the left gauge. (a) Starting with an arbitrary flux-free state, we apply gauge transformations in sequence (from right to left) to map the group labels to the identity. An edge without a label is assumed to be in the identity state $|1\rangle$. (b) Moving across the top row, we are left with some group element $g$ on the left most edge. We also leave behind group elements labeled with a tilde. (c) For each row, we can use gauge transformations to move the group elements on the horizontal edges to the leftmost edges. This gives us some group elements $g,h,k$. (d) The flux-free condition, and the fact that gauge transformations commute with the plaquette stabilizers, guarantees that the group elements around the plaquette multiply to the identity. This allows us to deduce that the group elements with a tilde are the identity. (e) Again, using the flux-free condition, we see that $g$, $h$, and $k$ must be equal.
• Figure 3: Example movement operators for the fluxes and charges. (a) A group-valued flux $m_p$ is moved to the right by applying the operator $\mathrm{C}\bar{\mathrm{R}}_{43}R_3^{m_p}\mathrm{C}{\mathrm{R}}_{43}$. The worldline of the flux is depicted to the right, with $m'_p=gm_p\bar{g}$, for some $g\in G$. (b) After the measurement outcome $R_{ij}$, the charge can be (probabilistically) moved to the right by applying the operator $Z^{R^*_{i'j'}}=\sum_{g \in G}R^*(g)_{i'j'}|g\rangle \langle g|$, for some choice of indices $i',j'$. For further details, we refer to Appendix \ref{['app: charge and flux']}.
• Figure 4: Unfolding the flux cube to use planar ribbon operators. $(A)$: the most general configuration of group elements on the cube. On the up and down plaquettes, $g_U$ and $g_D$ denote measurement outcomes for fluxes in the group basis. On the side faces, $g_L$, $g_F$, $g_R$ and $g_B$, if nontrivial, represent controlled operations on the physical edges in order to move the bottom flux $g_D$. $(B)$: Unfolded cube, with the projection point of view being the center of the cube. The dotted blue lines indicate matching copy tensors when the cube is assembled, and similarly for the side and top dotted red lines. From this perspective, the side group elements are on the same footing as the top/bottom group elements, and they represent the flux around a plaquette. The starting edge for the flux computation and the handedness (clockwise or anticlockwise) are indicated by the direction of the red arrow and handedness of the multiplication tensor. $(C)$: Group elements on each leg of the tensor network are assigned to physical edges of the lattice and time direction edges. This configuration resembles a spatial square lattice, except the handedness and site choice for each plaquette are not uniformly defined as is usually done for a square lattice hosting a quantum double model.
• Figure 5: Spacetime ribbon for a flux stationary in space, moving in the time direction. $(A)$: Ribbon path on the cube, where we only show faces crossed by the ribbon path. Here, $g_D$ and $g_U$ are the fluxes on the down and up plaquettes respectively. $(B)$: Explicit action of the $F^{(\bar{g}_D,k)}_{(t)}$ ribbon in the group basis on the path shown in $(A)$. To get the relevant ribbon in the irrep basis, we use the change of basis of Eq. \ref{['eq: change of basis equation for flux ribbons']}. $(C)$: Spacetime anyon wordline resulting from the application of the spacetime ribbon of $(A)-(B)$. We see that the internal state at the top is conjugated by $h_4$, which is the group element corresponding to the vertex measurement inbetween plaquette measurements.