Weight Distribution of Classical Codes Influences Robust Quantum Metrology
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Quantum metrology (QM) is expected to be a prominent use-case of quantum technologies. However, noise easily degrades these quantum probe states, and negates the quantum advantage they would have offered in a noiseless setting. Although quantum error correction (QEC) can help tackle noise, fault-tolerant methods are too resource intensive for near-term use. Hence, a strategy for (near-term) robust QM that is easily adaptable to future QEC-based QM is desirable. Here, we propose such an architecture by studying the performance of quantum probe states that are constructed from [n,k,d] binary block codes of minimum distance d ≥ t+1. Such states can be interpreted as a logical state of a CSS code whose logical X group is defined by the aforesaid binary code. When a constant, t, number of qubits of the quantum probe state are erased, using the quantum Fisher information (QFI) we show that the resultant noisy probe can give an estimate of the magnetic field with a precision that scales inversely with the variances of the weight distributions of the corresponding 2^t shortened codes. If C is any code concatenated with inner repetition codes of length linear in n, a quantum advantage in QM is possible. Hence, given any CSS code of constant length, concatenation with repetition codes of length linear in n is asymptotically optimal for QM with a constant number of erasure errors. We also explicitly construct an observable that when measured on such noisy code-inspired probe states, yields a precision on the magnetic field strength that also exhibits a quantum advantage in the limit of vanishing magnetic field strength. We emphasize that, despite the use of coding-theoretic methods, our results do not involve syndrome measurements or error correction. We complement our results with examples of probe states constructed from Reed-Muller codes.
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