Demonstration of Shor encoding on a trapped-ion quantum computer

Abstract

Fault-tolerant quantum error correction (QEC) is crucial for unlocking the true power of quantum computers. QEC codes use multiple physical qubits to encode a logical qubit, which is protected against errors at the physical qubit level. Here we use a trapped ion system to experimentally prepare m-qubit GHZ states and sample the measurement results to construct m × m logical states of the [[m 2 , 1, m]] Shor code, up to m = 7. The synthetic logical fidelity shows how deeper encoding can compensate for additional gate errors in state preparation for larger logical states. However, the optimal code size depends on the physical error rate and we find that m = 5 has the best performance in our system. We further realize the direct logical encoding of the [[9, 1, 3]] Shor code on nine qubits in a thirteen-ion chain for comparison, with 98.8(1)% and 98.5(1)% fidelity for state |± L , respectively.

Mitigating errors by quantum verification and post-selection

“…This allows−along with adequate ancilla qubit (syndrome) measurements and classical postprocessing−for the error rate affecting each logical qubit to be suppressed exponentially (in n) [15,17], provided that the error rates of individual physical qubits are below a constant value, named the fault-tolerance threshold [16]. Unfortunately, although QEC is fully scalable, the overhead of physical qubits required to run QEC, for circuit sizes where useful quantum advantage is expected, is still beyond the reach of current and nearterm quantum hardware; though promising results are beginning to surface in this direction [18][19][20].…”

Optimizing Stabilizer Parities for Improved Logical Qubit Memories

“…1 is the standard 9-qubit Shor's code. The logical state preparation and measurement of this code has been demonstrated with trapped ions [24,25] and photons [26]. The phase errors on a given GHZ state combine constructively, as can be seen for the states shown in Eq.…”