Entanglement distribution is a critical task in quantum networks. Since the distributed entanglement can suffer from noise in the channel, it is necessary to develop methods that distill higher quality entanglement from the shared noisy entangled states. In this work, we propose a protocol to distill multi-qubit Greenberger-Horne-Zeilinger (GHZ) states among the nodes of a network using quantum error correcting codes. The method builds upon a Bell state distillation protocol by Wilde et at. (2007) that we recently generalized to 3-qubit GHZ states. The key technical result that enables our protocol shows how measuring a Pauli operator, or in general a set of code stabilizers, on one subsystem of a multipartite GHZ state affects the other subsystems. The design and analysis of the protocol is driven by the stabilizer formalism for measurements, and we provide discussions to elucidate the steps of the protocol. A similar approach can be applied to distill other multipartite entangled states as long as they are stabilizer states.
Entanglement distribution is a critical task in quantum networks. Since the distributed entanglement can suffer from noise in the channel, it is necessary to develop methods that distill higher quality entanglement from the shared noisy entangled states. In this work, we propose a protocol to distill multi-qubit Greenberger-Horne-Zeilinger (GHZ) states among the nodes of a network using quantum error correcting codes. The method builds upon a Bell state distillation protocol by Wilde et at. (2007) that we recently generalized to 3-qubit GHZ states. The key technical result that enables our protocol shows how measuring a Pauli operator, or in general a set of code stabilizers, on one subsystem of a multipartite GHZ state affects the other subsystems. The design and analysis of the protocol is driven by the stabilizer formalism for measurements, and we provide discussions to elucidate the steps of the protocol. A similar approach can be applied to distill other multipartite entangled states as long as they are stabilizer states.
We introduce an entanglement distillation protocol that utilizes an arbitrary [[n,k,d]] stabilizer code to convert n raw copies of an N-qubit Greenberger-Horne-Zeilinger (GHZ) state into k logical copies in the presence of Pauli noise. Our explicit formulation of the scheme on a star network topology is efficiently scalable to arbitrary N, requires only local operations, and minimizes the impact of errors on idle physical qubits, enabling the distribution of high-fidelity entangled logical states between any number of quantum processors. We report the results of numerical experiments using the 5-qubit code, toric code, and [[144,12,12]] bivariate bicyle code to protect against independent single-qubit depolarizing noise on all qubits. Our encoding scheme improves the final GHZ state fidelity relative to a single bare GHZ state sent over the same noisy channel for depolarizing rates up to or above 6% for all codes tested. Additionally, we use the stabilizer formalism to prove that our protocol can be applied to any N-qubit Calderbank-Shor-Steane (CSS) stabilizer state, from which the GHZ state emerges as a special case. This result explicitly establishes the working principle of previous stabilizer-based Bell pair and GHZ state distillation schemes, and generalizes such protocols beyond the traditionally considered states. We also show that some stabilizer codes can distill any stabilizer state with our protocol.
Entanglement distillation is a well-studied problem in quantum information, where one typically starts with n noisy Bell pairs and distills k Bell pairs of higher fidelity. While distilling Bell pairs is the canonical setting, it is important to study the distillation of multipartite entangled states because these can be useful for realizing distributed algorithms on quantum networks. In this paper, we study the distillation of GHZ states using quantum error correcting codes (QECCs). Using the stabilizer formalism, we begin by explaining the QECC-based Bell pair distillation protocol in arXiv:0708.3699, which relies particularly on the transpose symmetry between Alice's and Bob's qubits in Bell states. Extending this idea, we show that, given n GHZ states, performing a matrix on Alice's qubits is equivalent to performing a "stretched" version of the transpose of the matrix on the qubits of Bob and Charlie. We call this mapping to the stretched version of the matrix the GHZ-map, and show that it is an algebra homomorphism. Using this property, we show that Alice projecting her qubits onto an [[n,k]] stabilizer code implies the simultaneous projection of Bob's and Charlie's qubits onto an induced [[2n,k]] stabilizer code. Guided by this insight, we develop a GHZ distillation protocol based on local operations and classical communication that uses any stabilizer code. Inspired by stabilizer measurements on GHZ states, we also develop a new algorithm to generate logical Pauli operators of any stabilizer code and use it in the protocol. Since quantum codes with finite rate and almost linear minimum distance have recently been discovered, this paper paves the way for high-rate high-output-fidelity GHZ distillation. We provide simulation results on the 5-qubit perfect code to emphasize the importance of the placement of a certain local Clifford operation in the protocol.
Recent constructions of quantum low-density parity-check (QLDPC) codes provide optimal scaling of the number of logical qubits and the minimum distance in terms of the code length, thereby opening the door to fault-tolerant quantum systems with minimal resource overhead. However, the hardware path from nearest-neighbor-connection-based topological codes to long-range-interaction-demanding QLDPC codes is likely a challenging one. Given the practical difficulty in building a monolithic architecture for quantum systems, such as computers, based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected medium-sized quantum processors. In such a setting, all syndrome measurements and logical operations must be performed through the use of high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. This protocol is applicable beyond the application of DQC since entanglement distribution and purification is a quintessential task of any quantum network. We use the min-sum algorithm (MSA) based iterative decoder with a sequential schedule for distilling 3 -qubit GHZ states using a rate 0.118 family of lifted product QLDPC codes and obtain an input fidelity threshold of ≈0.7974 under i.i.d. single-qubit depolarizing noise. This represents the best threshold for a yield of 0.118 for any GHZ purification protocol. Our results apply to larger size GHZ states as well, where we extend our technical result about a measurement property of 3 -qubit GHZ states to construct a scalable GHZ purification protocol.
We approach two interconnected problems of quantum information processing in networks: Conference key agreement and entanglement distillation, both in the so-called source model where the given resource is a multipartite quantum state and the players interact over public classical channels to generate the desired correlation. The first problem is the distillation of a conference key when the source state is shared between a number of legal players and an eavesdropper; the eavesdropper, apart from starting off with this quantum side information, also observes the public communication between the players. The second is the distillation of Greenberger-Horne-Zeilinger (GHZ) states by means of local operations and classical communication (LOCC) from the given mixed state. These problem settings extend our previous paper [IEEE Trans. Inf. Theory 68(2):976-988, 2022], and we generalise its results: using a quantum version of the task of communication for omniscience, we derive novel lower bounds on the distillable conference key from any multipartite quantum state by means of non-interacting communication protocols. Secondly, we establish novel lower bounds on the yield of GHZ states from multipartite mixed states. Namely, we present two methods to produce bipartite entanglement between sufficiently many nodes so as to produce GHZ states. Next, we show that the conference key agreement protocol can be made coherent under certain conditions, enabling the direct generation of multipartite GHZ states.
In this paper, we introduce a novel heuristic approach designed to optimize the performance of Greenberger-Horne- Zeilinger (GHZ) creation and distillation protocols under decoherence. Our methodology converts these protocols into a practical set of instructions, demonstrating, through simulations, the production of higher-quality GHZ states than previously known protocols. This advancement contributes to the field of distributed quantum computing by addressing the need for high-quality entanglement required for operations between different quantum computers.
We develop an efficient local operation and classical communication (LOCC) scheme for the distillation of Greenberger-Horne-Zeilinger (GHZ) states from tripartite systems subjected to both coherent and incoherent errors. The proposed method employs an iterative process that applies a postselection-based non-linear transformation to increase the entanglement of 3-qubit states. In contrast to traditional distillation protocols that require an exponential number of initial states as a resource, our method achieves subexponential convergence towards a pure GHZ state. The proposed scheme is practical in the sense that it employs a small set of relatively simple unitary operations and projective measurements in the computational basis. We systematically develop a double-iteration protocol by providing a mathematical framework for the transformation processes involved, emphasizing the role of unitary operations in correcting arbitrary small errors in the initial states. Through analytical derivations and numerical simulations, we demonstrate the protocol's ability to progressively eliminate noise and improve fidelity over subsequent iterations. Significantly, our protocol not only corrects for small arbitrary distortions in the GHZ states but also maintains operational simplicity, making it feasible for practical quantum computing applications.