Non-Clifford and Parallelizable Fault-Tolerant Logical Gates on Constant and Almost-Constant Rate Homological Quantum Low-Density Parity-Check Codes via Higher Symmetries

We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive a generic formula for a transversal <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>T</a:mi> </a:math> gate on color codes defined on general 3-manifolds, which acts as collective non-Clifford logical gates on any triplet of logical qubits with their logical- <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>X</c:mi> </c:math> membranes having a <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mrow> <e:mi mathvariant="double-struck">Z</e:mi> </e:mrow> <e:mn>2</e:mn> </e:msub> </e:math> triple intersection at a single point. The triple-intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory (TQFT): the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msubsup> <h:mrow> <h:mi mathvariant="double-struck">Z</h:mi> </h:mrow> <h:mn>2</h:mn> <h:mn>3</h:mn> </h:msubsup> </h:math> gauge theory. Moreover, the transversal <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>S</k:mi> </k:math> gate of the color code corresponds to a higher-form symmetry in TQFT supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical gates. A construction of constant-depth circuits of the above logical gates via cup-product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple-intersection invariants for 3-manifolds, with the structure encoded into an interaction hypergraph which determines the logical gate property and also corresponds to the hypergraph magic state that can be injected into the code without distillation (“”). We also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>k</m:mi> <m:mo>/</m:mo> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mi>O</m:mi> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>log</m:mi> <m:mo></m:mo> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">)</m:mo> </m:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>O</s:mi> <s:mo stretchy="false">(</s:mo> <s:mi>log</s:mi> <s:mo></s:mo> <s:mo stretchy="false">(</s:mo> <s:mi>n</s:mi> <s:mo stretchy="false">)</s:mo> <s:mo stretchy="false">)</s:mo> </s:math> distance; (2) A homological fiber-bundle code from twisting the product by an isometry of the surface based on the construction by Freedman-Meyer-Luo, with <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>O</y:mi> <y:mo stretchy="false">(</y:mo> <y:mn>1</y:mn> <y:mo>/</y:mo> <y:msup> <y:mi>log</y:mi> <y:mrow> <y:mstyle displaystyle="false" scriptlevel="0"> <y:mfrac> <y:mn>1</y:mn> <y:mn>2</y:mn> </y:mfrac> </y:mstyle> </y:mrow> </y:msup> <y:mo></y:mo> <y:mo stretchy="false">(</y:mo> <y:mi>n</y:mi> <y:mo stretchy="false">)</y:mo> <y:mo stretchy="false">)</y:mo> </y:math> rate and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>O</gb:mi> <gb:mo stretchy="false">(</gb:mo> <gb:msup> <gb:mi>log</gb:mi> <gb:mrow> <gb:mstyle displaystyle="false" scriptlevel="0"> <gb:mfrac> <gb:mn>1</gb:mn> <gb:mn>2</gb:mn> </gb:mfrac> </gb:mstyle> </gb:mrow> </gb:msup> <gb:mo></gb:mo> <gb:mo stretchy="false">(</gb:mo> <gb:mi>n</gb:mi> <gb:mo stretchy="false">)</gb:mo> <gb:mo stretchy="false">)</gb:mo> </gb:math> distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping-torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical- <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mi>X</ob:mi> </ob:math> measurements.