  Quantum state readout is a key component of quantum technologies, including
applications in sensing, computation, and secure communication. Readout
fidelity can be enhanced by repeating readouts. However, the number of repeated
readouts is limited by measurement backaction, which changes the quantum state
that is measured. This detrimental effect can be overcome by storing the
quantum state in an ancilla qubit, chosen to be robust against measurement
backaction and to allow error correction. Here, we protect the electronic-spin
state of a diamond nitrogen-vacancy center from measurement backaction using a
robust multilevel 14N nuclear spin memory and perform repetitive readout, as
demonstrated in previous work on bulk diamond devices. We achieve additional
protection using error correction based on the quantum logic of coherent
feedback to reverse measurement backaction. The repetitive spin readout scheme
provides a 13-fold enhancement of readout fidelity over conventional readout
and the error correction a 2-fold improvement in the signal. These experiments
demonstrate full quantum control of a nitrogen vacancy center electronic spin
coupled to its host 14N nuclear spin inside a ~25 nm nanodiamond, creating a
sensitive and biologically compatible platform for nanoscale quantum sensing.
Our error-corrected repetitive readout scheme is particularly useful for
quadrupolar nuclear magnetic resonance imaging in the low magnetic field regime
where conventional repetitive readout suffers from strong measurement
backaction. More broadly, methods for correcting longitudinal (bit-flip) errors
described here could be used to improve quantum algorithms that require
nonvolatile local memory, such as correlation spectroscopy measurements for
high resolution sensing.
