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arXiv:2311.00171 · Physical Review A

Coin dimensionality as a resource in quantum metrology involving discrete-time quantum walks

Simone Cavazzoni, Luca Razzoli, Giovanni Ragazzi, Paolo Bordone, Matteo G. A. Paris

Published Feb 22, 2024• 7 citations
arXivHTMLPDF

Abstract

We address metrological problems where the parameter of interest is encoded in the internal degree of freedom of a discrete-time quantum walker, and provide evidence that coin dimensionality is a potential resource to enhance precision. In particular, we consider estimation problems where the coin parameter governs rotations around a given axis and show that the corresponding quantum Fisher information (QFI) may increase with the dimension of the coin. We determine the optimal initial state of the walker to maximize the QFI and discuss whether, and to what extent, precision enhancement may be achieved by measuring only the position of the walker. Finally, we consider Grover-like encoding of the parameter and compare results with those obtained from rotation encoding.

DOI
10.1103/physreva.109.022432

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We address metrological problems where the parameter of interest is encoded in the internal degree of freedom of a discrete-time quantum walker, and provide evidence that coin dimensionality is a potential resource to enhance precision. In particular, we consider estimation problems where the coin parameter governs rotations around a given axis and show that the corresponding quantum Fisher information (QFI) may increase with the dimension of the coin. We determine the optimal initial state of the walker to maximize the QFI and discuss whether, and to what extent, precision enhancement may be achieved by measuring only the position of the walker. Finally, we consider Grover-like encoding of the parameter and compare results with those obtained from rotation encoding.
In discrete-time quantum walk (DTQW) the walker's coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to obtain the value of the coin parameter $\ensuremath{\theta}$ by performing measurements on the sole position space of the walker. In this paper, we evaluate the ultimate quantum limits to precision for this class of estimation protocols and use this result to assess measurement schemes having limited access to the position space of the walker in one dimension. We find that the quantum Fisher information (QFI) of the walker's position space ${H}_{w}(\ensuremath{\theta})$ increases with $\ensuremath{\theta}$ and with time, which, in turn, may be seen as a metrological resource. We also find a difference in the QFI of bounded and unbounded DTQWs and provide an interpretation of the different behaviors in terms of interference in the position space. Finally, we compare ${H}_{w}(\ensuremath{\theta})$ to the full QFI ${H}_{f}(\ensuremath{\theta})$, i.e., the QFI of the walkers position plus coin state, and find that their ratio is dependent on $\ensuremath{\theta}$, but saturates to a constant value, meaning that the walker may probe its coin parameter quite faithfully.
The quantum Fisher information of a quantum state with respect to a certain parameter quantifies the sensitivity of the quantum state to changes in that parameter. Maximizing the quantum Fisher information is essential for achieving the optimal estimation precision of quantum sensors. A typical quantum sensor involves a qubit(e.g. a spin-1/2) probe undergoing an unknown rotation, here the unknown rotation angle is the parameter to be estimated. A well known limitation is that if the rotation axis is unknown, the maximal quantum Fisher information is impossible to attain. This limitation has been lifted recently by leveraging entanglement between the probe qubit and an ancilla qubit. Namely, through measurement of the ancilla after the axis is revealed, one can prepare the probe that is optimal for any unknown rotation axis. This proposal, however, works only for a spin-1/2. Considering large spin probes can achieve a larger quantum Fisher information, offering enhanced metrological advantage, we here utilize the entanglement between a large spin probe and an ancilla to achieve optimal quantum Fisher information for estimating the rotation angle, without prior knowledge of the rotation axis. Different from the previous spin-1/2 case, achieving the optimal precision with large spins generally requires post-selection, resulting in a success probability dependent on the dimension of the Hilbert space. Furthermore, we extend the encoding state from the maximally entangled case to general entangled states, showing that optimal metrology can still be achieved with a certain success probability.
Discrete-time quantum walk (DTQW) represents a convenient mathematical framework for describing the motion of a particle on a discrete set of positions when this motion is conditioned by the values of certain internal degrees of freedom, which are usually referred to as the coin of the particle. As such, and owing to the inherent dependence of the position distribution on the coin degrees of freedom, DTQWs naturally emerge as promising candidates for quantum metrology. In this paper, we explore the use of DTQWs as quantum probes in scenarios where the parameter of interest is encoded in the internal degree of freedom of the walker and investigate the role of the topology of the walker's space on the attainable precision. In particular, we start considering the encoding of the parameter by rotations for a walker on the line and evaluate the quantum Fisher information (QFI) and the position Fisher information (FI), explicitly determining the optimal initial state in position space that maximizes the QFI across all encoding schemes. This allows us to understand the role of interference in the position space and to introduce an optimal topology, which maximizes the QFI of the coin parameter and makes the position FI equal to the QFI.
We address multiparameter quantum estimation for one-dimensional discrete-time quantum walks and its applications to quantum metrology. We use the quantum walker as a probe for the unknown parameters encoded in its coin degrees of freedom. We find an analytic expression of the quantum Fisher information matrix for the most general coin operator, and show that only two out of the three coin parameters can be accessed. We also prove that the resulting two-parameter coin model is asymptotically classical, i.e., the Uhlmann curvature vanishes. Finally, we apply our findings to relevant case studies, including the simultaneous estimation of charge and mass in the discretized Dirac model.
We investigate state estimation in discrete-time quantum walks with a single absorbing boundary. Using a spectral approach, we obtain closed expressions for the escape probability as a function of the initial coin state and the boundary position, together with the corresponding classical Fisher information for a binary absorption readout. Comparison with the single-copy quantum Fisher information reveals a clear complementarity: near boundaries carry broad information about the polar (Bloch-sphere) angle of the coin state, whereas moderate or distant boundaries reveal phase-sensitive regions. Because a single boundary probes only one information direction, combining two boundary placements yields, generically, a full-rank Fisher matrix and tight joint Cramér-Rao bounds while retaining a binary measurement without mode-resolved tomography. We also discuss a restricted-readout photonic implementation in which an on-chip sink realizes the absorber, and we frame the resulting advantage as a potential reduction in measurement-setting and reconfiguration overhead for low-dimensional parameter estimation tasks in architectures where direct projective access to the coin is unavailable. Our results show that absorption in quantum walks defines an analytically tractable restricted-access primitive for coin-state estimation.
Quantum walks have been shown to be important for quantum metrological tasks, in particular for the estimation of the evolution parameters of the walk. In this work, we address the enhancement of this parameter estimation using the driven discrete-time quantum walk (DDTQW), which is a variant of the discrete-time quantum walk with multiple walkers. DDTQW has two regimes based on the interference between the walker number, i.e., phase matched and phase mismatched. We derive an expression for the quantum Fisher information (QFI) in the phase-matched regime of DDTQW driven using the squeezing operator, demonstrating an exponential increase in QFI. In the phase-mismatched regime, QFI varies as t2{t}^{2}t2, consistent with previous studies. Our analysis shows that parameter estimation can be improved by driving the walk using squeezing operators.