Demonstration of quantum computation and error correction with a tesseract code

TL;DR

The paper tackles fault-tolerant quantum computation by introducing the $[[16,4,4]]$ tesseract subsystem color code, derived from the larger $[[16,6,4]]$ color code through deliberate qubit sacrifice to create gauge qubits. This design enables weight-four, fault-tolerant syndrome measurements and single-shot error correction, reducing overhead and enabling more efficient encoded operations. The authors implement and test encoded graph-state preparation and fault-tolerant CNOTs on Quantinuum H1/H2, demonstrating up to 12 logical qubits across three code blocks and five rounds of error correction with error rates significantly below unencoded baselines. They discuss the modularity of fault-tolerance gadgets, prospects for universality, and scaling to larger, more reliable quantum computations with trapped-ion hardware.

Abstract

A critical milestone for quantum computers is to demonstrate fault-tolerant computation that outperforms computation on physical qubits. The tesseract subsystem color code protects four logical qubits in 16 physical qubits, to distance four. Using the tesseract code on Quantinuum's trapped-ion quantum computers, we prepare high-fidelity encoded graph states on up to 12 logical qubits, beneficially combining for the first time fault-tolerant error correction and computation. We also protect encoded states through up to five rounds of error correction. Using performant quantum software and hardware together allows moderate-depth logical quantum circuits to have an order of magnitude less error than the equivalent unencoded circuits.

Figures (6)

• Figure 1: The $[[16,6,4]]$ color code on the 4D hypercube, or tesseract. Each of the $16$ vertices is a qubit. Cubes are $X$ and $Z$ stabilizers, and squares are logical operators, e.g., 0145.
• Figure 2: Encoded circuit to prepare the cube graph state, in two code blocks.
• Figure 3: It is convenient to arrange the tesseract code's $16$ qubits in a $4 \times 4$ grid. The $X$ and $Z$ stabilizers are supported on pairs of rows and pairs of columns; a set of generators is shown in black. Last, we highlight the supports of the logical operators in the basis we use. For example, logical $Z_2$ is $Z_0 Z_1 Z_2 Z_3$. Observe that the weight-four logical operators are not self dual, but come in three pairs of two.
• Figure 4: Circuits to measure weight-four operators. The circuit in (a) measures $X^{\otimes 4}$, but it is not fault tolerant, as a single $X$ fault (red) on the ancilla qubit can propagate to a weight-two error on the data. The circuit in (b), using one syndrome and two flag qubits, is fully fault tolerant. The corrections can be tracked classically, as part of the "Pauli frame." (c) This circuit's single flag qubit is enough to detect a possible correlated error, but not correct it. A subsequent measurement will have to take the flag into account to correct the answer. (d) We can also measure $X^{\otimes 4}$ and $Z^{\otimes 4}$ simultaneously. In our applications, the results should be uniformly random. But repeating the $X^{\otimes 4}$ and $Z^{\otimes 4}$ four times, on disjoint qubit sets (\ref{['f:ftmeasurements']}), if, e.g., one of the $Z^{\otimes 4}$ measurements disagrees with the other three, that can mean that either a weight-one $X$ error has been detected, or a weight-two correlated $X$ error has spread to the data.
• Figure 5: Relationships to other codes. (a) Removing a qubit from the $[[16,6,4]]$ code leaves the well-known $[[15,7,3]]$ Hamming code. (b) Applying CNOT gates between two halves of the 16-qubit code disentangles it into two $[[8,3,2]]$ color codes on the cube; the left color code has $X$ distance $2$ and $Z$ distance $4$, and the right color code vice versa. (c) Left: an encoding circuit for the $[[4,2,2]]$ color code. Stacking four copies of this color code, with the first in encoded ${|00\rangle}$ (a cat state) and the last in encoded ${|{+}{+}\rangle}$, and applying the same encoding circuit transversally yields our $[[16,4,4]]$ code with fixed gauge qubits BreuckmannBurton22foldtransversalclifford.