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Register a new userScattering as a quantum metrology problem: a quantum walk approach
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We address the scattering of a quantum particle by a one-dimensional barrier potential over a set of discrete positions. We formalize the problem as a continuous-time quantum walk on a lattice with an impurity, and use the quantum Fisher information as a mean to quantify the maximal possible accuracy in the estimation of the height of the barrier. We introduce suitable initial states of the walker and derive the reflection and transmission probabilities of the scattered state. We show that while the quantum Fisher information is affected by the width and central momentum of the initial wave packet, this dependency is weaker for the quantum signal-to-noise ratio. We also show that a dichotomic position measurement provides a nearly optimal detection scheme.
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Quantum metrology plays a fundamental role in many scientific areas. However, the complexity of engineering entangled probes and the external noise raise technological barriers for realizing the expected precision of the to-be-estimated parameter with given resources. Here, we address this problem by introducing adjustable controls into the encoding process and then utilizing a hybrid quantum-classical approach to automatically optimize the controls online. Our scheme does not require any complex or intractable off-line design, and it can inherently correct certain unitary errors during the learning procedure. We also report the first experimental demonstration of this promising scheme for the task of finding optimal probes for frequency estimation on a nuclear magnetic resonance (NMR) processor. The proposed scheme paves the way to experimentally auto-search optimal protocol for improving the metrology precision.
Continuous-time quantum walks (CTQWs) provide a valuable model for quantum transport, universal quantum computation and quantum spatial search, among others. Recently, the empowering role of new degrees of freedom in the Hamiltonian generator of CTQWs, which are the complex phases along the loops of the underlying graph, was acknowledged for its interest in optimizing or suppressing transport on specific topologies. We argue that the quantum-classical distance, a figure of merit which was introduced to capture the difference in dynamics between a CTQW and its classical, stochastic counterpart, guides the optimization of parameters of the Hamiltonian to achieve better quantum transport on cycle graphs and spatial search to the quantum speed limit without an oracle on complete graphs, the latter also implying fast uniform mixing. We compare the variations of this quantity with the 1-norm of coherence and the Inverse Participation Ratio, showing that the quantum-classical distance is linked to both, but in a topology-dependent relation, which is key to spot the most interesting quantum evolution in each case.
We investigate the behavior of coherence in scattering quantum walk search on complete graph under the condition that the total number of vertices of the graph is greatly larger than the marked number of vertices we are searching, $N gg v$. We find that the consumption of coherence represents the increase of the success probability for the searching,also the consumption of coherence is related to the efficiency of the algorithm represented by oracle queries.If no coherence is consumed, the efficiency of the algorithm will be the same as the classical blind search, implying that coherence is responsible for the speed up in this quantum algorithm over its classical counterpart. In case the initial state is incoherent, still $N gg v$ is assumed,the probability of success for searching will not change with time, indicating that this quantum search algorithm loses its power.We then conclude that the coherence plays an essential role and is responsible for the speed up in this quantum algorithm.
Numerical methods for the 1-D Dirac equation based on operator splitting and on the quantum lattice Boltzmann (QLB) schemes are reviewed. It is shown that these discretizations fall within the class of quantum walks, i.e. discrete maps for complex fields, whose continuum limit delivers Dirac-like relativistic quantum wave equations. The correspondence between the quantum walk dynamics and these numerical schemes is given explicitly, allowing a connection between quantum computations, numerical analysis and lattice Boltzmann methods. The QLB method is then extended to the Dirac equation in curved spaces and it is demonstrated that the quantum walk structure is preserved. Finally, it is argued that the existence of this link between the discretized Dirac equation and quantum walks may be employed to simulate relativistic quantum dynamics on quantum computers.
The study of quantum walks has been shown to have a wide range of applications in areas such as artificial intelligence, the study of biological processes, and quantum transport. The quantum stochastic walk, which allows for incoherent movement of the walker, and therefore, directionality, is a generalization on the fully coherent quantum walk. While a quantum stochastic walk can always be described in Lindblad formalism, this does not mean that it can be microscopically derived in the standard weak-coupling limit under the Born-Markov approximation. This restricts the class of quantum stochastic walks that can be experimentally realized in a simple manner. To circumvent this restriction, we introduce a technique to simulate open system evolution on a fully coherent quantum computer, using a quantum trajectories style approach. We apply this technique to a broad class of quantum stochastic walks, and show that they can be simulated with minimal experimental resources. Our work opens the path towards the experimental realization of quantum stochastic walks on large graphs with existing quantum technologies.
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