Fast reconstruction of high-qubit-number quantum states via low-rate measurements

Due to the exponential complexity of the resources required by quantum state tomography (QST), people are interested in approaches towards identifying quantum states which require less effort and time. In this paper, we provide a tailored and efficient method for reconstructing mixed quantum states up to 12 (or even more) qubits from an incomplete set of observables subject to noises. Our method is applicable to any pure or nearly pure state $\rho$ and can be extended to many states of interest in quantum information processing, such as a multiparticle entangled $W$ state, Greenberger-Horne-Zeilinger states, and cluster states that are matrix product operators of low dimensions. The method applies the quantum density matrix constraints to a quantum compressive sensing optimization problem and exploits a modified quantum alternating direction multiplier method (quantum-ADMM) to accelerate the convergence. Our algorithm takes $8,35$, and 226 seconds, respectively, to reconstruct superposition state density matrices of $10,11,\text{and}12$ qubits with acceptable fidelity using less than $1\%$ of measurements of expectation. To our knowledge it is the fastest realization that people can achieve using a normal desktop. We further discuss applications of this method using experimental data of mixed states obtained in an ion trap experiment of up to 8 qubits.

Figure 1

The reconstruction performances of $n=8\sim 12$ qubits using the proposed method are shown in terms of increasing number of iterations in different colors. The $x$ axis represents the number of iterations and the $y$ axis represents $1-D(\rho ,\widehat{\rho })$. The dashed line represents the $94.5\%$ accuracy in terms of the Hilbert-Schmidt norm difference. Each number in the figure is the average of 100 simulations. All curves reach an accuracy of $1-D(\rho ,\widehat{\rho })$ above $94.5\%$ within 50 iterations. Corresponding fidelity values can be referred to Table 1.

Figure 2

Run time for reconstruction of random $n$-qubit pure states subjected to Gaussian noise on Pauli measurements. We compare four techniques: The circle points are matlab's fminsearch minimizing $\text{Tr}\left[(\widehat{\rho }-\rho )\right]$ directly. Timings for a semidefinite programming method (SeDuMi) [24], the realization the iterative method of [25], and the efficient algorithm [13] are denoted as square, star, and diamond points, respectively. Our algorithm is shown with $*$. All timings were performed on a single core of a 3.6-GHz Intel i7-4790 CPU in matlab.

Figure 3

Absolute value of corresponding reconstructed density matrix of the experimentally realized W state. (a) Maximum likelihood estimate of full quantum state tomography $|{\rho }_{ML}|$ [26]. (b) Reconstruction $|\widehat{\rho }|$ using the method described in this article with sampling rate $\eta =15\%$ obtained after three iterations, 0.14 s.

Figure 4

The comparison of magnitudes of elements in density matrices of $|{\rho }_{ML}|$ and $|\widehat{\rho }|$ shown in Fig. 3: (a) the magnitudes of $|{\rho }_{ML}|$ and (b) the magnitudes of $|\widehat{\rho }|$. The algorithm can also recover details approaching full tomography maximum likelihood results.