Experimental Demonstration of Self-Guided Quantum Tomography

Traditional methods of quantum state characterization are impractical for systems of more than a few qubits due to exponentially expensive postprocessing and data storage and lack robustness against errors and noise. Here, we experimentally demonstrate self-guided quantum tomography performed on polarization photonic qubits. The quantum state is iteratively learned by optimizing a projection measurement without any data storage or postprocessing. We experimentally demonstrate robustness against statistical noise and measurement errors on single-qubit and entangled two-qubit states.

Figure 1

(a) The unknown quantum state that we want to characterize is shown as a red Bloch vector, and the current estimation is shown as a blue Bloch vector. (b) The algorithm estimates the gradient in a stochastically chosen direction by performing expectation value measurements with the green and purple projectors. (c) The algorithm steps in the direction of the highest expectation value, and the current estimate of the state is updated. (d)–(f) The gradient is estimated again and the process repeated for a set number of iterations.

Figure 2

We perform SGQT in a regime where only seven photons on average are used per iteration of the algorithm. (a) A single SGQT route for each target state is plotted on the Bloch sphere. (b) The red, green, and purple points show the average fidelity of SGQT for each target state. The red line shows the average fidelity across all target states. The blue points show the average fidelity of SQT across all target states for different total photon counts. (c) The table compares the fidelity of SGQT with $\sim 280$ photons to SQT with $\sim 280$, $\sim 3.9\times 10^3$ and $\sim 2\times 10^5$ photons used.

Figure 3

(a) Fidelity of SGQT with varying levels of experimental error. Points are the average of ten repetitions, and the band gives one standard deviation of error. SQT is performed with the same levels of experimental error, repeated ten times, and the fidelities shown as solid lines. (b) Table comparing the fidelity values for SQT after ten repetitions (40 measurements) and SGQT after 40 iterations.

Figure 4

We perform SGQT on two-qubit states by taking a random subset of Pauli measurements at each iteration. We present the state in terms of Pauli measurement expectation values, where element $i$, $j=\frac{1}{4}⟨{\mathrm{\Phi }}_k|P_i^{(1)}\otimes P_j^{(1)}|{\mathrm{\Phi }}_k⟩$ and $P^{(1)}=\{{\sigma }_I,{\sigma }_X,{\sigma }_Y,{\sigma }_Z\}$ are the one-qubit Pauli matrices. (a)–(e) The estimate of the state at different points of the algorithm run with eight measurements per iteration. The fidelity is measured against high-precision SQT. (f) The high-precision estimate of the state measured with SQT using a long integration time to reduce statistical noise. (g) The algorithm was run with eight, six, four, and two measurements per iteration, and the fidelity of the state estimate is plotted. The table below compares fidelities. (h) We compare performance of SGQT and SQT in the presence of high levels of statistical noise and measure the fidelity against the number of photons used. (i) We engineer measurement errors in wave plate rotations and measure the fidelity of SGQT and SQT with four levels of error. We repeat SQT to match the number of measurements of SGQT and average the results. Both (h) and (i) use eight measurements per iteration.