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Register a new userAccuracy of minimal and optimal qubit tomography for finite-length experiments
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Added by
Christian Kurtsiefer
Publication date
2008
fields
Physics
and research's language is
English
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Practical quantum state tomography is usually performed by carrying out repeated measurements on many copies of a given state. The accuracy of the reconstruction depends strongly on the dimensionality of the system and the number of copies used for the measurements. We investigate the accuracy of an experimental implementation of a minimal and optimal tomography scheme for one- and two-qubit states encoded in the polarization of photons. A suitable statistical model for the attainable accuracy is introduced.
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Given an experimental set-up and a fixed number of measurements, how should one take data in order to optimally reconstruct the state of a quantum system? The problem of optimal experiment design (OED) for quantum state tomography was first broached by Kosut et al. [arXiv:quant-ph/0411093v1]. Here we provide efficient numerical algorithms for finding the optimal design, and analytic results for the case of minimal tomography. We also introduce the average OED, which is independent of the state to be reconstructed, and the optimal design for tomography (ODT), which minimizes tomographic bias. We find that these two designs are generally similar. Monte-Carlo simulations confirm the utility of our results for qubits. Finally, we adapt our approach to deal with constrained techniques such as maximum likelihood estimation. We find that these are less amenable to optimization than cruder reconstruction methods, such as linear inversion.
Quantum process tomography --- a primitive in many quantum information processing tasks --- can be cast within the framework of the theory of design of experiment (DoE), a branch of classical statistics that deals with the relationship between inputs and outputs of an experimental setup. Such a link potentially gives access to the many ideas of the rich subject of classical DoE for use in quantum problems. The classical techniques from DoE cannot, however, be directly applied to the quantum process tomography due to the basic structural differences between the classical and quantum estimation problems. Here, we properly formulate quantum process tomography as a DoE problem, and examine several examples to illustrate the link and the methods. In particular, we discuss the common issue of nuisance parameters, and point out interesting features in the quantum problem absent in the usual classical setting.
In circuit-based quantum computing, the available gate set typically consists of single-qubit gates acting on each individual qubit and at least one entangling gate between pairs of qubits. In certain physical architectures, however, some qubits may be hidden and lacking direct addressability through dedicated control and readout lines, for instance because of limited on-chip routing capabilities, or because the number of control lines becomes a limiting factor for many-qubit systems. In this case, no single-qubit operations can be applied to the hidden qubits and their state cannot be measured directly. Instead, they may be controlled and read out only via single-qubit operations on connected control qubits and a suitable set of two-qubit gates. We first discuss the impact of such restricted control capabilities on the quantum volume of specific qubit coupling networks. We then experimentally demonstrate full control and measurement capabilities in a superconducting two-qubit device with local single-qubit control and iSWAP and controlled-phase two-qubit interactions enabled by a tunable coupler. We further introduce an iterative tune-up process required to completely characterize the gate set used for quantum process tomography and evaluate the resulting gate fidelities.
We develop a practical quantum tomography protocol and implement measurements of pure states of ququarts realized with polarization states of photon pairs (biphotons). The method is based on an optimal choice of the measuring schemes parameters that provides better quality of reconstruction for the fixed set of statistical data. A high accuracy of the state reconstruction (above 0.99) indicates that developed methodology is adequate.
Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we design optimal probe states for detector estimation based on the minimum upper bound of the mean squared error (UMSE) and the maximum robustness. We establish the minimum UMSE and the minimum condition number for quantum detectors and provide concrete examples that can achieve optimal detector tomography. In order to enhance estimation precision, we also propose a two-step adaptive detector tomography algorithm and investigate how this adaptive strategy can be used to achieve efficient estimation of quantum detectors. Moreover, the superposition of coherent states are used as probe states for quantum detector tomography and the estimation error is analyzed. Numerical results demonstrate the effectiveness of both the proposed optimal and adaptive quantum detector tomography methods.
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