Heterogeneous multipartite entanglement purification for size-constrained quantum devices

The entanglement resource required for quantum information processing comes in a variety of forms, from Bell states to multipartite GHZ states or cluster states. Purifying these resources after their imperfect generation is an indispensable step towards using them in quantum architectures. While this challenge, both in the case of Bell pairs and more general multipartite entangled states, is mostly overcome in the presence of perfect local quantum hardware with unconstrained qubit register sizes, devising optimal purification strategies for finite-size realistic noisy hardware has remained elusive. Here we depart from the typical purification paradigm for multipartite states explored in the last 20 years. We present cases where the hardware limitations are taken into account and, surprisingly, find that smaller “sacrificial” states, such as Bell pairs, can be more useful in the purification of multipartite states than additional copies of these same states. This drastically simplifies the requirements and presents a fundamentally different pathway to leverage near-term networked quantum hardware.

Figure 1

Purification circuits for three-qubit GHZ states. On the left is the standard two-stage circuit, out of which the hashing method can be built. It sacrifices an entire GHZ state at each stage. We denote the stages as P1 and P2 (a full implementation of the hashing method would sacrifice an asymptotic number of GHZ states in repeated applications of P1 and P2). On the right is our much simpler circuit, sacrificing only two Bell pairs for the same final fidelity and higher success probability (or yield). Our circuit not only requires much simpler resources (four qubits in two Bell pairs instead of six qubits in two GHZ states) but, as it is shorter, it is less susceptible to gate errors. The vertical classical communication (double lines) between the measurement symbols denotes the sets of classical bits whose parity needs to be checked in order to detect errors. Stars denote the qubits that need to be transmitted over the network, thus being subjected to noise. Measuring stabilizers that involve mostly the noisier qubits transmitted over the network provide for higher error detection efficiency.

Figure 2

Performance comparison of various purification circuits. From left to right, the columns represent purification of (a) a single three-qubit GHZ state, (b) a four-qubit GHZ state, and (c) a four-qubit linear cluster state. The top figures give the fidelity of the purified state, while the bottom figures give the probability of success for the purification circuit. In purple $(\mathrm{P1}+\mathrm{P2})$ is the typical “one block of the hashing method” considered in the literature, which we compare against. As the circuits are heterogeneous, one cannot provide a well-defined yield. Rather, the main observation is that for short circuits, performing the sacrifice of a small number of Bell pairs can provide higher final fidelity and a better probability of success than performing the sacrifice of the same number of large, expensive multipartite entangled states. The horizontal axis corresponds to the initial fidelity of the state to be purified. All circuits assume all entangled resources are generated locally on one of the nodes and then distributed over the network. All qubits sent over the network suffer the same rate of depolarization noise. The P1 and P2 purification circuits are the two stages of purification used in the standard hashing method. “$\mathrm{P1}+\mathrm{P2}$” denotes both stages being performed (and two multipartite states being sacrificed). In the case of purifying four-qubit states, we have included curves for the best circuits using two Bell sacrifices, but we do not discuss them in the main text as they do not show exceptional performance (they are available in the documentation of [40]).

Figure 3

Purification circuits for the four-qubit entangled states. Left to right, the circuits include our heterogeneous circuit for purification of a four-qubit GHZ state; the standard two-stage circuit for the purification of the four-qubit cluster state (we refer to each stage by the labels P1 and P2—out of these stages, an asymptotic hashing purification protocol can be built); and our simpler heterogeneous circuit for purification of the four-qubit cluster state. Our circuit provides higher fidelity of the final purified state and requires simpler sacrificial resources. Moreover, as our circuit is shorter, it is less susceptible to uncorrected gate errors. The vertical classical communication (double lines) between the measurement symbols denotes sets of classical bits whose parity needs to be checked in order to detect errors. Our circuits specifically target the measurement of stabilizers that are more probable to be flipped by network errors, given the particularities of the network (one qubit per state is never transmitted over the network and stays at the entanglement generation node).

Figure 4

The effect of local gate infidelity is minimized in our purification circuits. Throughout this text, we stressed the importance of designing small circuits for the sake of lowering resource overhead. This has the additional benefit of limiting the deleterious effects due to the finite fidelity with which local gates can be performed. This figure showcases the drastic improvement in purification fidelity in our circuits, simply by virtue of performing fewer imperfect operations. For this example, we pick the circuits for purification of three-qubit GHZ states. We compare a single round of the state-of-the-art hashing method (in purple) vs our smaller and simpler purification circuit (in green). With different line styles (solid, dashed, and dotted), we represent the infidelity (depolarization rate) of the local gates performing the purification.