16-qubit IBM universal quantum computer can be fully entangled

Entanglement is an important evidence that a quantum device can potentially solve problems intractable for classical computers. In this paper, we prepare connected graph states involving 8 to 16 qubits on ibmqx5, a 16-qubit superconducting quantum processor accessible via IBM cloud,using low-depth circuits. We demonstrate that the prepared state is fully entangled, i.e. the state is inseparable with respect to any fixed partition.


I. INTRODUCTION
Quantum computation has been an active research topic since middle 90s with the invention of the Shor's algorithm and many other important discoveries such as quantum error correction [1]. For the last two decades, physical implementations of quantum computation have achieved significant progress. The fidelity of single and two-qubit gates exceeds 99%, reaching the threshold of fault-tolerant quantum computing [2]. The number of qubits in both superconducting and trapped ions quantum computers are both greater than 20 now [3,4]. It is projected that the number of qubits will approach to 50 or more in the next few years. At that time, the quantum computer may become more powerful than the fastest classical computer for some specific tasks, into the regime of the so called quantum supremacy [5].
The IBM Q is a quantum cloud service released by IBM. Its present backend devices include two processors with 5 superconducting qubits (ibmqx2 and ibmqx4), one 16 qubit processor (ibmqx5) and one 20 qubit processor (QS1 1) [4]. IBM recently announced that they have successfully built and tested a 20-qubit and a 50-qubit machine [4]. The quantum cloud service of IBM provides high fidelity quantum gate operations and measurements.
Entanglement is considered to be the most nonclassical manifestation of quantum physics [13]. It is also a critical resource for quantum information processing. Highly entangled states such as Bell states, GHZ states and cluster states [14] have been applied in quantum teleportation, super-dense coding, one-way quantum computing [15] and various quantum algorithms. The ability to produce highly-entangled states is, therefore, one important step to demonstrate quantumness for quantum processors like ibmqx5. This task is, however, highly non-trivial due to the error accumulation of faulty gates.
In this paper, we wish to assess the quantumness and performance of the 16 qubit ibmqx5 device via the production of highly entangled states, namely the graph states, which is an important class of many-body entangled states that are widely used in one-way quantum computing, quantum error correction [15,16]. We generate graph states that correspond to rings involving 8 to 16 qubits via IBM Q cloud service (ibmqx5), using optimized low-depth circuits that are tailored to the universal get set on ibmqx5. We detect full entanglement up to 16 qubits, using an entanglement criterion based on reduced density matrices. Qubits are fully entangled in the sense that the state involves all physical qubits and is inseparable with respect to any fixed partition.

A. Graph states and entanglement
Graph state [17] is a generalization of cluster state introduced in 2001 [14], which is the resource state of one-way quantum computing [15] and quantum error correction [16]. GHZ state is an example of graph state and has been demonstrated in superconducting qubit system [18]. However, GHZ state is fragile. Some other graph states are very robust to local operations, such as local measurements and noises. In order to disentangle the cluster state of N qubits, N/2 local measurements are needed [14]. Because of this nice feature, we decide to generate and detect linear cluster states in the IBM cloud service ibmqx5.
where U ab is a control-Z operator acting on qubits a and b [19], and are eigenvectors of the X operator.
An equivalent definition is, the graph state that corresponds to G (V, E) is the unique common eigenvector (of eigenvalue 1) of the set of independent commuting operators: where the eigenvalues to K a are +1 for all a ∈ V , and N a denotes the set of neighbor vertices of a in G [19]. As implied by the first definition, a n-qubit graph state can be prepared by the following steps. can be written as where ρ i A and ρ i B are quantum states of the system A and B, respectively, with p i ≥ 0 and For a state ρ of a many-body system, for any fixed bipartition AB of the system, if ρ is entangled with respect to the partition AB, then the entanglement of the many-body state ρ can also be examined via its subsystems. That is, if the subsystems are all entangled, the whole system must be also entangled.
To be more concrete, consider a 4-qubit subsystem ρ A,B,C,D in an n qubit system. Suppose that we perform two local operations O A and O D on qubit A and D respectively, and then obtain the reduced density matrix of qubit B and C by tracing out qubit A and D. The reduced density matrix for qubits B and C reads The entanglement of ρ B,C can be determined by using entanglement monotones such as negativity and concurrence, which, in the 2 qubit case, has non-zero values if and only if the system is entangled [13,22]. If ρ B,C is entangled, we can conclude that in the original system, there could not exist a separation with qubit B and C on different sides. In other words, if the original system is biseparable with respect to a fixed partition, the qubit B and C must be on the same side. Otherwise, we will be able to create entanglement between the two separable parties with only local operations, which is not possible [13].
For an n qubit system {q 1 , q 2 , ..., q n }, if we can show that among the n qubit pairs (q 1 , q 2 ),...,(q n−1 , q n ),(q n , q 1 ), n − 1 of them must be on the same side in a separation, then we may conclude that there is no possible separation, and that the system is a n-qubit entangled state (meaning that the state is not biseparable w.r.t. a fixed partition, and that it involves all qubits). The (minimal) number of circuit configurations needed in this approach is 3 4 (n − 1), which grows linear with respect to n. This method is far more efficient compared to a full n-qubit tomography, which requires exponential number of configurations.

B. Graph states on ibmqx5
ibmqx5 is a 16-qubit superconducting quantum processor. It allows independent single qubit operations with fidelity > 99% and control operations with fidelity 95 − 97% (see Fig.   2) marked as the edges in the connectivity map (see Fig.1). That is, CNOT operations with qubit a as the control qubit and b as the target is allowed if and only if a → b is an edge in the map.
In our experiment, as shown in Fig. 3, the following five graph states are employed. The first state is a 8 qubit graph state involving qubits q5 − q12 that corresponds to a ring of length 8; the second one is a 10 qubit state involving qubits q4 − q13 corresponding to a ring of length 10; the third one involves qubits q3 − q14 and corresponds to a ring of length 12; the 4th one involves qubits q2 − q15 and corresponds to a ring of length 14; the 5th one involves all the 16 qubits. We employ these particular graph states based on the following considerations. First, these states are genuinely entangled and will remain entangled after tracing out a large number of qubits. Second, research has shown that 1-D cluster states are robust against decoherence, meaning that it would be more likely to find entanglement in a rather large graph state close to a 1-D chain, compared to GHZ states and 2-D graph states [24]. At last, even rings are two-edge-colorable; as a result, on the 16-qubit ibmqx5, these "even-ring" states could be prepared using low-depth circuits (See Fig. 4).
To prepare the desired graph state, we start from the circuit implied by the definition of graph states (see Fig. 4(a)). The control-Z gates are implemented using a CNOT gate and two Hadamard gates. We then optimize this circuit by adjusting the order of commuting gates and removing redundant Hadamard gates (see Fig. 4(b)). The circuit that we implemented are shown in Fig. 4(b) and Fig. 5(a)-Fig. 5(d).
Therefore, for a state close to this graph state, we should expect the resulting 2-qubit state to have a negativity significantly greater than zero. The results are plotted in Fig.7.
For the 8-qubit graph state, the measured negativities are all significantly greater than 0. For the 10-qubit graph state, 9 out of 10 measured negativities are significantly greater than 0. Based on our argument in Sec. II A, both the 8-qubit state and 10-qubit state are fully entangled.
Therefore, this possibility is ruled out with very high confidence. We can now conclude that the 12 qubit graph state is fully entangled.
In the 14 qubit case, as shown in Fig. 7(d) It may be noted that the subsystem of qubits {q8, q9, q10, q11} yields close-to-zero negativity in 3 out of 4 experiments. This can be due to relatively high readout errors or gate errors involving these qubits, which is compatible with the measured parameters provided by IBM's website [4] (see Fig. 2). For instance, the CNOT gate between q10 and q11 has the largest error among all possible CNOT gates, while the readout error of q10 and q11 are also above the average level (6.5%).

D. Further Exploration of the 16-qubit State
The results above could be understood as an ability to generate localized entanglement on physically neighboring qubits [26]. That is, neighboring qubits can be put into entanglement by performing ideal local operations on the 16-qubit state. Using the same data obtained above, we will show that localized entanglement on qubits with distance 2 and 3 could also be generated.
Suppose {E, A, B, C, D, F } is a six-qubit subsystem that forms a chain. We first apply Based on data obtained in previous experiments, we have calculated the corresponding negativity for each 6-qubit subsystem and shown them in Table. 1. Using this method, we have identified localized entanglement in 13 out of 16 pairs of qubits with distance 2.
To generate localized entanglement on qubits with distance 3, we may apply the same O E and O F , and then apply O B = X B +I 2 and O C = X C +I 2 (see Fig.9).