Learning quantum tomography from incomplete measurements

TL;DR

It is demonstrated that neural networks can effectively learn the underlying geometry of multi-qubit states using this for their reconstruction and outperform standard maximum likelihood estimation and also scale to larger 3- and 4-qubit systems.

Abstract

We revisit quantum tomography in an informationally incomplete scenario and propose improved state reconstruction methods using deep neural networks. In the first approach, the trained network predicts an optimal linear or quadratic reconstructor with coefficients depending only on the collection of (already taken) measurement operators. This effectively refines the undercomplete tomographic reconstructor based on pseudoinverse operation. The second, based on an LSTM recurrent network performs state reconstruction sequentially. It can also optimize the measurement sequence, which suggests a no-free-lunch theorem for tomography: by narrowing the state space, we gain the possibility of more efficient tomography by learning the optimal sequence of measurements. Numerical experiments for a 2-qubit system show that both methods outperform standard maximum likelihood estimation and also scale to larger 3- and 4-qubit systems. Our results demonstrate that neural networks can effectively learn the underlying geometry of multi-qubit states using this for their reconstruction.

Figures (6)

• Figure 1: Scheme for the neural-based measurements selection and reconstruction network. Having previous selection $\Pi_{l-1}$, the goal for the recurrent autoregressive LSTM$_\mathrm{S}$ unit is to propose the next measurement operator $\Pi_l$, which is used to measure the value $m_l$. Simultaneously, a pair $(\Pi_l,m_l)$ is used as an input to LSTM$_\mathrm{R}$ cell, which outputs increasingly better reconstructions $\rho_l$, where $l$ denotes the number of already performed measurements.
• Figure 2: Undercomplete 1-qubit density matrix tomography with two measurements $(\nu,\nu')$. (a-c) Element-wise reconstruction error $\mathcal{L}_{\alpha\beta}(S_\mathrm{test})$ using: (a) pseudoinverse reconstruction, (b) NN corrector, (c) best possible analytical formula. (d) Bures distance to target matrix averaged over $S_\mathrm{test}$ for the different techniques with increasing number of measurements.
• Figure 3: Undercomplete tomography of 2-qubit random mixed states: $S_\mathrm{test}$-averaged Bures distance $\langle\mathcal{B}\rangle$ to target matrix for the different reconstruction techniques with increasing number of measurement outcomes $M$.
• Figure 4: Positive semidefiniteness of density matrices reconstructed using different techniques. Shown are the average lowest (intense color) and second-lowest (faded color) eigenvalues, with error bars indicating the standard deviation.
• Figure 5: Undercomplete (a) 3-qubit and (b) 4-qubit tomography of random mixed states: average Bures distance $\langle\mathcal{B}\rangle$ to a target matrix for the different reconstruction techniques with increasing number of measurement outcomes $M$: (a) $M=1,\dots,4^3$, and (b) $M=1,\dots,4^4$. Additional dotted red curve in (b) presents results for the smaller LSTM with adjusted basis model.