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New Journal of Physics

Hamiltonian tomography: the quantum (system) measurement problem

Jared H. Cole

Published Sep 30, 2015• 7 citations
Publisher

Abstract

Abstract To harness the power of controllable quantum systems for information processing or quantum simulation, it is essential to be able to accurately characterise the system's Hamiltonian. Although in principle this requires determining less parameters than full quantum process tomography, a general and extendable method for reconstructing a general Hamiltonian has been elusive. In their recent paper, Wang et al (2015 New J. Phys. 17 093017 ) apply dynamical decoupling to the problem of Hamiltonian tomography and show how to reconstruct a general many-body Hamiltonian comprised of arbitrary interactions between qubits.

DOI
10.1088/1367-2630/17/10/101001

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Abstract To harness the power of controllable quantum systems for information processing or quantum simulation, it is essential to be able to accurately characterise the system's Hamiltonian. Although in principle this requires determining less parameters than full quantum process tomography, a general and extendable method for reconstructing a general Hamiltonian has been elusive. In their recent paper, Wang et al (2015 New J. Phys. 17 093017 ) apply dynamical decoupling to the problem of Hamiltonian tomography and show how to reconstruct a general many-body Hamiltonian comprised of arbitrary interactions between qubits.
Characterization of qubit couplings in many-body quantum systems is essential for benchmarking quantum computation and simulation. We propose a tomographic measurement scheme to determine all the coupling terms in a general many-body Hamiltonian with arbitrary long-range interactions, provided the energy density of the Hamiltonian remains finite. Different from quantum process tomography, our scheme is fully scalable with the number of qubits as the required rounds of measurements increase only linearly with the number of coupling terms in the Hamiltonian. The scheme makes use of synchronized dynamical decoupling pulses to simplify the many-body dynamics so that the unknown parameters in the Hamiltonian can be retrieved one by one. We simulate the performance of the scheme under the influence of various pulse errors and show that it is robust to typical noise and experimental imperfections.
Abstract Quantum process tomography conventionally uses a multitude of initial quantum states and then performs state tomography on the process output. Here we propose and study an alternative approach which requires only a single (or few) known initial states together with time-delayed measurements for reconstructing the unitary map and corresponding Hamiltonian of the time dynamics. The overarching mathematical framework and feasibility guarantee of our method is provided by the Takens embedding theorem. We explain in detail how the reconstruction of a single-qubit Hamiltonian works in this setting and provide numerical methods and experiments for general few-qubit and lattice systems with local interactions. In particular, the method allows to find the Hamiltonian of a two qubit system by observing only one of the qubits.
The parameter estimation of the Hamiltonian of an unknown quantum system is essential in quantum information processing. However, the general methods based on quantum state tomography and process tomography are resource demanding and challenging in large quantum systems. On the other hand, if the interaction form of a system is known, one can adopt the quantum quench protocol to reconstruct the generic many-body Hamiltonian with local interactions, requiring significantly lower complexity and measurement cost compared to the general tomography technique. Here we report an experimental demonstration of the quench method to determine the scalar coupling constants in a three-spin chain. In the experiments, we demonstrate that how the quality of the estimation is affected by the duration of the quench and the number of input states. Our results show that the quantum quench protocol is a promising strategy to implement Hamiltonian estimation for many-body systems, especially when the cost of measurement is crucial in practical cases.
PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY-type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data. Quantum information processing requires the effective measurement of quantum states. However, a single quantum measurement can only obtain partial information of a quantum state. The reconstruction of a quantum state requires measuring a complete set of observables on an ensemble of identically prepared copies of the system. This method called quantum state tomography [1], is very important because any unknown state can be ascertained by tomographic measurements. Moreover, the full description of qubit states can increase the accuracy of quantum operations. Tomographic measurements have been experimentally implemented for, e.g., the nuclear spin state of an NMR system [2], the electromagnetic field and photon
Full characterization of quantum states and processes is a fundamental requirement for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because of the exponential growing with the system size of the number of measurements and the amount of computational power required to process them. A new approach for quantum state tomography and quantum process tomography, requiring a limited number of measurements, has been recently introduced by Maciel et al. [Quantum Inf. Comput. 12, 0442 (2012); Int. J. Mod. Phys. C 22, 1361 (2011).], namely variational quantum-process tomography (VQT). Here we adopt the VQT approach for the experimental characterization of two-qubit quantum processes and compare the reconstructed maps with those obtained by standard tomographic methods. The results demonstrate the high performance of this approach and propose VQT as a powerful alternative to the standard quantum process tomography.
Reconstructing the full quantum state of a many-body system requires the estimation of a number of parameters that grows exponentially with system size. Nevertheless, there are situations in which one is only interested in a subset of these parameters, and a full reconstruction is not needed. A paradigmatic example is a scenario in which one aims at determining all the reduced states only up to a given size. Overlapping tomography provides constructions to address this problem with a number of product measurements much smaller than what is obtained when performing independent tomography of each reduced state. There are, however, many relevant physical systems with a natural notion of locality where one is mostly interested in the reduced states of neighboring particles. In this work, we study this form of local overlapping tomography. First of all, we show that, contrary to its full version, the number of product-measurement settings needed for local overlapping tomography does not grow with system size. Then, we present strategies for qubit and fermionic systems in selected lattice geometries. The developed methods find a natural application in the estimation of many-body systems prepared in current quantum simulators or quantum computing devices, where interactions are often local.