Abstract
The ability to prepare sizeable multiqubit entangled states with full qubit control is a critical milestone for physical platforms upon which quantum computers are built. We investigate the extent to which entanglement is found within a prepared graph state on the 20qubit superconducting quantum computer IBM Q Poughkeepsie. We prepared a graph state along a path consisting of all twenty qubits within the device and performed full quantum state tomography on all groups of four connected qubits along this path. We determined that each pair of connected qubits was inseparable and hence the prepared state was entangled. Additionally, a genuine multipartite entanglement witness was measured on all qubit subpaths of the graph state and we found genuine multipartite entanglement on chains of up to three qubits. These results represent a demonstration of entanglement in one of the largest solidstate qubit arrays to date and indicate the positive direction of progress towards the goal of implementing complex quantum algorithms relying on such effects.
Introduction
Quantum entanglement describes nonclassical correlations between different subsystems^{1}. It is regarded as one of the key hallmarks separating quantum from classical systems. Its significance is fundamental, being the subject of the ‘spooky’ correlations noted by Einstein, Podolsky and Rosen (EPR)^{2} and used by Bell^{3,4,5} to rule out nonlocal hidden variable descriptions of quantum mechanics. More recently the utility of entanglement has become apparent, with quantum entanglement viewed as a useful resource to aid information processing tasks^{6}. The simplest form of entanglement, EPR pairs, enable tasks such as quantum cryptography^{7}, superdense coding^{8}, teleportation^{9} and entanglement swapping^{10,11}. More complex, multiqubit examples of entanglement enable oneway quantum computation^{12}, entanglement assisted error correction^{13}, and as a tomographic resource^{14}.
Quantum entanglement often is seen as a key ingredient if quantum computers are to demonstrate an advantage over classical computers. In particular, if a quantum system is not highly entangled it can often be simulated efficiently on a classical computer^{15,16,17}. Multiqubit entanglement is therefore a fundamental property for potential quantum computers to demonstrate, if they are to ultimately outperform classical computation^{18}. To this end large multiqubit entangled states have been demonstrated in a number of experimental systems. On systems with full qubit control, entanglement has been shown on up to a 16qubit superconducting system^{19} and a 20qubit ion trap system^{20,21}, while genuine multipartite entanglement has been shown on up to an 18qubit photonic system^{22}, 12photon system^{23,24} and 12qubit superconducting system^{25,26}. Genuine multipartite entanglement has also been shown on arrays of up to 22 atomic ensembles however they are not convenient for the realisation of quantum computation^{27}.
Over the past few years, a series of quantum devices have been released by IBM that comprise five to twenty superconducting qubits and can be accessed via their cloud service^{28,29}. Of particular interest here is the device IBM Q Poughkeepsie, which exhibits improved error rates over previous devices. In this paper we consider entanglement generation and quantification on the Poughkeepsie device. We show that Poughkeepsie can be fully entangled. To do this, the system was prepared into a graph state along a path through all twenty qubits and full quantum state tomography performed on all groups of four connected qubits along this path. Using these measurements, we evaluated the negativity^{30,31} to detect entanglement between each pair of qubits along the chain. By showing that every pair is entangled we conclude that the graph state is not separable and hence fully entangled. The magnitude of the entanglement measured between each pair of qubits in the graph state, and the number of qubits entangled, surpasses the entanglement previously measured between pairs within the 16qubit IBM Q Ruschlikon (ibmqx5) device^{19}. In addition, we consider genuine multipartite entanglement within the 20qubit graph state. To quantify this, we make use of an entanglement witness^{32}, and show there is genuine multipartite entanglement on chains of three qubits. During the submission process, we were made aware of recent works that demonstrate genuine multipartite entanglement on 18 and 20qubit GreenbergerHorneZeilinger (GHZ) states^{33,34}, and in particular on an 18qubit GHZ state prepared on the IBM Q System One device^{35}.
Results
Entanglement measures
A pure state ρ_{p} is separable if there exists a qubit bipartition A and B such that
where ρ^{A} and ρ^{B} are pure states of A and B respectively. A pure state is entangled if it is not separable. Similarly, a mixed state ρ over n qubits is separable (see Fig. 1a) if it can be expressed as a probabilistic mixture of separable pure states with respect to a fixed qubit bipartition A and B. That is,
where \({\rho }_{i}^{A}\) and \({\rho }_{i}^{B}\) are pure states of A and B, N is the number of pure states over all qubits in the composition and the probabilities satisfy p_{i} ≥ 0 and \(\sum _{i}\,{p}_{i}=1\). A mixed state is entangled if it is not separable.
Entanglement between bipartitions A and B can be determined by calculating an entanglement measure^{36,37}, such as the negativity, of the state. For a given state ρ, the negativity \({\mathscr{N}}(\rho )\) is given by
where λ_{i} are the eigenvalues of the partial transpose of ρ with respect to the system B^{38}. If the negativity is greater than zero, then the two partitions are entangled. In the case of a 2qubit mixed state, the negativity is zero if and only if the two qubits are separable^{39}. So although other entanglement measures exist such as concurrence^{40}, for a 2qubit system, the negativity alone is a sufficient condition.
For a state with more than two qubits, things are more complicated because each pure state can be separated into different partitions. A mixed state ρ is biseparable (see Fig. 1b) if it can be expressed as
where N is the number of pure states over all qubits in the composition, \({\rho }_{i}^{{a}_{i}}\) and \({\rho }_{i}^{{b}_{i}}\) are pure states of qubit bipartitions a_{i} and b_{i}, and probabilities satisfy p_{i} ≥ 0 and \(\sum _{i}\,{p}_{i}=1\). An entangled mixed state is bipartite entangled if it is biseparable and genuine multipartite entangled otherwise as shown in Fig. 1c.
Detecting genuine multipartite entanglement using full quantum tomography can be a prohibitively expensive process. However, an entanglement witness^{41}, which is an observable that detects the presence of entanglement, can require far fewer measurements. An entanglement witness has a nonnegative expectation value for all separable states and a negative value for some entangled states. A negative expectation value implies the presence of entanglement, however the converse is not necessarily true. Here, we use a genuine multipartite entanglement witness^{32} which is defined on a graph state G_{n}〉 of n qubits as
where \({S}_{i}^{({G}_{n})}\) is the i^{th} generator of the stabiliser group of G_{n}〉.
To show that all qubits within a quantum computer can be entangled, we first aim to prepare them into a highly entangled state. We use the graph state since it has been shown to have entanglement that is more robust against local measurements and noise than GHZ states^{42}. A graph state is defined in relation to a graph where each qubit corresponds to a vertex and is prepared in the state \(+\rangle \equiv (0\rangle +1\rangle )/\sqrt{2}\), then controlledphase gates are applied between adjacent qubits within the graph. The state is genuinely multipartite entangled in the absence of errors and decoherence. The resulting state has the form
where E is the edge set of the graph G_{n} corresponding to the n qubit graph state, and \({{\rm{CZ}}}_{\beta }^{\alpha }\) represents a controlledphase gate between adjacent qubits α and β. The set of generators of the stabiliser group, which we refer to as stabilisers, for the graph state can be written as
where N_{α} is the set of neighbouring qubits of α.
Graph states have the property that by projecting all but 2 qubits to the Zbasis we are in principle left with maximally entangled Bell states (up to local transformations). The entanglement between these remaining qubits may then be determined by measuring the negativity.
As an example, consider a 4qubit graph state on qubits 1, 2, 3 and 4 which is stabilised by the operators
Since K_{1} anticommutes with Z_{1}I_{2}I_{3}I_{4} while the other stabilisers commute, projecting qubit 1 onto the Zbasis corresponds to replacing K_{1} with \({K}_{1}^{^{\prime} }={(1)}^{{m}_{1}}{Z}_{1}{I}_{2}{I}_{3}{I}_{4}\) ^{43} where m_{i} ∈ {0, 1} is the projected state of qubit i. Similarly, projecting qubit 4 onto the Zbasis corresponds to replacing K_{4} with \({K}_{4}^{^{\prime} }={(1)}^{{m}_{4}}{I}_{1}{I}_{2}{I}_{3}{Z}_{4}\). The stabilisers can then be simplified to
which stabilise the Bell states (up to local transformations)
Since the dimension of the Hilbert space doubles for each qubit, performing full quantum state tomography on a 20qubit system would require approximately 3.5 billion measurements. However, to show that a graph state is entangled, we can show that for any given bipartition of the graph, qubits in one partition are entangled with those in the other. For a connected graph, any bipartition will have at least one qubit in one partition that has a neighbour in the other partition. Thus, if we can show that there is entanglement between every pair of connected qubits, then any two partitions must contain a qubit in one partition that is entangled with a qubit from the other partition. Using local operations, we project the pair into a Bell state, and then perform quantum tomography on each pair of connected qubits α and β with their neighbours N_{α} and N_{β}, then calculate the negativity between them to establish entanglement between each pair.
20qubit entanglement
The strategy to detect entanglement along a chain of qubits was implemented on physical hardware, specifically that of IBM Q Poughkeepsie. The device consists of 20 superconducting qubits that are capable of coherence times of T_{1}, T_{2} ~ 100 μs^{44}. A path consisting of all qubits was embedded onto the device layout as shown in Fig. 2 and a corresponding graph state was prepared using the circuit shown in Fig. 3.
To determine that each pair of connected qubits are entangled, we perform full quantum state tomography on each quad – the pair and their neighbours – as described in the previous section. When constructing the tomography circuits, measurements that include the I basis can be postprocessed, totalling just 3^{n} circuit configurations instead of the full 4^{n} configurations. A total of 2048 shots were used for each measurement. To ensure that the constructed density matrix has physical (nonnegative) eigenvalues, the nearest physical state under the 2norm is determined using the efficient procedure introduced by Smolin et al.^{45}.
The negativity for each Zbasis state projection was calculated and the results are plotted in Fig. 4 and tabulated in Table 1. The negativity ranges between 0 and 0.5 where 0 indicates no entanglement and 0.5 indicates maximum entanglement. There are four different, equally likely, outcomes for Zmeasurements on the neighbouring qubits in the chain (and two different outcomes for the pairs at the end of each chain). Each of these measurement outcomes leads to a different Bell state, which, being noisy, can have a different negativity. Figure 4 shows for every pair of neighbouring qubits in the chain (a) the negativity of the 00 projection/measurement result, (b) the maximum of the four negativities and (c) the mean of the four negativities. Our results, in all three cases, indicate that the Poughkeepsie device has been fully entangled.
The negativities for the zero state projection shown in Fig. 4a were used to compare entanglement with the previous results found within the 16qubit IBM Q Ruschlikon (ibmqx5) device^{19}. Rather than embedding a path, they embed a loop which has the benefit that instead of requiring all connected pairs to have statistically significant nonzero negativity, it allows up to one pair to have a nonsignificant value. For the 16qubit graph state, 15 of the 16 connected pairs had statistically significant nonzero negativity, so we compare with those. We found that the magnitude of entanglement between pairs of qubits in the IBM Q Poughkeepsie device surpasses the Ruschlikon device. We found that the minimum and maximum negativities and 95% confidence intervals calculated on the IBM Q Poughkeepsie device were 0.082 (0.054, 0.103) and 0.329 (0.310, 0.349) while the Ruschlikon device had 0.034 (0.012, 0.053) and 0.241 (0.229, 0.261) respectively. The 95% confidence intervals are calculated using bootstrapping methods^{46}. While the results imply entanglement in the 20qubit system they do not allow us to rule out the possibility that the state is biseparable.
Genuine multipartite entanglement
To further investigate the entanglement and biseparability of the graph state, a genuine multipartite entanglement witness is measured. Using the tomography results obtained for the 20qubit graph state, the genuine multipartite entanglement witness in Eq. 5 was calculated for all chains of qubits along the graph state. The results are shown in Fig. 5, where negative values imply genuine multipartite entanglement and the 95% confidence intervals are estimated using bootstrapping methods^{46}. Up to 3qubit chains were found to be genuinely multipartite entangled and are visually summarised in Fig. 6. A 4qubit chain between qubits 15 and 18 was found to have a negative witness however was nonsignificant. Figure 5a shows all values below zero, while Fig. 5b has nonsignificant and redundant values omitted for clarity, e.g. if (86) are genuinely multipartite entangled then (87) must also be genuinely multipartite entangled making the result for (87) redundant. The witness found no entanglement for all pairs of qubits between qubit 10 and 19 even though they were shown to be entangled by the negativity analysis. This is due to the witness detecting genuine multipartite entanglement in only some genuinely multipartite entangled states. However in the case of negativity, for all 2qubit states, nonzero negativity is equivalent to entanglement. These results are inconclusive as to whether the graph state is genuinely multipartite entangled. To investigate further, a more tailored experiment could be used such as calculating the fidelity^{47} or entanglement structure^{48} of a GHZ state.
Discussion
By preparing a graph state along a path consisting of all 20 qubits within the IBM Q Poughkeepsie device, we were able to determine that entanglement is present between any bipartition of the system in this state. To do this we performed full quantum state tomography over each connected pair and neighbours. By calculating the negativity between every pair, we determined that each pair possessed nonzero entanglement. We found that the level of entanglement of the full system surpasses previous results found in the 16qubit IBM Q Ruschlikon (ibmqx5) device^{19}. Additionally, we calculated a genuine multipartite entanglement witness along all qubit subpaths of the 20qubit graph state, finding genuine multipartite entanglement in up to 3qubit chains. These results indicate that the ability to entangle qubits in these devices is steadily improving as required for the physical implementation of complex quantum algorithms.
Data Availability
The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by the University of Melbourne through the establishment of an IBM Network Q Hub at the University. We would like to thank Anna Phan and Frank Suits for reading through this paper and providing valuable feedback.
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G.J.M., C.D.H. and L.C.L.H. conceived the experiment, G.J.M. obtained, processed and analysed the results, and wrote the manuscript with input from C.D.H. and L.C.L.H.
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The authors are supported by the University of Melbourne through the establishment of an IBM Network Q Hub at the University.
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Mooney, G.J., Hill, C.D. & Hollenberg, L.C.L. Entanglement in a 20Qubit Superconducting Quantum Computer. Sci Rep 9, 13465 (2019). https://doi.org/10.1038/s41598019498057
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