Below-threshold execution of fault-tolerant ripple-carry addition on a trapped-ion quantum computer

We deploy a universal fault-tolerant quantum computing (FTQC) architecture based on error-detecting $[\text{\hspace{0.17em}⁣}[8,3,2]\text{\hspace{0.17em}⁣}]$ and $[\text{\hspace{0.17em}⁣}[4,2,2]\text{\hspace{0.17em}⁣}]$ codes on Quantinuum’s Helios-1 trapped-ion quantum computer. We compile and execute ripple-carry adder (RCA) circuits for classical inputs up to five bits on this architecture. Uncomputing a workspace qubit provides an additional ``algorithmic check’’ to detect errors. While available at both the physical and logical levels, this check synergizes with our architecture by increasing the effective fault distance of many circuit locations from two to four. Combining this algorithmic check with the error-detecting codes, we observed only a single logical failure in one-bit addition after burst-error filtering, yielding an estimated logical failure rate of $(6.0\pm 5.9)\times {10}^{-5}$. This is well below the $(3.5\pm 1.3)\times {10}^{-3}$ failure probability observed for one-bit addition on the same hardware in unencoded form. Two- and three-bit adders also showed performance improvements, while the longer four- and five-bit adders executed just above threshold.