Achieving the Heisenberg limit in quantum metrology using quantum error correction (1706.02445v2)

Abstract: Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available, and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, a quantum error-correcting code can be constructed which suppresses the noise without obscuring the signal; the optimal code, achieving the best possible precision, can be found by solving a semidefinite program.

• The paper establishes that applying QEC under the HNLS condition enables Heisenberg-limit scaling in quantum metrology despite Markovian noise.
• It introduces a sequential scheme using rapid error corrections with probes and noiseless ancillas to isolate the Hamiltonian signal from noise.
• The framework has significant implications for enhancing precision in quantum sensors, benefiting applications like frequency spectroscopy and gravitational wave detection.

Achieving the Heisenberg Limit in Quantum Metrology Using Quantum Error Correction

The studied paper focuses on a pivotal challenge in quantum metrology: enhancing measurement precision despite the presence of noise, using quantum error correction (QEC) to reach the Heisenberg limit (HL) of precision. The HL represents the ultimate bound on measurement precision in quantum mechanics, scaling inversely with the total probing time or the number of probes. Although attainable in noiseless systems, the presence of noise usually restricts achievable precision to the standard quantum limit (SQL), which scales inversely with the square root of the number of probes. This research formulates explicit conditions under which QEC can be leveraged to restore HL scaling in the presence of Markovian noise, assuming the availability of noiseless ancillas and rapid quantum processing.

Summary of Results and Methods

The paper establishes the necessary and sufficient condition for achieving HL precision as the Hamiltonian-not-in-Lindblad-span (HNLS) condition. When utilizing quantum probes subject to Markovian noise, HL scaling in parameter estimation is possible if the system Hamiltonian is not in the linear span of the noise Lindblad operators. This is substantiated by constructing a specific QEC code that isolates the Hamiltonian evolution from noise effects. Conversely, if the HNLS condition is not met, the quantum Cramer-Rao bound indicates that SQL scaling cannot be surpassed.

The sequential scheme, a pivotal concept discussed, involves a single probe making parameter estimations over multiple rounds interspersed with quantum error corrections. This scheme, employing rapid quantum operations on probes and ancillas, ensures error correction after each infinitesimal time step, enabling effective decoupling of noise effects and preservation of the Hamiltonian signal.

Practical Implications and Future Directions

The practical impact of this work lies in the enhancement of precision in quantum sensors. By defining when and how QEC can benefit metrology, this research offers a framework that combines noise control with signal detection, potentially benefiting fields like frequency spectroscopy and gravitational wave detection. The methodology is especially applicable to settings where quantum controls outpace decoherence rates, as demonstrated in superconducting circuits, thus pushing experimental precision closer to fundamental physical limits.

Future exploration could address several avenues:

1. Optimal QEC Code Construction: Although the paper offers approaches for code optimization, further exploration into more computationally efficient algorithms for specific quantum systems remains valuable.
2. Non-Markovian and Realistic Noise Models: Extending the work to encompass non-Markovian environments and more intricate noise models would enhance its applicability, particularly in complex or hybrid systems.
3. Error-Corrected Metrological Enhancement Beyond HNLS: Consideration of strategies for partial noise corrections that still offer measurement improvements, even when HNLS is not satisfied, can broaden the utility of QEC in less controlled or noisier scenarios.

In summary, the findings delineate the boundaries within which QEC can effectively enable HL scaling in quantum metrology, providing a nuanced understanding crucial for the development of high-precision quantum technologies.