Abstract
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 16qubit superconducting quantum processor accessible via IBM cloud, using lowdepth circuits. We demonstrate that the prepared state is fully entangled, i.e., the state is inseparable with respect to any fixed partition.
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Introduction
Quantum computation has been an active research topic since the 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 twoqubit gates exceeds 99%, reaching the threshold of faulttolerant 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 socalled 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 16qubit processor (ibmqx5) and one 20qubit processor (QS1_1).^{4} IBM recently announced that they have successfully built and tested a 20qubit and a 50qubit machine.^{4} The quantum cloud service of IBM provides high fidelity quantum gate operations and measurements. Hence, after the launch of the IBM Q, many groups tested it and performed quantum computing experiments on the cloud (for instance, see refs. ^{6,7,8,9,10,11,12}).
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 (Greenberger–Horne–Zeilinger) states and cluster states^{14} have been applied in quantum teleportation, superdense coding, oneway 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 nontrivial due to the error accumulation of faulty gates.
In this paper, we wish to assess the quantumness and performance of the 16qubit ibmqx5 device via the production of highly entangled states, namely the graph states, which is an important class of manybody entangled states that are widely used in oneway 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 lowdepth 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.
Results
Graph states and entanglement
Graph state^{17} is a generalization of cluster state introduced in 2001,^{14} which is the resource state of oneway 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.
X, Y, Z denote the Pauli operators. An undirected graph G(V, E) includes a set of vertices V and a set of edges E. A graph state that correspond to an undirected graph G(V, E) is a \(\left V \right\)qubit state that has the form
where U_{ab} is a controlZ operator acting on qubits a and b,^{19} and
are eigenvectors of the X operator.
An equivalent definition, 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 nqubit graph state can be prepared by the following steps.

1.
Initialize the state to \(\left + \right\rangle ^{ \otimes n}\) by applying n Hadamard gates to \(\left 0 \right\rangle ^{ \otimes n}\);

2.
For every (a, b) ∈ E, apply a controlZ gate on qubits a and b; the order can be arbitrary.
Entanglement of general mixed states was discussed by Werner in 1989.^{20} Since then, many entanglement criteria were proposed; among them the widely used ones include the partial transpose criterion^{13,21,22} and the symmetric extension criterion.^{23}
A bipartite state ρ_{AB} on the Hilbert space \({\cal H} = {\cal H}_A \otimes {\cal H}_B\) is said to be separable if ρ_{AB} can be written as
where \(\rho _A^i\) and \(\rho _B^i\) are quantum states of the system A and B, respectively, with p_{i} ≥ 0 and \(\mathop {\sum}\nolimits_i {\kern 1pt} p_i = 1\). Otherwise ρ_{AB} is entangled. For a state ρ of a manybody system, for any fixed bipartition AB of the system, if ρ is entangled with respect to the partition AB, then the entanglement of the manybody 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 4qubit subsystem ρ_{A,B,C,D} in an nqubit 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 \(\rho _{B,C}^\prime\) can be determined by using entanglement monotones such as negativity and concurrence, which, in the 2qubit case, has nonzero values if and only if the system is entangled.^{13,22} If \(\rho _{B,C}^\prime\) 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 nqubit system {q_{1}, q_{2}, …, q_{n}}, if we can show that among the nqubit 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 nqubit entangled state (meaning that the state is not biseparable with respect to (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 nqubit tomography, which requires exponential number of configurations.
Graph states on ibmqx5
ibmqx5 is a 16qubit superconducting quantum processor. It allows independent singlequbit operations with fidelity >99% and control operations with fidelity 95–97% (see Fig. 1) marked as the edges in the connectivity map (see Fig. 2). That is, controlled NOT (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 8qubit graph state involving qubits q5–q12 that corresponds to a ring of length 8; the second one is a 10qubit 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 fourth one involves qubits q2–q15 and corresponds to a ring of length 14; the fifth 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 onedimensional (1D) 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 1D chain, compared to GHZ states and twodimensional (2D) graph states.^{24} At last, even rings are twoedge colorable; as a result, on the 16qubit ibmqx5, these “evenring” states could be prepared using lowdepth circuits (see Fig. 4).
To prepare the desired graph state, we start from the circuit implied by the definition of graph states (see Fig. 4a). The controlZ 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. 4b). The circuit that we implemented are shown in Fig. 4b and Fig. 5a–d.
Experimental results
For each nqubit ring state, n partial tomographies are performed for every subsystem with 4 qubits that forms a chain in the ring. For example, for the 8qubit graph state, the 8 subsystems are (q5, q6, q7, q8), (q6, q7, q8, q9), …, (q12, q5, q6, q7). For every state, 3^{4}n experimental configurations are used; 2048 measurements are taken under each configuration. The n 4qubit reduced density matrices are obtained using the maximum likelihood method proposed by Smolin et al.^{25}
Due to Eq. (3), for a ring graph state, each 4qubit density matrix of neighboring four qubits, as illustrated in Fig. 6, is given by
Then, for each 4qubit density matrix, we apply the local operations \(O_A = {\textstyle{{Z_A + I} \over 2}}\) and \(O_D = {\textstyle{{Z_D + I} \over 2}}\) and calculate the negativity of the resulting 2qubit subsystem. For instance, we may choose (q5, q6, q7, q8) as our subsystem; after applying O_{A} and O_{D} to q5 and q8 respectively, we will trace out q5 and q8, and measure the negativity of the remaining subsystem, (q6, q7). We choose \(O_A = {\textstyle{{Z_A + I} \over 2}}\) and \(O_D = {\textstyle{{Z_D + I} \over v}}\) for the following reason. If ρ is graph state, and the 4qubit subsystem corresponds to 4 vertices that form a chain in the graph, then the resulting 2qubit state is a maximally entangled state
Therefore, for a state close to this graph state, we should expect the resulting 2qubit state to have a negativity significantly greater than 0. The results are plotted in Fig. 7.
For the 8qubit graph state, the measured negativities are all significantly greater than 0. For the 10qubit graph state, 9 out of 10 measured negativities are significantly greater than 0. Based on our argument above, both the 8qubit state and 10qubit state are fully entangled.
In the 12qubit case, as shown in Fig. 7c, 10 out of 12 measured negativites are significantly nonzero. The two zeros come from (q9, q10) and (q14, q3) pairs. Therefore, there is only one possible separation, namely {q10, q11, q12, q13, q14}  {q3, q4, q5, q6, q7, q8, q9}. Should this be true, the reduced density matrix of qubits q8,q9,q10,q11 should also be separable with the separation {q8, q9}  {q10, q11}. In that case, its partial transpose with respect to qubit q8 and q9 must be positive. However, with respect to this partial transpose, ρ_{q8,q9,q10,q11} has negativity 0.0391 ± 0.0039 (standard deviation estimated via bootstrapping). Therefore, this possibility is ruled out with very high confidence. We can now conclude that the 12qubit graph state is fully entangled.
In the 14qubit case, as shown in Fig. 7d, 12 out of 14 measured negativites are significantly greater than 0. Here, we may apply the same trick again. The only possible separation is {q2, q3, q4, q5, q6, q7, q8, q9, q12, q13, q14}  {q10, q11}. In this case, subsystem {q8, q9, q10, q11} should have zero negativity with respect to the partial transpose on q8 and q9. However, the measured negativity is 0.0698 ± 0.0048 (standard deviation estimated via bootstrapping). Hence, this possibility is ruled out with very high confidence. We may conclude that this state is fully entangled.
In the 16qubit case, as shown in Fig. 7e, 15 out of 16 measured negativites are significantly greater than 0. As argued above, this means that this state is not biseparable w.r.t. a fixed partition, thereby showing that all 16 qubits in ibmqx5 are in full entanglement.
It may be noted that the subsystem of qubits {q8, q9, q10, q11} yields closetozero 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 website^{4} (see Fig. 1). For instance, the CNOT gate between q10 and q11 has the largest error among all possible CNOT gates, while the readout errors of q10 and q11 are also above the average level (6.5%).
Further exploration of the 16qubit 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 16qubit 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 6qubit subsystem that forms a chain. We first apply \(O_E = {\textstyle{{Z_E \pm I} \over 2}}\) and \(O_F = {\textstyle{{Z_F \pm I} \over 2}}\) on E and F respectively (four possibilities). On our data, this can effectively be done by first postselecting 0s on qubits E and F before calculating the tomography of {A, B, C, D}. Next, \(O_B = {\textstyle{{X_B + I} \over 2}}\) and \(O_D = {\textstyle{{Z_D + I} \over 2}}\) are performed (see Fig. 8). At last, B, E, F and D are traced out, while the negativity in subsystem {A, C} is calculated. If the 16qubit state is perfect, this resulting system would be maximum entangled.
Based on data obtained in previous experiments, we have calculated the corresponding negativity for each 6qubit 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 \(O_B^\prime = {\textstyle{{X_B + I} \over 2}}\) and \(O_C^\prime = {\textstyle{{X_C + I} \over 2}}\) (see Fig. 9). The negativity of subsystem {A, D} would be calculated. Again, if the 16qubit state is perfect, these two qubits would be maximum entangled; therefore, we should expect a nonzero negativity if the actual state is close to the theoretical one.
Among 16 pairs of qubits with distance 3, we have identified localized entanglement in 6 pairs of them. The results based on our data is presented in Table 2.
Discussion
We have prepared graph states of 8, 10, 12, 14 and 16 qubits on the 16qubit ibmqx5 processor and demonstrated that these graph states are not biseparable w.r.t. any fixed partition. In particular, we have realized full entanglement using all 16 qubits. Moreover, we have demonstrated the ability to create localized entanglement on qubit pairs with distance 3 and 4 from this 16qubit entangled state. In our approach of detecting nonseparability, we only have to measure the reduced density matrix of up to 4 qubits, and the size of reduced density matrix does not scale with the total qubit number, i.e., our method is efficient and scalable. In our experiments, graph states do not have high fidelity because of the large number of qubits, e.g., the fidelity of the 12qubit graph state is lower than 0.44. (This upperbound is obtained by computing the fidelity between each 4qubit subsystem and the theoretical result and taking the minimum.) However, the negativity of 4qubit reduced density matrix decays gently with respect to the qubit number, which implies that the error per qubit weakly depends on the qubit number. It is a strong evidence that ibmqx5 is capable of generating highly entangled states and demonstrates the computer’s quantumness. In computational tasks such as oneway quantum computing, graph state with decaying fidelity is acceptable, and the computing is fault tolerant as long as the error per qubit is lower than a threshold.^{27,28}
Data availability
The experimental data that support the findings of this study^{29} are available in figshare with the identifier https://doi.org/10.6084/m9.figshare.6790781.
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Acknowledgements
We gratefully acknowledge the IBM Q team for providing us access to their 16qubit platform. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum Experience team. Y.L. is supported by NSAF (Grant No. U1730449). Z.Q.Y. is supported by the National Natural Science Foundation of China Grants 61771278, 11574176 and 11474177. B.Z. is supported by NSERC and CIFAR.
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Y.L., Z.Q.Y. and B.Z. designed and conceived the study. Y.W. designed quantum circuits and performed the experiments. All authors wrote the manuscript.
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Wang, Y., Li, Y., Yin, Zq. et al. 16qubit IBM universal quantum computer can be fully entangled. npj Quantum Inf 4, 46 (2018). https://doi.org/10.1038/s415340180095x
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DOI: https://doi.org/10.1038/s415340180095x
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