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Entanglement is essential for quantum information processing but is limited by noise. We address this by developing high-yield entanglement distillation protocols with several advancements. (1) We extend the 2-to-1 recurrence entanglement distillation protocol to higher-rate n-to-(n-1) protocols that can correct any single-qubit errors. These protocols are evaluated through numerical simulations focusing on fidelity and yield. We also outline a method to adapt any classical error-correcting code for entanglement distillation, where the code can correct both bit-flip and phase-flip errors by incorporating Hadamard gates. (2) We propose a constant-depth decoder for stabilizer codes that transforms logical states into physical ones using single-qubit measurements. This decoder is applied to entanglement distillation protocols, reducing circuit depth and enabling protocols derived from advanced quantum error-correcting codes. We demonstrate this by evaluating the circuit complexity for entanglement distillation protocols based on surface codes and quantum convolutional codes. (3) Our stabilizer entanglement distillation techniques advance quantum computing. We propose a fault-tolerant protocol for constant-depth encoding and decoding of arbitrary quantum states, applicable to quantum low-density parity-check (qLDPC) codes and surface codes. This protocol is feasible with state-of-the-art reconfigurable atom arrays and surpasses the limits of conventional logarithmic depth encoders. Overall, our study integrates stabilizer formalism, measurement-based quantum computing, and entanglement distillation, advancing both quantum communication and computing.

All-photonic quantum repeaters use multi-qubit photonic graph states, called repeater graph states (RGS), instead of matter-based quantum memories, for protection against predominantly loss errors. The RGS comprises tree-graph-encoded logical qubits for error correction at the repeaters and physical \em link qubits to create entanglement between neighboring repeaters. The two methods to generate the RGS are probabilistic stitching -- using linear optical Bell state measurements (fusion) -- of small entangled states prepared via multiplexed-probabilistic linear optical circuits fed with single photons, and a direct deterministic preparation using a small number of quantum-logic-capable solid-state emitters. The resource overhead due to fusions and the circuit depth of the quantum emitter system both increase with the size of the RGS. Therefore engineering a resource-efficient RGS is crucial. We propose a new RGS design, which achieves a higher entanglement rate for all-photonic quantum repeaters using fewer qubits than the previously known RGS would. We accomplish this by boosting the probability of entangling neighboring repeaters with tree-encoded link qubits. We also propose a new adaptive scheme to perform logical BSM on the link qubits for loss-only errors. The adaptive BSM outperforms the previous schemes for logical BSM on tree codes when the qubit loss probability is uniform. It reduces the number of optical modes required to perform logical BSM on link qubits to improve the entanglement rate further.

Bell-state measurement (BSM) on entangled states shared between quantum repeaters is the fundamental operation used to route entanglement in quantum networks. Performing BSMs on Werner states shared between repeaters leads to exponential decay in the fidelity of the end-to-end Werner state with the number of repeaters, necessitating entanglement distillation. Generally, entanglement routing protocols use \emphprobabilistic distillation techniques based on local operations and classical communication. In this work, we use quantum error correcting codes (QECCs) for \emphdeterministic entanglement distillation to route Werner states on a chain of repeaters. To maximize the end-to-end distillable entanglement, which depends on the number and fidelity of end-to-end Bell pairs, we utilize global link-state knowledge to determine the optimal policy for scheduling distillation and BSMs at the repeaters. We analyze the effect of the QECC's properties on the entanglement rate and the number of quantum memories. We observe that low-rate codes produce high-fidelity end-to-end states owing to their excellent error-correcting capability, whereas high-rate codes yield a larger number of end-to-end states but of lower fidelity. The number of quantum memories used at repeaters increases with the code rate as well as the classical computation time of the QECC's decoder.

Multi-qubit photonic graph states are necessary for quantum communication and computation. Preparing photonic graph states using probabilistic stitching of single photons using linear optics results in a formidable resource requirement due to the need of multiplexing. Quantum emitters present a viable solution to prepare photonic graph states, as they enable controlled production of photons entangled with the emitter qubit, and deterministic two-qubit interactions among emitters. A handful of emitters often suffice to generate useful photonic graph states that would otherwise require millions of single photon sources using the linear-optics method. But, photon loss poses an impediment to this method due to the large depth, i.e., age of the oldest photon, of the graph state, given the typically large number of slow and noisy two-qubit CNOT gates required on emitters. We propose an algorithm that can trade the number of emitters with the graph-state depth, while minimizing the number of emitter CNOTs. We apply our algorithm to generating a repeater graph state (RGS) for all-photonic repeaters. We find that our scheme achieves a far superior rate-vs.-distance performance than using the least number of emitters needed to generate the RGS. Yet, our scheme is able to get the same performance as the linear-optics method of generating the RGS where each emitter is used as a single-photon source, but with orders of magnitude fewer emitters.

Stabilizer states along with Clifford manipulations (unitary transformations and measurements) thereof -- despite being efficiently simulable on a classical computer -- are an important tool in quantum information processing, with applications to quantum computing, error correction and networking. Cluster states, defined on a graph, are a special class of stabilizer states that are central to measurement based quantum computing, all-photonic quantum repeaters, distributed quantum computing, and entanglement distribution in a network. All cluster states are local-Clifford equivalent to a stabilizer state. In this paper, we review the stabilizer framework, and extend it, by: incorporating general stabilizer measurements such as multi-qubit fusions, and providing an explicit procedure -- using Karnaugh maps from Boolean algebra -- for converting arbitrary stabilizer gates into tableau operations of the CHP formalism for efficient stabilizer manipulations. Using these tools, we develop a graphical rule-book and a MATLAB simulator with a graphical user interface for arbitrary stabilizer manipulations of cluster states, a user of which, e.g., for research in quantum networks, will not require any background in quantum information or the stabilizer framework. We extend our graphical rule-book to include dual-rail photonic-qubit cluster state manipulations with probabilistically-heralded linear-optical circuits for various rotated Bell measurements, i.e., fusions (including new `Type-I' fusions we propose, where only one of the two fused qubits is destructively measured), by incorporating graphical rules for their success and failure modes. Finally, we show how stabilizer descriptions of multi-qubit fusions can be mapped to linear optical circuits.

Quantum networks will be able to service consumers with long-distance entanglement by use of quantum repeaters that generate Bell pairs (or links) with their neighbors, iid with probability $p$ and perform Bell State Measurements (BSMs) on the links that succeed iid with probability $q$. While global link state knowledge is required to maximize the rate of entanglement generation between any two consumers, it increases the protocol latency due to the classical communication requirements and requires long quantum memory coherence times. We propose two entanglement routing protocols that require only local link state knowledge to relax the quantum memory coherence time requirements and reduce the protocol latency. These protocols utilize multi-path routing protocol and time multiplexed repeaters. The time multiplexed repeaters first generate links for $k$-time steps before performing BSMs on any pairs of links. Our two protocols differ in the decision rule used for performing BSMs at the repeater: the first being a static path based routing protocol and second a dynamic distance based routing protocol. The performance of these protocols depends on the quantum network topology and the consumers' location. We observe that the average entanglement rate and the latency increase with the time multiplexing block length, $k$, irrespective of the protocol. When a step function memory decoherence model is introduced such that qubits are held in the quantum memory for an exponentially distributed time with mean $\mu$, an optimal $k$ ($k_\text{opt}$) value appears, such that for increasing $k$ beyond $k_{\rm opt}$ hurts the entanglement rate. $k_{\rm opt}$ decreases with $p$ and increases with $\mu$. $k_{\rm opt}$ appears due to the tradeoff between benefits from time multiplexing and the increased likelihood of previously established Bell pairs decohering due to finite memory coherence times.

Measurement-Based Quantum Computing (MBQC), proposed in 2001 is a model of quantum computing that achieves quantum computation by performing a series of adaptive single-qubit measurements on an entangled cluster state. Our project is aimed at introducing MBQC to a wide audience ranging from high school students to quantum computing researchers through a Tangram puzzle with a modified set of rules, played on an applet. The rules can be understood without any background in quantum computing. The player is provided a quantum circuit, shown using gates from a universal gate set, which the player must map correctly to a playing board using polyominos. Polyominos or 'puzzle blocks' are the building blocks of our game. They consist of square tiles joined edge-to-edge to form different colored shapes. Each tile represents a single-qubit measurement basis, differentiated by its color. Polyominos rest on a square-grid playing board, which signifies a cluster state. We show that mapping a quantum circuit to MBQC is equivalent to arranging a set of polyominos, each corresponding to a gate in the circuit on the playing board, subject to certain rules, which involve rotating and deforming polyominos. We state the rules in simple terms with no reference to quantum computing. The player has to place polyominos on the playing board conforming to the rules. Any correct solution creates a valid realization of the quantum circuit in MBQC. A higher-scoring correct solution fills up less space on the board, resulting in a lower-overhead embedding of the circuit in MBQC, an open and challenging research problem.

In a quantum network that successfully creates links, shared Bell states between neighboring repeater nodes, with probability $p$ in each time slot, and performs Bell State Measurements at nodes with success probability $q<1$, the end to end entanglement generation rate drops exponentially with the distance between consumers, despite multi-path routing. If repeaters can perform multi-qubit projective measurements in the GHZ basis that succeed with probability $q$, the rate does not change with distance in a certain $(p,q)$ region, but decays exponentially outside. This region where the distance independent rate occurs is the supercritical region of a new percolation problem. We extend this GHZ protocol to incorporate a time-multiplexing blocklength $k$, the number of time slots over which a repeater can mix-and-match successful links to perform fusion on. As $k$ increases, the supercritical region expands. For a given $(p,q)$, the entanglement rate initially increases with $k$, and once inside the supercritical region for a high enough $k$, it decays as $1/k$ GHZ states per time slot. When memory coherence time exponentially distributed with mean $\mu$ is incorporated, it is seen that increasing $k$ does not indefinitely increase the supercritical region; it has a hard $\mu$ dependent limit. Finally, we find that incorporating space-division multiplexing, i.e., running the above protocol independently in up to $d$ disconnected network regions, where $d$ is the network's node degree, one can go beyond the 1 GHZ state per time slot rate that the above randomized local link-state protocol cannot surpass. As $(p,q)$ increases, one can approach the ultimate min-cut entanglement generation capacity of $d$ GHZ states per slot.

Graph states are a central resource in measurement-based quantum information processing. In the photonic qubit architecture based on Gottesman-Kitaev-Preskill (GKP) encoding, the generation of high-fidelity graph states composed of realistic, finite-energy approximate GKP-encoded qubits thus constitutes a key task. We consider the finite-energy approximation of GKP qubit states given by a coherent superposition of shifted finite-squeezed vacuum states, where the displacements are Gaussian distributed. We present an exact description of graph states composed of such approximate GKP qubits as a coherent superposition of a Gaussian ensemble of randomly displaced ideal GKP-qubit graph states. We determine the transformation rules for the covariance matrix and the mean displacement vector of the Gaussian distribution of the ensemble under tools such as GKP-Steane error correction and fusion operations that can be used to grow large, high-fidelity GKP-qubit graph states. The former captures the noise in the graph state due to the finite-energy approximation of GKP qubits, while the latter relates to the possible absolute displacement errors on the individual qubits due to the homodyne measurements that are a part of these tools. The rules thus help in pinning down an exact coherent error model for graph states generated from truly finite-energy GKP qubits, which can shed light on their error correction properties.

Quantum communications capacity using direct transmission over length-$L$ optical fiber scales as $R \sim e^{-\alpha L}$, where $\alpha$ is the fiber's loss coefficient. The rate achieved using a linear chain of quantum repeaters equipped with quantum memories, probabilistic Bell state measurements (BSMs) and switches used for spatial multiplexing, but no quantum error correction, was shown to surpass the direct-transmission capacity. However, this rate still decays exponentially with the end-to-end distance, viz., $R \sim e^{-s{\alpha L}}$, with $s < 1$. We show that the introduction of temporal multiplexing - i.e., the ability to perform BSMs among qubits at a repeater node that were successfully entangled with qubits at distinct neighboring nodes at \em different time steps - leads to a sub-exponential rate-vs.-distance scaling, i.e., $R \sim e^{-t\sqrt{\alpha L}}$, which is not attainable with just spatial or spectral multiplexing. We evaluate analytical upper and lower bounds to this rate, and obtain the exact rate by numerically optimizing the time-multiplexing block length and the number of repeater nodes. We further demonstrate that incorporating losses in the optical switches used to implement time multiplexing degrades the rate-vs.-distance performance, eventually falling back to exponential scaling for very lossy switches. We also examine models for quantum memory decoherence and describe optimal regimes of operation to preserve the desired boost from temporal multiplexing. Quantum memory decoherence is seen to be more detrimental to the repeater's performance over switching losses.

Photonics is the platform of choice to build a modular, easy-to-network quantum computer operating at room temperature. However, no concrete architecture has been presented so far that exploits both the advantages of qubits encoded into states of light and the modern tools for their generation. Here we propose such a design for a scalable and fault-tolerant photonic quantum computer informed by the latest developments in theory and technology. Central to our architecture is the generation and manipulation of three-dimensional hybrid resource states comprising both bosonic qubits and squeezed vacuum states. The proposal enables exploiting state-of-the-art procedures for the non-deterministic generation of bosonic qubits combined with the strengths of continuous-variable quantum computation, namely the implementation of Clifford gates using easy-to-generate squeezed states. Moreover, the architecture is based on two-dimensional integrated photonic chips used to produce a qubit cluster state in one temporal and two spatial dimensions. By reducing the experimental challenges as compared to existing architectures and by enabling room-temperature quantum computation, our design opens the door to scalable fabrication and operation, which may allow photonics to leap-frog other platforms on the path to a quantum computer with millions of qubits.

We develop a protocol for entanglement generation in the quantum internet that allows a repeater node to use $n$-qubit Greenberger-Horne-Zeilinger (GHZ) projective measurements that can fuse $n$ successfully-entangled \em links, i.e., two-qubit entangled Bell pairs shared across $n$ network edges, incident at that node. Implementing $n$-fusion, for $n \ge 3$, is in principle not much harder than $2$-fusions (Bell-basis measurements) in solid-state qubit memories. If we allow even $3$-fusions at the nodes, we find---by developing a connection to a modified version of the site-bond percolation problem---that despite lossy (hence probabilistic) link-level entanglement generation, and probabilistic success of the fusion measurements at nodes, one can generate entanglement between end parties Alice and Bob at a rate that stays constant as the distance between them increases. We prove that this powerful network property is not possible to attain with any quantum networking protocol built with Bell measurements and multiplexing alone. We also design a two-party quantum key distribution protocol that converts the entangled states shared between two nodes into a shared secret, at a key generation rate that is independent of the distance between the two parties.