Recursive Magic State Distillation on the Surface Code

Abstract

I reduce the cost to prepare magic states with lattice surgery operations on the surface code by using a recursive implementation of 15-to-1 magic state distillation. On a rotated surface code with distance $d$, $|T\rangle$ preparation requires a $d$-by-$3 d$ grid of data qubits for up to $15 d$ error correction cycles, and $|CCZ\rangle$ preparation requires a $3 d$-by-$2 d$ grid for up to $10.5 d$ cycles. However, a significantly lower physical error threshold than that of the underlying surface code is required to match the error probability of the output magic state with the logical error rate of the output surface code at large code distances.

Figures (10)

• Figure 1: Diagrammatic examples of deformation operations on surface code patches. (a) A surface code tile can be moved by an intermediate deformation into a patch connecting its initial and final location, and (b) it can also be rotated if the patch contains an odd number of turns. Boundaries of a surface code patch can be deformed if they are connected by (c) timelike or (d) spacelike boundaries that preserves the code distance. A domain wall can cover (e) an entire surface code patch and switch all boundary types or (f) part of a surface code patch if twist defects in the interior of the patch partition it into regions of preserved and switched boundary types. Each type of domain wall applies a different map to the logical Pauli operators of the patch.
• Figure 2: Diagrammatic examples of lattice surgery operations on surface code patches. Patches can be (a) merged along a smooth boundary to measure the product of their logical Pauli $Z$ operators or (b) split to copy their state in the eigenbasis of $Z$, $|x\rangle$. The output state of a merge depends on a choice of error correction conventions. (c) A smooth merge and split can be combined into a projective $Z \otimes Z$ measurement if a Pauli $X$ correction is applied to the patch that is altered by the merge after outcome 1. (d) A smooth merge and split with an intermediate connection between multiple patches projectively measures the product of their $Z$ operators. (e) Intermediate ancilla patches can also be used to delay part of a Pauli product measurement. (f) These Pauli measurement operations generalize to patches of varying code distance. Similarly, merges and splits along a rough boundary measure products of $X$ and act on its eigenbasis, $H|x\rangle$ (i.e. $|+\rangle = H|0\rangle$ and $|-\rangle = H|1\rangle$).
• Figure 3: Single-qubit Clifford gates for all nontrivial elements of the Clifford group up to global phases and Pauli corrections: (a) $S$, (b) $H$, (c) $SH$, (d) $HS$, and (e) $SHS$. These implementations have a common spatial footprint and preserve the initial position and orientation of the surface code patch. The timelike twist defects at the corners of the initial and final patches are labeled to visualize their distinct permutation by each Clifford gate.
• Figure 4: Multi-qubit Clifford gates (a) $CX$, (b) $CZ$, (c) $CZ^2$ with targets on one side, and (d) $CZ^2$ with targets on two sides. Pairs of connected twist defects are labeled before and after each gate.
• Figure 5: Teleportation of non-Clifford gates (a) $T$ and (b) $CCZ$ by consuming the corresponding magic states $|T\rangle$ and $|CCZ\rangle$. In both cases, the patch merging operation performs the logical $X$ correction on the magic state to stabilize the merged patch. Magic states may also have a non-trivial Clifford frame that modifies the Clifford correction. The trivial Clifford frame is $a = b = c = 0$.