Fault-Tolerant Gates on Hypergraph Product Codes

Quantum computers have the capacity to change the landscape of computing. To make these devices robust to experimental imperfections, we require some form of quantum error correction. Current approaches focus on topological codes, as they appear to be in the realm of experimental capabilities. These techniques will likely lead to the first proof of principle of fault tolerance in the next 5 yr. In the long run, however, these approaches require a very large number of physical qubits to construct a fault-tolerant quantum circuit that can run interesting algorithms. As an alternative, it is proposed that quantum low-density parity-check (LDPC) codes could serve as an architecture that circumvents this problem. The best candidate class of quantum LDPC codes were discovered by Tillich and Zémor and are called hypergraph product codes. These codes have a constant encoding rate and were recently shown to have a constant fault-tolerant error threshold. With these features, they asymptotically offer a smaller overhead compared to topological codes. Here, we demonstrate how to perform Clifford gates on this class of codes using code deformation. To this end, we introduce wormhole defects on hypergraph product codes. Together with state injection, we can perform a universal set of gates within a single block of the class of hypergraph product codes.

Popular Summary

Quantum computers will change the landscape of computing, but first must be fault tolerant, that is, robust to experimental imperfections. This can be achieved using quantum error-correcting codes, but not all codes are created equal, and some leading contenders require an enormous number of physical quantum bits (qubits) to construct a fault-tolerant circuit that can run interesting algorithms. One class of codes, known as hypergraph product codes, could circumvent this problem, but it is unclear yet whether this architecture is worth the experimental effort. Here, we demonstrate how to perform a set of fundamental “building block” operations, or universal gates, on this class of codes.

Hypergraph product codes have several key properties that make them promising: a constant encoding rate and a constant fault-tolerant error threshold. With these features, they promise a smaller overhead compared to current techniques. To realize a set of universal gates, we use a broad framework called code deformation. This entails modifying small pieces of the code to string together basic operations. Each small step ensures that nothing drastic can happen to the encoded information.

Our novel framework proposes how to perform a fault-tolerant set of universal gates, which in turn will inform the construction of hypergraph product codes in the future and be one key factor in code design. This framework will be crucial in developing more efficient routes to scalable quantum computers.

Figure 1

A generic bipartite graph with two types of vertices $C$ (green square vertices) and $V$ (blue circular vertices). Edges run only between blue vertices and green vertices and never between two vertices of the same color.

Figure 2

(a) Hypergraph product representation of the surface code with smooth boundaries above and below and rough boundaries on the sides. (b) The action of $X$ stabilizers on $VV$ qubits and (c) $CC$ qubits on the right. Filled nodes represent nodes involved in the stabilizer generator.

Figure 3

Schematic of factor graphs. (a) denotes the subgraph induced by $S$. $N$ is its neighborhood, and $A$ is its ancestor. (b) denotes the subgraph induced by $T$. $M$ is its neighborhood, and $B$ is its ancestor.

Figure 4

Schematic for the puncture. The subgraphs selected by $T$ and $S$ are flattened and placed below and to the left. Their product is represented using four quadrants. The qubits are in the North-West and South-East quadrants. The $Z$ stabilizers are in the South-West quadrant. The $X$ stabilizers are in the North-East quadrant. (a) Smooth puncture defined by $T$ and $S$. The $Z$ stabilizers $T\times S$ are completely within the puncture and are, thus, not broken. (b) Rough puncture defined by $S$ and $T$. The $X$ stabilizers $S\times T$ are completely within the puncture and are, thus, not broken.

Figure 5

Subcode created by $T\times S$. Shaded nodes are part of the puncture.

Figure 6

(a) and (b) feature a smooth puncture defined on a lattice with only smooth boundaries. The strings of $X$ operators defined only on $VV$ qubits [as in (a)] or only on $CC$ qubits [as in (b)] are equivalent. (c) and (d) feature a lattice with smooth and rough boundaries, with a smooth punctured carved out from the inside. The logical $X$ shown running from the smooth puncture to the boundary in (c) and (d) are equivalent up to an embedded logical $X$.

Figure 7

Schematic to denote the wormhole. Smooth puncture on the left and rough puncture on the right.

Figure 8

A pair of punctures used to encode a single logical qubit on the surface code.

Figure 9

A measurement-based circuit to perform controlled-$Z$. We introduce an ancilla prepared in the $|0⟩$ state. The double boxes indicate a nondestructive projective measurement. The labels $PQ$ on these measurements indicate that the projection is performed along the $+1$ and $-1$ eigenstates of the two-qubit Pauli operator $PQ$. Finally, we perform a Hadamard and destructively measure the ancilla qubit in the computational basis.

Figure 10

A puncture crossing its own path. The puncture first leaves a trail of $Z$’s and passes through a wormhole. Upon crossing $Z$, the logical $X$ is mapped to $XZ$. This operator is no longer guaranteed to be needle measurable.

Figure 11

A pointlike puncture centered at $c$, $v$. The check $c^{\prime }$ is connected to $c$ via the variable node $u$.