Higher-group symmetry of (3+1)D fermionic $\mathbb{Z}_2$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry
Maissam Barkeshli, Po-Shen Hsin, Ryohei Kobayashi
Abstract
It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D \mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> gauge theory with an emergent fermion, and point out that pumping chiral p+ip <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> topological states gives rise to a \mathbb{Z}_{8} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mn>8</mml:mn> </mml:msub> </mml:math> 0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization of T^3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> (3-torus) and T^2 \rtimes_{C_2} S^1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mo>⋊</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:msub> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> (2-torus bundle over the circle) respectively, and pumping p+ip <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> states. Our considerations also imply the possibility of a logical T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>T</mml:mi> </mml:math> gate by placing the code on \mathbb{RP}^3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mrow> <mml:mi>ℝ</mml:mi> <mml:mi>ℙ</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> and pumping a p+ip <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> topological state.