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Quantum error correction protects fragile quantum information by encoding it into a larger quantum system1,2. These extra degrees of freedom enable the detection and correction of errors, but also increase the control complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while controlling the logical qubit, and are essential for realizing error suppression in practice3-6. Although fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. Here we experimentally demonstrate fault-tolerant circuits for the preparation, measurement, rotation and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault-tolerant protocols, we see significant reductions in the error rates ofthe logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6 per cent and a Clifford gate error of 0.3 per cent after offline error correction. In addition, we prepare magic states with fidelities that exceed the distillation threshold7, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant control. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.
Quantum computers are promising for solving models of important physical processes, optimizing complex cost functions and challenging cryptography in ways that are intractable using current computers8-12. However, realistic quantum component failure rates are typically too high to achieve these goals13,14. These applications will therefore almost certainly require quantum error-correction schemes to greatly suppress errors4,6.
Quantum error-correcting codes combine multiple physical qubits into logical qubits that robustly store information within an entangled state1,2,15. However, these codes are not enough on their own. Fault-tolerant (FT) operations, which limit the ways in which errors can spread throughout the system, must also be used. Without them, the logical error rate may be limited by faults at critical circuit locations that cascade into logical failures, negating the advantage of error correction.
FT state preparation, detection and operations have been demonstrated using quantum error-detecting codes with four data qubits16-20. These codes can identify when errors have occurred, but do not extract enough information to...