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arXiv:1502.06459 · New Journal of Physics

Quantum metrology for the Ising Hamiltonian with transverse magnetic field

Michalis Skotiniotis, Pavel Sekatski, Wolfgang Dür

Published Jul 31, 2015• 60 citations
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Abstract

We consider quantum metrology for unitary evolutions generated by parameter-dependent Hamiltonians. We focus on the unitary evolutions generated by the Ising Hamiltonian that describes the dynamics of a one-dimensional chain of spins with nearest-neighbour interactions and in the presence of a global, transverse, magnetic field. We analytically solve the problem and show that the precision with which one can estimate the magnetic field (interaction strength) given one knows the interaction strength (magnetic field) scales at the Heisenberg limit, and can be achieved by a linear superposition of the vacuum and N free fermion states. In addition, we show that Greenberger–Horne–Zeilinger-type states exhibit Heisenberg scaling in precision throughout the entire regime of parameters. Moreover, we numerically observe that the optimal precision using a product input state scales at the standard quantum limit.

DOI
10.1088/1367-2630/17/7/073032

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We consider quantum metrology for unitary evolutions generated by parameter-dependent Hamiltonians. We focus on the unitary evolutions generated by the Ising Hamiltonian that describes the dynamics of a one-dimensional chain of spins with nearest-neighbour interactions and in the presence of a global, transverse, magnetic field. We analytically solve the problem and show that the precision with which one can estimate the magnetic field (interaction strength) given one knows the interaction strength (magnetic field) scales at the Heisenberg limit, and can be achieved by a linear superposition of the vacuum and N free fermion states. In addition, we show that Greenberger–Horne–Zeilinger-type states exhibit Heisenberg scaling in precision throughout the entire regime of parameters. Moreover, we numerically observe that the optimal precision using a product input state scales at the standard quantum limit.
We show how quantum metrology protocols that seek to estimate the parameters of a Hamiltonian that exhibits a quantum phase transition can be efficiently simulated on an exponentially smaller quantum computer. Specifically, by exploiting the fact that the ground state of such a Hamiltonian changes drastically around its phase-transition point, we construct a suitable observable from which one can estimate the relevant parameters of the Hamiltonian with Heisenberg scaling precision. We then show how, for the one-dimensional Ising Hamiltonian with transverse magnetic field acting on NNN spins, such a metrology protocol can be efficiently simulated on an exponentially smaller quantum computer while maintaining the same Heisenberg scaling for the squared error, i.e., $O({N}^{\ensuremath{-}2})$ precision, and derive the explicit circuit that accomplishes the simulation.
The aim of quantum metrology is to estimate target parameters as precisely as possible. In this paper, we propose a protocol for quantum metrology based on symmetry-protected adiabatic transformation. We introduce a ferromagnetic Ising model with a transverse field as a probe and consider the estimation of a longitudinal field. Without the transverse field, the ground state of the probe is given by the Greenberger-Horne-Zeilinger state, and thus the Heisenberg limit estimation of the longitudinal field can be achieved through parity measurement. We find that full information of the longitudinal field encoded on parity can exactly be mapped to global magnetization by symmetry-protected adiabatic transformation, i.e., parity measurement can be replaced with global magnetization measurement. Our scheme requires neither accurate control of individual qubits nor that of interaction strength.
We investigate superpositions of quantum Ising ground states generated through a quantum interaction, in which the nearest-neighbor coupling strength is promoted to an operator acting on a central auxiliary spin. We prepare the central spin in a superposition, which allows the chain to evolve coherently under different effective interaction strengths. The states are prepared by adiabatic evolution of the joint system comprising the central spin and the many-body spin chain, followed by a measurement on the central spin and post-selection of a specific outcome. It is possible to prepare superpositions of transverse-field quantum Ising ground states belonging to the three magnetic phases, namely, the paramagnetic, ferromagnetic, and anti-ferromagnetic phases. For a spin-1/2 central system, it is possible to prepare superpositions of ground states belonging to any two phases, while for the spin-1 case, the superpositions may contain three ground states, each belonging to one of the three phases. We analytically derive expressions for the magnetization, spin correlations, and entanglement of such states. The magnetization and spin correlations show distinct features in different superposition sectors. The nearest-neighbor entanglement is largest where the transverse field and Ising interaction compete, and its behavior differs from that of the magnetic observables. We also derive the quantum Fisher information relevant to estimating the Ising interaction strength. And it exhibits multiple peaks near the critical points of the constituent ground states, with the maximum value showing Heisenberg scaling, for the parameters and system sizes examined.
Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-12\frac{1}{2}21​ particle and compare the results for estimating the amplitude and the direction of the magnetic field.
We study in this work the metrology precision of a probe system coupled to an ancillary qubit. Restricting the probe-qubit coupling along only one or two directions is found to be a sufficient criterion for the effective dynamical generator to achieve the Heisenberg limit in precision. Under the criterion, the quantum Fisher information (QFI) about the to-be-estimated parameter becomes the expectation value of mean square of the effective generator with respect to the initial state of the composite system. Our criterion is justified in two distinct systems. For a bosonic probe, QFI about the displacement estimation is found to be proportional to the mean excitation number of probe. It renders a counterintuitive result that quantum metrology sensitivity can be enhanced by increasing the temperature of the probe system. For a spin-ensemble probe, QFI about both rotation-phase and magnetic-field estimation exhibit a quadratic dependence on the probe-spin number. It is found that even when the spin-ensemble is prepared as a finite-temperature state, faraway from the so-called resource states, e.g., the squeezed state or the Greenberger-Horne-Zeilinger state, QFI can still manifest a Heisenberg scaling behavior.
Abstract A central qubit coupled to an Ising ring of N qubits, operating close to a critical point is investigated as a potential precision quantum magnetometer for estimating an applied transverse magnetic field. We compute the quantum Fisher information for the central, probe qubit with the Ising chain initialized in its ground state or in a thermal state. The non-unitary evolution of the central qubit due to its interaction with the surrounding Ising ring enhances the accuracy of the magnetic field measurement. Near the critical point of the ring, Heisenberg-like scaling of the precision in estimating the magnetic field is obtained when the ring is initialized in its ground state. However, for finite temperatures, the Heisenberg scaling is limited to lower ranges of N values.