A Better Neural Network For Quantum State Reconstruction

Quantum computers which work on the principles of quantum physics have received a lot of attention among researchers and industries due to their promising potential to perform computations that are nearly impossible by classical computers.

One of the important aspects of the quantum computers is that a qubit, the quantum analog of the classical bit, can exist in superposition of two computational states (0-state and 1-state) as opposed to classical bits existing in either of the two states. This means that quantum computers can manipulate the 0-state and 1-state of a qubit simultaneously as opposed to manipulating one state at a time. However, the limitation in using this tremendous computational ability comes in the form of limitation in reading the state of the qubit. Any attempt to read the (superposition) state of a qubit, will destroy this superposition and yield one of the 2 computational states. Well, the good news is that this collapse is probabilistic. By experimentally estimating the probabilities with which the state collapses to different computational states, one can estimate the actual (superposition) state of the qubit.

Such a superposition state of n-qubits is mathematically represented by a complex matrix called the density matrix. The process of estimating the quantum state from the probabilities of its collapse to computational states is called the quantum state tomography.

However, reconstruction of a state has two major problems:

1. It requires an informationally complete set of measurements which is exponential in the number of qubits.

2. The estimated state may not be positive semidefinite due to the measurement noise in estimated probabilities.

Most of the existing works focus on estimating the density matrix using a compressive sensing approach, with only a few measurements required than that required for a tomographically complete set, with the assumption that the true state has a low rank. One of the most popular methods used to estimate the state of a system is the use of the Singular Value Thresholding (SVT) algorithm.

In recent times, the capability of classical computing systems to store and process a huge amount of data and the capability of neural networks to learn very complicated functions have enabled neural networks to find their roots in almost all fields including quantum state tomography. Neural networks are computing systems inspired by biological neural networks. In this study conducted by Mr. Siva Shanmugam and Prof. Sheetal Kalyani from the Department of Electrical Engineering, Indian Institute of Technology (IIT) Madras, Chennai, India, a machine learning approach has been done to estimate the quantum state of an n-qubit system by unrolling the iterations of the SVT which overcomes the aforementioned problems in state reconstruction. This method has been named Learned Quantum State Tomography (LQST).

In this approach, a custom neural network whose architecture has been inspired from the iterations of SVT has been designed and tailored specifically for quantum state tomography. Here the quantum state tomography of a 4-qubit system has been numerically simulated and compared with the SVT algorithm. It was found that LQST had several advantages over the commonly used SVT method.

Firstly, LQST was found to outperform SVT in terms of fidelity between the true and estimated states. For instance, LQST estimates the state of a 4-qubit system with a fidelity of 0.91 whereas SVT estimates the same with a fidelity of 0.87. Secondly, the computational complexity of LQST is much less than that of SVT as a 3-layered LQST outperforms SVT which converges in 1632 iterations. It is worth noting here the trade-off that the LQST network was trained using 60,000 instances of ground truth data before using the network for estimating the state which is not the case in SVT.  Thirdly, LQST directly reconstructs the density matrix of a general quantum state from an informationally incomplete set of measurements, whereas the other existing generative models learn only the measurement probabilities of a particular quantum state from the complete set of measurements.

The estimation of a 2 qubit quantum Bell state was also performed using LQST in this study. It is observed that with 10 measurements, the network estimates Bell state with a Fidelity of 0.91. When an informationally complete set of measurements is observed (16 measurements) LQST estimates the Bell state with a fidelity of 0.97.

Article by Akshay Anantharaman
Click here for the original link to the paper


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