PHYSICAL REVIEW A 108, 042607 (2023) Phase estimation in driven discrete-time quantum walks Shivani Singh, * Craig S. Hamilton, † and Igor Jex ‡ FNSPE, Czech Technical University in Prague, Brêhová 7, 119 15 Praha 1, Czech Republic (Received 27 June 2023; accepted 25 September 2023; published 12 October 2023) 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 quan- tum 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 t 2 , consistent with previous studies. Our analysis shows that parameter estimation can be improved by driving the walk using squeezing operators. DOI: 10.1103/PhysRevA.108.042607 I. INTRODUCTION Quantum walks (QWs) are the subject of numerous studies in the field of quantum information processing applications [1–3]. Theoretically, QWs have been shown to be universal for quantum computation [4–6] and have been used in vari- ous search algorithms [7,8]. They are also a popular tool for quantum simulations, e.g., photosynthesis [9], Anderson lo- calization [10–12], Dirac cellular automata [13–15], neutrino oscillation [16], topological phenomena [17–19], and many more [20,21]. Experimentally, they have been implemented on various physical systems such as nuclear magnetic resonance (NMR) [22], photonics [23–25], trapped ions [26–28], and waveguide arrays [29,30]. They are the quantum analog of classical random walks (CRWs) [31–33], where the quantum walker can be in a superposition of position states on the defined spatial network of sites. One of the most common features of QWs is that they can spread faster than CRWs in space due to interference phenomena of the walker. A QW has two main forms: continuous-time quantum walk (CTQW) and discrete-time quantum walk (DTQW) [34–38]. CTQW is described on a position Hilbert space and by a Hamiltonian that generally represents a graph topology (of edges and vertices) that the walker can move on. DTQW is defined on combined Hilbert spaces, i.e., the position and coin space (internal degree of freedom) of the walker. The latter variant usually evolves iteratively, first by applying a unitary coin operator that “flips” the coin state, and then a conditional shift operator that moves the walker in position space, dependent upon its coin state. The dynamics of the walk are generally controlled by the coin operation, which, for a one-dimensional (1D) walk with two internal coin states, is * singhshi@fjfi.cvut.cz † hamilcra@fjfi.cvut.cz ‡ igor.jex@fjfi.cvut.cz represented by an SU(2) operator [39], with three independent parameters. The driven discrete-time quantum walk (DDTQW) [40,41] is a variation of DTQW, motivated by the experimental realizations of photonic QWs in waveguide arrays with a nonlinear down-conversion process and optical delay loops pumped with laser light [42–44]. In DDTQW, walkers can be coherently created and destroyed at each time step of the walk, possibly interfering with walkers already present in the system. This is represented by the addition of extra terms in the iterative evolution, which in previous studies were either displacement or squeezing operators, representing the pumping by coherent and squeezed light, respectively. It was shown that DDTQW can have very different dynamics than the original DTQW, primarily due to phase-matching conditions between the pumped terms and eigenmodes of the system. Parameter estimation has applications in various fields of modern science, e.g., gravitational wave detection, mi- croscopy and imaging, Hamiltonian estimation, and general sensing technologies [45–48]. The measurement uncertainty can be characterized by the quantum Cramer-Rao bound, which gives a bound over precision of the estimated parameter in terms of quantum Fisher information (QFI) [49], which a detection scheme would seek to maximize. Typically, interfer- ometric methods are used to measure the parameter, which is encoded in the path difference of the interferometer. Quantum walks (QWs) can be considered as a multipath interferometer [50], and their use for parameter estimation has been explored in the past [51–53]. Here we are improving the parameter estimation by maximizing the QFI using a variant of the QW using squeezed state driving. Quantum metrological schemes have been proposed in the past and precision benchmarks have been obtained for both CTQWs [51] and DTQWs [52,53], where in the lat- ter they estimate the phase parameter of the SU(2) operator representing the coin. There it was shown that the QFI for the SU(2)-parameter estimation increases quadratically with 2469-9926/2023/108(4)/042607(11) 042607-1 ©2023 American Physical Society