Session N52: Theoretical Advances in Quantum Metrology

Variational quantum metrology for multiparameter estimation under noise

We introduce a hybrid quantum-classical variational approach designed to elevate precision in quantum metrology. In the scheme, both the initial state and the measurement basis in the quantum part are parameterized and optimized via the classical part. This integrated approach facilitates the maximization of information acquired regarding the measured quantity. We explore its practical applications in the context of 3D magnetic field sensing, accounting for various dephasing noise scenarios. Our results showcase its capacity to simultaneously estimate all relevant parameters, surpassing the constraints of the standard quantum limit. This innovation emerges as a robust and invaluable tool for a wide range of metrological applications.   

Exploring Measurement and Feedforward-Assisted Quantum Metrology

A multitude of recent studies have proposed and demonstrated the use of mid-circuit measurements and feedforward to efficiently prepare entangled states on digital quantum platforms. Here, we examine the possible role of mid-circuit measurements and feedforward in a quantum metrology context. We consider the classic phase shift estimation problem, which is standard quantum limited on single qubits but enjoys Heisenberg scaling on GHZ states with collective measurements. It is known that unitary GHZ state preparation requires linear circuit depth on a linear nearest-neighbour qubit connectivity topology, but this complexity can be reduced to constant depth with measurements and feedforward. More broadly, we investigate circuit structures that interpolate between the unitary and measurement-based extremes, on both the pre- and post-processing portions of the experiment circuits. We show in a simplified setting that different families of such circuits are preferable for quantum metrology depending on noise characteristics. We investigate the applicability of measurement-assisted circuits for metrology on present-day quantum devices.   

Analysis of the optimal solution of quantum metrology with quantum decoherence

Optimal control theory is applied to the quantum parameter estimation in the presence of decoherence. Concretely we look for the optimal control that maximizes quantum Fisher information for “twist and turn” problem, and the optimal solutions are confirmed by testing the optimality conditions.

Although the resulting optimal controls appear complicated, we identify two classes of problem-specific characterisitic states, the Heisenberg-limit (HL) state and the decoherence-free subspace, which are crucial in understanding the optimal quantum trajectories.

For the intemediate evolution time where the quantum decoherence is still small, the optimal control takes the system to the HL state. When the evolution time is long, the optimal control takes the system to the decoherence free subspace.   

Scaling of quantum Fisher information for quantum exceptional point sensors

In recent years, significant progress has been made in utilizing the spectrum divergence at the exceptional point for sensing in classical systems. However, the use and characterization of quantum exceptional points for sensing have been largely unexplored. This raises the question of the relationship between the order of the quantum exceptional point and the scaling of quantum Fisher information, an essential quantity for characterizing quantum sensor. Here we investigate multi-mode quartic Bosonic systems, which exhibit higher-order exceptional point dynamics, but possess Hermitian Hamiltonians without Langevin noise.  We derive an exact analytic formula for the quantum Fisher information, from which we establish a scaling relation between the quantum Fisher information and the order of the exceptional point. Our work builds the connection among three important fields: non-Hermitian exceptional point dynamics, quantum sensing, and entangled squeezed states, and may find important applications in quantum sensing and quantum non-Hermitian physics.   

Quantum metrology with finite-energy GKP states

Finite-energy GKP states have been shown to be sensitive single-mode displacements sensors [1]. Moreover, quantum error correction has been proposed as a tool to enhanced the accuracy of metrology protocols [2]. Here, we explore the connection between these concepts and study the use of stabilized quantum error corrected finite-energy GKP states as quantum sensors. We analyze their use as single-mode displacement sensors, and provide protocols to improve the measurement sensitivity achievable with uncorrected finite-energy GKP states. Going beyond the single-mode scenario, we explore multi-mode GKP sensitivity improvements over their single-mode counterparts. Finally, we explore potential applications.

[1] Duivenvoorden, K., Terhal, B. M., & Weigand, D. Phys. Rev. A 95, 012305 (2017).

[2] Kessler, E. M., Lovchinsky, I., Sushkov, A. O., & Lukin, M. D. Phys. Rev. Lett. 112, 150802 (2014).   

Sensing applications utilizing the quantum Bayesian minimum square error

We consider the genuine quantum sensing Bayesian approach, i.e., the unknown parameter is a random value that follows a prior probability distribution function (pPDF). We compute the minimum mean square error (MMSE) which is always attainable by a measurement. We consider sensing of loss rate and optical phase.

For sensing loss rate, we first consider the two-point pPDF and we prove analytically the optimality of the Fock state and photon-counting measurement. For the beta pPDF our numerical investigation provides evidence on the optimality of the Fock state and photon-counting.

For our optical phase sensing probe, we consider the generic fixed-photon state. For the NOON state, which is a special case of our probe state and for a flat pPDF we find that the Heisenberg scaling is lost and the attainable MMSE is found to be proportional to -n^-2, which is a manifestation of the fundamental difference between the Fisherian and Bayesian approaches. Then, we consider an adaptive technique: We start with a flat pPDF, and for each next step we use as pPDF the posterior probability of the previous step, while for each step we update the optimal state and optimal measurement. We compute the Bayesian MMSE per step. We draw conclusions on the performance of optimal probe states.   

Variational quantum sensing

Sensing strategies that leverage quantum correlations, such as entanglement, can reach precisions beyond the fundamental limits of classical techniques, and are broadly relevant, e.g., in optical and atomic interferometry. However, constraints imposed in near-term quantum devices, such as noise processes, limited number of qubits, and limited sampling rates, motivate the need for quantum sensing protocols that are adaptive to device capabilities. In recent years, learning techniques have found wide-spread application in quantum technologies; through variational quantum algorithms and classical machine-learning techniques trained on data from quantum devices. In this work, towards protocol adaptivity, we frame each step of a quantum sensing protocol as a variational learning problem. First, the maximum achievable precision, bounded by the Cramér–Rao bound, is optimized by tuning the parameters of the probe state preparation and detection. Next, a classical neural network is trained on simulated, single shot data from the optimized device and used to efficiently estimate the unknown parameter. Using this approach, we design sensing protocols that mitigate the effects of noisy state preparation, and reach precisions that surpass the standard quantum limit. Our framework provides a promising avenue for designing entanglement-enhanced sensing protocols with realistic, near-term quantum devices.   

Optimal omni-directional field sensing with entangled states

A qubit (spin-half system) subject to an unknown rotation serves as a paradigmatic setup for studying fundamental limits in quantum metrology. The optimal quantum Fisher information about the parameter is obtained when the spin is prepared in a state that maximizes the variance of the operator that induces the rotation. However, if the rotation angle is unknown an optimal single-qubit sensor cannot be prepared. In this work, inspired by simulations of closed time-like curves, we circumvent this limitation, achieving the optimal quantum Fisher information about a rotation angle for any rotation axis. To achieve this result, we prepare the probe qubit in an entangled state with an ancilla qubit, then measure the pair in an entangled basis, obtaining more information about the rotation than achievable with a single-qubit sensor. We demonstrate this metrological advantage using a two-superconducting-qubit quantum processor.   

Mitigating Markovian Noise in DC Magnetometry via Zero-Noise Extrapolation

Quantum sensors seek to leverage the sensitivity of quantum systems to their environment as a resource to achieve advantages over classical sensors. However, the viability of quantum sensing is challenged in practical settings where noise can lead to undesired evolution and limitations on sensitivity. Within the field of quantum computation, many methods have been developed to address errors induced by noise. Zero-noise extrapolation (ZNE) is one such approach commonly utilized in currently available quantum processors to reduce the effect of noise during the estimation of expectation values. ZNE involves artificially amplifying noise in a controlled way and using measurement outcomes to extrapolate to the zero-noise limit. While ZNE has been extensively studied in quantum computation, it has not been considered in the domain of quantum sensing. In this work, we adapt ZNE to the sensing problem and investigate its effectiveness in mitigating noise in DC magnetometry. We focus on Markovian noise environments and develop unitary folding protocols for the estimation of magnetic field parameters. We demonstrate that ZNE can be used to improve sensing accuracy relative to the standard Ramsey protocol.   

Improving Quantum Sensors Subject to Correlated Noise Using Zero-noise Extrapolation

We apply zero-noise extrapolation (ZNE), a well-known error mitigation technique for noisy, intermediate scale quantum computers, to the domain of quantum sensing. We evaluate its effectiveness for the prototypical examples of DC and AC magnetometry using entangled qubit sensors. We focus on the realistic case of spatiotemporally correlated noise, employing the filter function formalism to analyze the impact on sensor performance. Our results suggest that ZNE is a viable method for enhancing quantum sensing capabilities in the presence of highly-correlated noise.   

Achieving the Heisenberg limit with Dicke states in noisy quantum metrology

Going beyond the standard quantum limit in noisy quantum metrology is a very challenging task. Here we show how Dicke states can be used to surpass the standard quantum limit and achieve the Heisenberg limit in open quantum systems. The system we study has qubits symmetrically coupled to a resonator and our objective is to estimate the coupling between the qubits and the resonator. The time-dependent quantum Fisher information with respect to the coupling is studied for this open quantum system where the same decay rates are assumed on all qubits. We show that when the system is initialized to a Dicke state with an optimal excitation number one can go beyond the standard quantum limit and achieve the Heisenberg limit even for finite values of the decays on the qubit and the resonator, particularly when the qubits and resonator are strongly coupled. For comparison, we find that the highly entangled GHZ state performs quite poorly. Our results show that one must consider not only the degree of entanglement of an initial probe state, but its resilience to noise in order to achieve optimum sensing performance.   

Constraints of Average Hamiltonian Theory for Quantum Sensing

Average Hamiltonian theory (AHT) approximates the evolution of a closed quantum system by defining a time-independent effective Hamiltonian, obtained through a series expansion solution of the Schrodinger equation. By generating analytical descriptions that provide insights into complex dynamical systems, AHT has demonstrated broad use in the fields of nuclear magnetic resonance (NMR) and quantum information science (QIS). More recently, application of AHT in quantum sensing has resulted in advanced magnetometry protocols for solid-state spin defects (Choi et al. 2020, Zhou et al. 2020). However, this operating regime does not meet the established criteria for convergence of the Magnus series expansion, which forms the basis of AHT. In this talk, we discuss the accuracy of AHT for (1) predictions of unitary evolution, and (2) determination of the effective gyromagnetic ratio for pulsed magnetometry protocols. These considerations are relevant to a variety of broadband and narrowband sensing sequences, and informs the attainable sensitivity associated with each protocol.   

Optimal Strategies in Post-Selected Quantum Metrology

Post-selected quantum metrology uses filters to allow detectors to operate at lower intensities without reducing the input rate of quantum information about unknown parameters of interest [1, 2, 3]. In this talk, I will present the optimal family of filters that achieves this lossless compression of information [4]. I will also show that optimal post-selection can always increase the (Fisher) information per output state, even in the presence of strong depolarising noise. This is true irrespectively of whether detector saturation or post-processing costs are dominant. Our optimal filter depends on the underlying parameters to be estimated. Thus, the best way to distil quantum information involves an adaptive strategy [5]. I will present our efforts towards constructing such an optimal adaptive strategy. Finally, I will explore potential applications of post-selection in metrology.   

Heisenberg-limited metrology with perturbing interactions

We show that it is possible to perform Heisenberg-limited metrology on GHZ-like states, in the presence of generic spatially local interactions during the measurement process. An explicit protocol, which relies on measurements and feedback based on polynomial-time classical computation, achieves the Heisenberg limit. In one dimension, matrix product state methods can be used to perform this classical calculation, while in higher dimensions the cluster expansion underlies the efficient calculations. The latter approach is based on an efficient classical sampling algorithm for short-time quantum dynamics, which may be of independent interest.   

Improved scaling of variational quantum metrology with midcircuit adaptivity

Phase estimation with only limited prior knowledge of the value of the phase has many applications. For example, atomic clocks are operated at the largest interrogation time at which phase wraps can still be neglected and quantum algorithms based on phase estimation require the ability to estimate a phase of any value. Typically, the averaged mean squared error will decrease as a power of the system size. It was previously found that variational metrology schemes using only one or two one-axis twists exhibit a reduced scaling exponent compared to the scaling exponent associated with the variance alone. Recently, midcircuit measurements have been demonstrated in neutral atom array, superconducting qubit, and trapped ion systems. Inspired by this, we show that with only a few rounds of adaptivity variational schemes with only a single state preparation twist and no measurement twists can achieve significantly improved scaling. We further investigate the robustness of these strategies to noise and the performance of adaptive strategies utilizing more entangling resources.