Ancilla-Free Quantum Error Correction Codes for Quantum Metrology

Abstract

Quantum error correction has recently emerged as a tool to enhance quantum sensing under Markovian noise. It works by correcting errors in a sensor while letting a signal imprint on the logical state. This approach typically requires a specialized error-correcting code, as most existing codes correct away both the dominant errors and the signal. To date, however, few such specialized codes are known, among which most require noiseless, controllable ancillas. We show here that such ancillas are not needed when the signal Hamiltonian and the error operators commute; a common limiting type of decoherence in quantum sensors. We give a semidefinite program for finding optimal ancilla-free sensing codes in general, as well as closed-form codes for two common sensing scenarios: qubits undergoing dephasing, and a lossy bosonic mode. Finally, we analyze the sensitivity enhancement offered by the qubit code under arbitrary spatial noise correlations, beyond the ideal limit of orthogonal signal and noise operators.

Distributed quantum sensing enhanced by continuous-variable error correction

“…There has been various proposals to improve the sensing precision with error correction codes in DV systems [39][40][41][42][43][44][45]. Most of these works consider a Hamiltonian parameter estimation scenario, where frequent error correction steps are applied to suppress the noise without at the same time suppressing the signal.…”

Optimal approximate quantum error correction for quantum metrology

“…The task involves preparing a suitable initial state of the system, allowing it to evolve under quantum controls for a specific time, performing a suitable measurement, and inferring the value of the unknown system parameter from the measurement outcome. To enhance the estimation precision, a variety of quantum strategies have been proposed, such as squeezing the initial state [7][8][9][10][11][12], optimizing the probing time [13], monitoring the environment [14][15][16], exploiting non-Markovian effects [17][18][19], optimizing the control Hamiltonian [20][21][22] and quantum error correction [23][24][25][26][27][28][29][30][31][32][33][34].Quantum mechanics places a fundamental limit on estimation precision, the Heisenberg limit (HL), where the estimation precision scales like 1/N for N probes; or equivalently, 1/t for a total probing time t. In the noiseless case, the HL is achievable using the maximally entangled state among probes [1,35]. In practice, decoherence plays an indispensible role.…”

“…In the noisy case, it was proven that the HL is achievable if and only if H / ∈ S (the HNLS condition) and there exists a QEC strategy achieving the HL [30,31].The HNLS condition holds usually when the noise has a special structure, e.g. rank-one noise [29] or spacially correlated noise [32,33]. For generic noise, however, the HNLS condition is often violated.…”

“…Using the vectorization of matrices |· = jk j| (·) |k |j |k , let.According to the Cauchy-Schwarz inequality,and the optimal |C = B −1 |H h . Here −1 means the Moore-Penrose pseudoinverse.Highly-biased noise.-We consider a special case where noises are separated into two groups -strong ones and weak ones [31][32][33]. To be specific, we consider the following quantum master equationwhere the indices of Lindblad operators {L i } r i=1 are separated intos and s, representing weak and strong noises respectively.…”

“…where |0 L and |1 L are the logical zero and one states. Applying the AQEC quantum operation P + R • P ⊥ infinitely fast, the effective evolution would be (up to the first order of dt [31,33])…”