Gauge color codes in two dimensions

We present a family of quantum error-correcting codes that support a universal set of transversal logic gates using only local operations on a two-dimensional array of physical qubits. The construction is a subsystem version of color codes where gauge fixing through local measurements dynamically determines which gates are transversal. Although the operations are local, the underlying code is not topological in structure, which is how the construction circumvents no-go constraints imposed by the Bravyi-König and Pastawski-Yoshida theorems. We provide strong evidence that the encoding has no error threshold in the conventional sense, though it is still possible to have logical gates with error probability much lower than that of physical gates.

Figure 1

Boundary measurement between color codes. (a) Codes $D_0$ and $D_1$ are projected into logical Bell state by measuring weight-$4 X$ and $Z$ operator shown in dashed lines. (b) Codes $D_1$ and $D_2$ are entangled through two boundary operators. Green check operators along the boundary are merged together, as indicated by the two-sided arrow. (c) Logical $X_L{X}_L$ between two $D_2$ color codes (note $X$'s inside boundary operators) will only merge green check operators of $Z$ type.

Figure 2

Examples of boundary measurement between triangular color codes. The grid of examples shows orders $t=1,2,3$ by row and red/green/blue boundaries by column. Larger codes follow these patterns, with squares or octagons (depending on color) as operators in the center of the boundary, and one weight-2 or weight-6 operator at an endpoint of the boundary.

Figure 3

Examples of boundary measurement between color codes of orders $(t-1)$ and $t$. The grid shows orders $t=1,2,3$ by row and red, green, and blue boundaries by column.

Figure 4

Example of fusion in bilayer of two $D_2$ code blocks, using [4.8.8] triangular color codes [8]. (a) Bilayer stack of two code blocks, showing two examples of the weight-$4 Z$ gauge operators that are measured. Our proposed approach is to measure weight-four $Z$ operators for every matching pairs of edges in the [4.8.8] lattices. (b) The $X$ stabilizers of the two code blocks are fused by the gauge fixing operation.

Figure 5

2D gauge color code $T_2$ in a bilayer structure. (a) Top view, showing the lateral connections through boundary operators (purple). (b) Side view, showing the bilayer structure with vertical connections as black lines. Each weight-$4 Z$ measurement in fusion consists of two neighboring pairs of qubits connected by these vertical lines. Notice that the boundary operators alternate between top and bottom layer, following the recursion in Eq. (6).

Figure 6

Logical cnot using initialization and measurement of an ancilla code block and the boundary measurement protocol described in the text. (a) Measuring the dashed-line operators at a boundary labeled with $Z$'s ($X$'s) projectively measures the operator $Z_L{Z}_L \left(X_L{X}_L\right)$, as indicated in the circuit at right. $X$ stabilizers along the boundary are fused together during $Z_L{Z}_L$ measurement, and vice versa, as indicated by the two-sided arrows. The middle color code is the ancilla logical qubit. (b) Circuit for logical cnot through parity measurements. Logical Pauli corrections shown in dashed boxes are conditioned on the measurement results $(a,b,c)$, which are each $\pm 1$.

Figure 7

Code expansion showing how to convert a triangular color code from order 2 to 3. (a) The existing order-2 code is supplemented with 14 additional physical qubits needed for the expanded code, and they are prepared as Bell pairs along every diagonal edge, as drawn here. (b) Measuring the red squares in both $X$ and $Z$ stabilizers will project into the order-3 code. (c) The remaining blue and green stabilizers are measured for error detection. The procedure detects errors because the Bell states are stabilized by $XX$ and $ZZ$, and one can form any of the new blue and green stabilizers from products of prior ones along the bottom boundary and weight-2 operators from the Bell pairs, enabling error detection.

Figure 8

Example configuration of $Z$ errors to initialize the triply even syndrome decoder. The decomposition of the triply even code into doubly even codes is shown, with the connections through logical Bell states (“Bell”) and fusion (“ZZ”) labeled. Note that this corresponds to $T_2$ in Fig. 5 of the main text. As this is a soft decoder, two configurations are considered with (a) zero $Z$ errors on $D_0$ and (b) one $Z$ error on $D_0$, as shown by the number in that box. By processing the fused doubly even codes, errors are located on vertical edges, but there is ambiguity as to whether the error should be in the top or bottom code (represented by grayed boxes with question marks). The circled number between the two copies of the same code ($D_1$ or $D_2$ here) show an example configuration of located vertical-edge errors that could occur after processing the type-F stabilizer measurements as syndromes in a doubly even code.

Figure 9

Final error configurations in the decoder after cascading through the type-B stabilizers. For this example, both have been assigned $-1$ eigenvalue, which corresponds to odd parity of errors. (a) The ambiguous vertical edge in $D_1$ is assigned to the top code. Since the bottom $D_1$ has even error parity while the type-B stabilizer is odd, the bottom $D_2$ must also have an odd number of errors. To achieve this, a vertical pair is inserted, placing one error each in top and bottom $D_2$. The total weight of this configuration is now 3. (b) The ambiguous vertical edge is placed in the bottom $D_1$ block, because $D_0$ has an error and the first type-B stabilizer has odd parity, so top $D_1$ must be even. Placing zero errors in top and bottom $D_2$ satisfies the second type-B stabilizer. The weight of this configuration is 2. This example highlights the importance of the soft decoder, as sometimes the minimal-weight matching changes when vertical pairs must be inserted, as must be done in this example for case (a).