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Moscow University Physics Bulletin

Simulation of Quantum Tomography Process of Biphoton Polarization States on a Quantum Computer

Sofya Manko, Dmitry N. Frolovtsev, S.A. Magnitsky

Published Mar 1, 2021• 1 citation
Publisher

Abstract

The process of quantum tomography of polarization-entangled photon pairs was simulated on quantum computers of different generations of the IBM Q Experience platform. The simulation was carried out for all four two-qubit Bell states. It turned out that the contribution of basis states that are absent in Bell states does not exceed 10 $$\%$$ even on quantum computers of the first generations, and is reduced to several percent on quantum computers of the latest generation. The simulation quality was evaluated using Fidelity. The Fidelity values were in a range of 68.6–94.9 $$\%$$ , depending on the generation of computers. It was shown that for Bell states Fidelity can be calculated using three tomographic measurements instead of the nine required for determination of the density matrix using quantum state tomography.

DOI
10.3103/s0027134921020065

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The process of quantum tomography of polarization-entangled photon pairs was simulated on quantum computers of different generations of the IBM Q Experience platform. The simulation was carried out for all four two-qubit Bell states. It turned out that the contribution of basis states that are absent in Bell states does not exceed 10 %\%% even on quantum computers of the first generations, and is reduced to several percent on quantum computers of the latest generation. The simulation quality was evaluated using Fidelity. The Fidelity values were in a range of 68.6–94.9 %\%% , depending on the generation of computers. It was shown that for Bell states Fidelity can be calculated using three tomographic measurements instead of the nine required for determination of the density matrix using quantum state tomography.
The method and prototype of a device for characterizing of biphoton light sources based on spontaneous parametric downdonversion by quantum tomography are described. The prototype is an experimental implementation of a specialized quantum tomograph designed to measure the quantum polarization states of radiation generated by biphoton sources. Specially developed software will determine the statistical characteristics of the measured quantum state, calculate the tomographic and likelihood estimations of the density matrix, calculate the measurement errors of the density matrix elements and evaluate the quality of the quantum state of biphotons.
Quantum computation depends on quantum entanglement, a correlation between subsystems that cannot occur classically. A variety of theoretical measures exist for quantifying the degree entanglement in such schemes, all of which are functions of the system density matrix. How can the entanglement be measured experimentally? Using quantum tomography techniques developed for two photon entangled states, the density matrix can be reconstructed from the appropriate experimental data. In this case the state tomography gives the complete characterization of the physical system (for the relevant degree of freedom, such as spin or polarization). It gives information on both the degree of nonclassical correlation, that is entanglement, as well as the amount of decoherence in the system. In this proceedings we discuss the general state tomography procedure required to characterize a few qubit quantum computer, for any architecture.
We compare reconstructed quantum state images of a birefringent sample using direct quantum state tomography and inverse numerical optimization technique. Qubits are used to characterize birefringence in a flat transparent plastic sample by means of polarization sensitive measurement using density matrices of two-level quantum entangled photons. Pairs of entangled photons are generated in a type-II nonlinear crystal. About half of the generated photons interact with a birefringent sample, and coincidence counts are recorded. Coincidence rates of entangled photons are measured for a set of sixteen polarization states. Tomographic and inverse numerical techniques are used to reconstruct the density matrix, the degree of entanglement, and concurrence for each pixel of the investigated sample. An inverse numerical optimization technique is used to obtain a density matrix from measured coincidence counts with the maximum probability. Presented results highlight the experimental noise reduction, greater density matrix estimation, and overall image enhancement. The outcome of the entanglement distillation through projective measurements is a superposition of Bell states with different amplitudes. These changes are used to characterize the birefringence of a 3M tape. Well-defined concurrence and entanglement images of the birefringence are presented. Our results show that inverse numerical techniques improve overall image quality and detail resolution. The technique described in this work has many potential applications.
Quantum state tomography is used in quantum communication, computing, and cryptography to characterize quantum states enabling assessment and control of quantum systems. Rapidly scaling quantum technologies require fast, accurate, and simple techniques, but current tomography methods are bulky, hard to scale, and slow. To overcome these limitations, we experimentally demonstrate Fourier tomography [1], of single-photon and entangled photon pairs generated by a quantum dot. As shown in Fig. 1(a) for two-photon tomography, the setup consists of a continuously rotating quarter waveplate, a fixed linear polarizer and a single photon detector per qubit. As the waveplates rotates with different speed, we record the coincidences rate between the two detectors. This signal can be expressed as a Fourier series whose coefficients can be related to the polarization state of light. Fig. 1(b) shows the simulated coincidence rate as the waveplate rotates for input Bell states∣Φ+⟩\vert \Phi^{+}\rangle∣Φ+⟩and∣Φ−⟩\left\vert\Phi^{-}\right\rangle∣Φ−⟩. By capturing the coincidences variations as a Fourier series through a single rotating element and fixed polarizer, we simplify the tomography process, providing an accurate, and simple approach suitable for scalable quantum systems.
Demonstration of quantum entanglement, a key resource in quantum computation arising from a nonclassical correlation of states, requires complete measurement of all states in varying bases. By using simultaneous measurement and state tomography, we demonstrated entanglement between two solid-state qubits. Single qubit operations and capacitive coupling between two super-conducting phase qubits were used to generate a Bell-type state. Full two-qubit tomography yielded a density matrix showing an entangled state with fidelity up to 87%. Our results demonstrate a high degree of unitary control of the system, indicating that larger implementations are within reach.
We performed process tomography of a coherent state transfer from light polarization into electron spin polarization in a semiconductor quantum well. The density matrix of the generated electron spin state was reconstructed by a three-dimensional tomographic Kerr rotation measurement so that we could quantitatively evaluate the transfer process. We found that the process fidelity of the state transfer was 81% and clarified that the phase-flip error caused by mixing with another Zeeman sublevel at the hole state was the major contributor to fidelity degradation.