"Achieving the Heisenberg limit using ancilla-free quantum error-correcting codes"

Quantum error correction is theoretically capable of achieving the ultimate estimation limits in noisy quantum metrology. However, existing quantum error-correcting codes designed for quantum sensing under Markovian noise generally contain noiseless ancillas, one for each probe, which is one of the major obstacles to implementing it in practice. Here we successfully lift the requirement of noiseless ancillas by explicitly constructing two types of new quantum error-correcting codes, where the first code entangles multiple probes with an exponentially small ancilla and the second one is an ancilla-free random code. We show that, for general Hamiltonian estimation under Markovian noise, whenever the Heisenberg limit is achievable using quantum strategies, our new codes can both achieve the Heisenberg limit and the optimal rate at which the estimation error scales with the number of probes.