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arXiv:1908.05047 · Physical Review Letters

Graph States as a Resource for Quantum Metrology

Nathan Shettell, Damian Markham

Published Mar 17, 2020• 85 citations
arXivHTMLPDF

Abstract

By using highly entangled states, quantum metrology guarantees a precision impossible with classical measurements. Unfortunately such states can be very susceptible to noise, and it is a great challenge of the field to maintain quantum advantage in realistic conditions. In this Letter we investigate the practicality of graph states for quantum metrology. Graph states are a natural resource for much of quantum information, and here we characterize their quantum Fisher information for an arbitrary graph state. We then construct families of graph states which approximately achieves the Heisenberg limit, we call these states bundled graph states. We demonstrate that bundled graph states maintain a quantum advantage after being subjected to independent and identically distributed dephasing or finite erasures. This shows that these graph states are good resources for robust quantum metrology. We also quantify the number of n qubit stabilizer states that are useful as a resource for quantum metrology.

DOI
10.1103/physrevlett.124.110502

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By using highly entangled states, quantum metrology guarantees a precision impossible with classical measurements. Unfortunately such states can be very susceptible to noise, and it is a great challenge of the field to maintain quantum advantage in realistic conditions. In this Letter we investigate the practicality of graph states for quantum metrology. Graph states are a natural resource for much of quantum information, and here we characterize their quantum Fisher information for an arbitrary graph state. We then construct families of graph states which approximately achieves the Heisenberg limit, we call these states bundled graph states. We demonstrate that bundled graph states maintain a quantum advantage after being subjected to independent and identically distributed dephasing or finite erasures. This shows that these graph states are good resources for robust quantum metrology. We also quantify the number of n qubit stabilizer states that are useful as a resource for quantum metrology.
Quantum metrology exploits quantum mechanical effects to increase the precision of measurements of physical quantities. A wide variety of applications are currently being developed for scientific and technological purposes, however, most research relies on the use of highly entangled resource states that are challenging to generate and control in a given physical system. Here, we study the use of weighted graph states as more accessible resources for quantum metrology, which yield a favorable precision beyond the classical limit, approaching the Heisenberg limit. We find a notable robustness to variation in weights and less challenging weight requirements compared to standard graph states, which require a maximal weight at all edges. Both of these aspects reduce the practical demands in a physical setup, with the latter implying significantly less entanglement is required to gain a quantum advantage in metrology. We study the quantum Fisher information and optimized estimator variance of two identified sub classes of weighted graph states for an arbitrary number of N qubits, providing analytical forms and investigating their scaling. Our work opens up opportunities for using weakly entangled states in quantum-enhanced metrology.
Quantum correlations can be used to improve sensing precision beyond classical limits, yet the correlated resource states needed can be challenging to prepare and preserve. Here, we study the family of weighted hypergraph states and identify subclasses characterized by their underlying graph topology that exhibit enhanced metrological performance. Using the quantum Fisher information and a novel Dicke state basis representation, we show that these subclasses outperform both separable and standard weighted graph state benchmarks over broad ranges of edge weights corresponding to weak controlled operations between qubits representing the vertices of the graph. We provide analytical formulas for an arbitrary number NNN of qubits, supplemented by a general study, while focusing a detailed analysis on the example case of N=4N=4N=4. By identifying the critical probabilities for local dephasing, we show that adding higher-order weighted hyperedges for NNN qubits can enhance metrological robustness, allowing the resulting states to retain a quantum advantage at higher noise levels. These probe states exhibit favorable scaling, enhanced precision, and noise tolerance. Our findings establish weighted hypergraph states as viable, structurally tunable and robust resources for quantum-enhanced metrology.
Quantum effects in metrology can in principle enhance measurement precision from the so-called standard quantum limit to the Heisenberg limit. Further advancements in quantum metrology largely rely on innovative metrology protocols that can avoid a number of known obstacles, including the challenge of preparing entangled states with sufficient fidelity, the readout noise in measuring highly entangled states, and no-go theorems for quantum metrology under noisy environments. In this Letter, exploiting some peculiar but experimentally feasible dynamical features of a collection of spins with all-to-all time-periodic interactions, we propose a metrology protocol that can circumvent all three mentioned obstacles and yet still make good use of time as a resource for metrology. Specifically, by mapping the dynamics of such a periodically driven spin system to that of a paradigm of quantum chaos but tuned to some high-order quantum resonance, it is shown that a simple SU(2) coherent state can, after evolving to highly entangled states in the ensuing dynamics, be dynamically brought back to the same initial coherent state. The associated quantum Fisher information is found to exhibit quadratic scaling with both the number of spins and the duration of the metrology protocol. The achieved Heisenberg scaling can also largely survive in the presence of Markovian noise. Representing a previously unknown strategy for quantum metrology, the protocol proposed here can be tested on available experimental platforms.
Fragile quantum features such as entanglement are employed to improve the precision of parameter estimation and as a consequence the quantum gain becomes vulnerable to noise. As an established tool to subdue noise, quantum error correction is unfortunately overprotective because the quantum enhancement can still be achieved even if the states are irrecoverably affected, provided that the quantum Fisher information, which sets the ultimate limit to the precision of metrological schemes, is preserved and attained. Here, we develop a theory of robust metrological schemes that preserve the quantum Fisher information instead of the quantum states themselves against noise. After deriving a minimal set of testable conditions on this kind of robustness, we construct a family of 2t+12t+12t+1 qubits metrological schemes being immune to ttt-qubit errors after the signal sensing. In comparison at least five qubits are required for correcting arbitrary 1-qubit errors in standard quantum error correction.
Quantum metrology is an auspicious discipline of quantum information which is currently witnessing a surge of experimental breakthroughs and theoretical developments. The main goal of quantum metrology is to estimate unknown parameters as accurately as possible. By using quantum resources as probes, it is possible to attain a measurement precision that would be otherwise impossible using the best classical strategies. For example, with respect to the task of phase estimation, the maximum precision (the Heisenberg limit) is a quadratic gain in precision with respect to the best classical strategies. Of course, quantum metrology is not the sole quantum technology currently undergoing advances. The theme of this thesis is exploring how quantum metrology can be enhanced with other quantum techniques when appropriate, namely: graph states, error correction and cryptography.
Quantum metrology is a promising practical use case for quantum technologies, where physical quantities can be measured with unprecedented precision. In lieu of quantum error correction procedures, near term quantum devices are expected to be noisy, and we have to make do with noisy probe states. We prove that, for a set of carefully chosen symmetric probe states that lie within certain quantum error correction codes, quantum metrology exhibits an advantage over classical metrology even after the probe states are corrupted by a constant number of erasure and dephasing errors. These probe states prove useful for robust metrology not only in the NISQ regime, but also in the asymptotic setting where they achieve Heisenberg scaling. This brings us closer towards making robust quantum metrology a technological reality.