Pairwise tomography networks for many-body quantum systems

We introduce the concept of pairwise tomography networks to characterize quantum properties in many-body systems and demonstrate an efficient protocol to measure them experimentally. Pairwise tomography networks are generators of multiplex networks where each layer represents the graph of a relevant quantifier, such as, e.g., concurrence, quantum discord, purity, quantum mutual information, or classical correlations. We propose a measurement scheme to perform two-qubit tomography of all pairs showing exponential improvement in the number of qubits $N$ with respect to previously existing methods. We illustrate the usefulness of our approach by means of several examples revealing its potential impact to quantum computation, communication, and simulation. We perform a proof-of-principle experiment demonstrating pairwise tomography networks of $W$ states on IBM Q devices.

Figure 1

Illustration of steps 2(a) in the measurement scheme for $N=10$ qubits. Here, $L=\lceil {log}_{3}10\rceil =3$. Each figure depicts the letter assignment to each qubit (represented by three different colors: $a$ in red, $b$ in green, and $c$ in gray) for $l=1\text{–}3$ (from left to right). The connections represent the pairs of qubits for which the relevant observables are measured. Note that the letters $a,b,\mathrm{and}c$ are assigned according to the result of $\lfloor i/3^{l-1}\rfloor mod3$, where $i$ is the qubit index (with $i=0$ being the topmost qubit and the indices increasing in clockwise order).

Figure 2

Circuit for the generation of a five-qubit W state [46] used in the experiments reported in Fig. 3, optimized for the connectivity layout of the IBM Q devices used in the experiments, shown in Fig. 3. The $R_y$ gates are rotations around the $y$ axis by the angle written below (in radians).

Figure 3

(a)–(e) Pairwise concurrence network for a five-qubit $W$ state generated experimentally on five different IBM Q devices. The edges represent the concurrence between the two qubits, represented as nodes, that it intersects. The weight of an edge is represented by its width and color, which correspond to the color bar on the right. The size of each node is proportional to its strength, that is, the sum of the weights of the links reaching it. The edge between qubits 1 and 4 in ibmq_london has been filtered out, being statistically insignificant, as explained at the end of Sec. 2. For all other missing links, the reconstructed two-qubit states have zero concurrence. (f) The T-shaped connectivity layout of the IBM Q devices used in the experiment. The arrows represent the pairs of qubits that can be coupled with cnot gates.

Figure 4

Collisional model at time $\lambda t=1000$ and with entangling interaction strength $\theta =2\pi /3$. Qubits 1, 3, 5, and 7 are the emitters 2, 4, 6, and 8, the ancillae, whereas 9 is the system qubit. The concurrence reveals pairwise entanglement between the system and the ancillae only, whereas there are quantum discord, classical correlations, and mutual information among the ancillae as well. The emitters are in the ground state since they are not correlated with any other qubit, whereas they form a clique in the purity layer; similarly, note that there are no connections among emitters in the entropy layer.

Figure 5

(a) Concurrence network of the XX ground state of an $N=9$ spin chain for different values of $k$. (b) Pairwise multiplex for the ground state in the $k=2$ zone showing the differences in the two-spin properties between bulk pairs, e.g., spins 5 and 6, and edge pairs, e.g., spins 1 and 2.