Efficient tomography of microwave photonic cluster states

Abstract

Entanglement among a large number of qubits is a crucial resource for many quantum algorithms. Such many-body states have been efficiently generated by entangling a chain of itinerant photonic qubits in the optical or microwave domain. However, it has remained challenging to fully characterize the generated many-body states by experimentally reconstructing their exponentially large density matrices. Here, we develop an efficient tomography method based on the matrix-product-operator formalism and demonstrate it on a cluster state of up to 35 microwave photonic qubits by reconstructing its $2^{35} \times 2^{35}$ density matrix. The full characterization enables us to detect the performance degradation of our photon source which occurs only when generating a large cluster state. This tomography method is generally applicable to various physical realizations of entangled qubits and provides an efficient benchmarking method for guiding the development of high-fidelity sources of entangled photons.

Figures (24)

• Figure 1: Efficient tomography of sequentially-emitted entangled photonic qubits. (a) Sequential emission of entangled photonic qubits by a $d$-level system (qudit). (b) Mixed-state quantum-circuit representation, which can be interpreted as a tensor network corresponding to the density matrix of the emitted photonic qubits. (c) Matrix product operator (MPO) obtained by contracting the tensors in groups. This is an efficient parameterization of the many-body density matrix of the generated photonic qubits because the number of parameters only increases proportionally with the number of photonic qubits, $N$. (d) Efficient choice of measurement bases which makes the number of measurement settings independent of $N$. Here, we measure the complete set of local correlations of up to five consecutive photonic qubits, which is sufficient for reconstructing the density matrix of a linear cluster state.
• Figure 2: Generation and measurement of a microwave photonic linear cluster state. (a) Photograph of the disassembled (front) and assembled (back) microwave photon sources, which consist of a superconducting transmon qubit and a microwave resonator on a silicon substrate inside an aluminum cylindrical cavity. The threaded connector with a long center pin is used to couple a coaxial cable to the resonator. (b) Circuit model of the photon source and a schematic of the measurement chain, which includes a Josephson parametric amplifier (JPA) for phase-sensitive quantum-limited amplification. (c) Energy-level diagram of the photon source. The numbered arrows correspond to the steps comprising the conditional photon emission operation. (d) Measured mode function $u(t)$ of the photonic qubit. (e) Probability density $p(q_1, q_2)$ of measuring the quadrature values $q_1$ and $q_2$ for the first and second photonic qubits, respectively, of the generated five-qubit cluster state. The solid and dashed lines are the contour lines for the measured data and an ideal cluster state under the effect of the same measurement inefficiency, respectively. (f) Pulse sequence for generating an $N$-qubit linear cluster state. (g) Measured photon flux of the cluster states. The traces are offset vertically by units of 20 $\mu$s$^{-1}$ for clarity.
• Figure 3: Estimating the required bond dimension of the MPO representation using the measured local correlations. (a) Schematic of the procedure. (b), (c) Five largest singular values $\sigma_1, \ldots, \sigma_{5}$ of the four-qubit local correlation matrices $B_s$ ($s = 1, \ldots, N-3$) measured for the $N$-qubit linear cluster states with $N = 10$ and 35, respectively. Error bars represent the standard errors of the singular values estimated by the first-order propagation of the statistical errors of the correlation data. Because only four singular values of any $B_s$ are significantly larger than zero, the bond dimension of these cluster states can be estimated as $D = 4$.
• Figure 4: Corners of the reconstructed density matrices of the (a) 10-qubit and (b) 35-qubit cluster states. Absolute values are plotted in the diagonal and the lower-left triangle, and the complex arguments in the upper-right triangle. See Appendix \ref{['app:10-qubit']} for the full density matrix of the 10-qubit cluster state.
• Figure 5: Quantum state fidelities and purities of the generated $N$-qubit cluster states. (a) Fidelities $\mathcal{F}$ of the generated cluster states to the ideal cluster states (blue circle). Also plotted are the fidelities calculated using numerical models with only photon loss and dephasing errors, where the error probabilities are individually fit to each qubit for each $N$ (orange square) or are extrapolated from the average values for $N = 5$ (green triangle). (b) Same plot for the purity $\gamma$. (c) Relative uncertainty of $\mathcal{F}$ calculated by propagating the uncertainties of the quadrature measurements. (d) Same plot for the purity $\gamma$.