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arXiv:2302.07908 · PRX Quantum

Linear Optical Logical Bell State Measurements with Optimal Loss-Tolerance Threshold

Paul Hilaire, Yaron Castor, Edwin Barnes, Sophia E. Economou, Frédéric Grosshans

Published Nov 6, 2023• 12 citations
arXivHTMLPDF

Abstract

Quantum threshold theorems impose hard limits on the hardware capabilities to process quantum information.We derive tight and fundamental upper bounds to loss-tolerance thresholds in different linearoptical quantum information processing settings through an adversarial framework, taking into account the intrinsically probabilistic nature of linear optical Bell measurements.For logical Bell state measurements-ubiquitous operations in photonic quantum information-we demonstrate analytically that linear optics can achieve the fundamental loss threshold imposed by the no-cloning theorem even though, following the work of Lee et al. [Phys.Rev. A 100, 052303 (2019)] the constraint was widely assumed to be stricter.We spotlight the assumptions of the latter publication and find their bound holds for a logical Bell measurement built from adaptive physical linear-optical Bell measurements.We also give an explicit even stricter bound for nonadaptive Bell measurements.

DOI
10.1103/prxquantum.4.040322

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Quantum threshold theorems impose hard limits on the hardware capabilities to process quantum information.We derive tight and fundamental upper bounds to loss-tolerance thresholds in different linearoptical quantum information processing settings through an adversarial framework, taking into account the intrinsically probabilistic nature of linear optical Bell measurements.For logical Bell state measurements-ubiquitous operations in photonic quantum information-we demonstrate analytically that linear optics can achieve the fundamental loss threshold imposed by the no-cloning theorem even though, following the work of Lee et al. [Phys.Rev. A 100, 052303 (2019)] the constraint was widely assumed to be stricter.We spotlight the assumptions of the latter publication and find their bound holds for a logical Bell measurement built from adaptive physical linear-optical Bell measurements.We also give an explicit even stricter bound for nonadaptive Bell measurements.
Bell measurements (BMs) are ubiquitous in quantum information and technology. They are basic elements for quantum commmunication, computation, and error correction. In particular, when performed on logical qubits encoded in physical photonic qubits, they allow for a read-out of stabilizer syndrome information to enhance loss tolerance in qubit-state transmission and fusion. However, even in an ideal setting without photon loss, BMs cannot be done perfectly based on the simplest experimental toolbox of linear optics. Here we demonstrate that any logical BM on stabilizer codes can always be mapped onto a single physical BM perfomed on any qubit pair from the two codes. As a necessary condition for the success of a logical BM, this provides a general upper bound on its success probability, especially ruling out the possibility that the stabilizer information obtainable from only partially succeeding, physical linear-optics BMs could be combined into the full logical stabilizer information. We formulate sufficient criteria to find schemes for which a single successful BM on the physical level will always allow to obtain the full logical information by suitably adapting the subsequent physical measurements. Our approach based on stabilizer group theory is generally applicable to any stabilizer code, which we demonstrate for quantum parity, five-qubit, standard and rotated planar surface, tree, and seven-qubit Steane codes. Our schemes attain the general upper bound for all these codes, while this bound had previously only been reached for the quantum parity code.
This paper investigates a critical challenge at the intersection of quantum optics and fundamental physics: the strict physical constraints governing Bell state measurement (BSM) efficiency within linear-optical frameworks. Without the integration of ancilla photons, conventional setups relying on standard linear-optical components are bounded by a fundamental efficiency threshold that cannot be exceeded. We conduct an analytical analysis of the physical roots of this barrier using Hong-Ou-Mandel photon interference phenomena and multi-photon state space geometry. Furthermore, this study models advanced physical techniques to break past this threshold, focusing on non-linear optical mediums—such as parametric down-conversion setups—and polarization-entangled ancilla flying qubits. These analytical insights deliver foundational physical parameters necessary for engineering fault-tolerant optical quantum computing networks and scalable quantum internet repeaters.
Fault tolerance is essential for scalable quantum technologies and is enabled by quantum error-correction codes. Bell-state measurements (BSMs) are a fundamental building block for modern quantum technologies such as measurement-based quantum computation and fusion-based quantum computation, as well as quantum networks. Therefore, performing BSMs on error-corrected qubits is a necessary step for achieving fault tolerance in these applications. In this work, we realise a logical BSM using linear optics, based on a two-qubit repetition code, an instance of a quantum parity code that allows detection of bit-flip errors, and experimentally achieve a mean success probability of (70.8 +/- 0.4)%. While standard linear-optical BSMs are fundamentally limited to a maximum success probability of 50%, this increased success probability enables higher secure key rates in quantum communication and facilitates the generation of large graph states for quantum computation. Since fault-tolerant schemes require error-correction codes regardless, this improvement comes at no additional resource overhead. Our results demonstrate that error-correction codes can be used to surpass the linear-optics limit of BSMs, which is an important step towards practical, fault-tolerant, and scalable photonic quantum technologies.
Bell state measurements (BSMs) play a significant role in quantum information and quantum computing, in particular in fusion-based quantum computing (FBQC). The FBQC model is a framework for universal quantum computing, provided that we are able to perform entangling measurements, called fusions, on qubits within small entangled resource states. Here, we analyze the usage of different linear-optical BSM circuits as fusions in the FBQC schemes and numerically evaluate hardware requirements for fault-tolerance in this framework. We examine and compare the performance of several BSM circuits with varying additional resources and estimate the requirements on losses for every component of the linear-optical realization of fusions under which errors in fusion networks caused by these losses can be corrected. Our results show that fault-tolerant quantum computing in the FBQC model is possible with currently achievable levels of optical losses in an integrated photonic implementation, provided that we can create and detect single photons of the resource states with a total marginal efficiency higher than 0.973.
Several emerging quantum technologies, including quantum networks, and modular and fusion-based quantum computing, rely crucially on the ability to perform photonic Bell state measurements. Therefore, photon losses and the 50% success probablity upper bound of Bell state measurements pose a critical limitation to photonic quantum technologies. Here, we develop protocols that overcome these two key challenges through logical encoding of photonic qubits. Our approach uses a tree graph state logical encoding, which can be produced deterministically with a few quantum emitters, and achieves near-deterministic logical photonic Bell state measurements while also protecting against errors including photon losses, with a record loss-tolerance threshold.