Analytical fidelity calculations for photonic linear cluster state generation

By precisely timed optical excitation of their spin, optical emitters such as semiconductor quantum dots or atoms can be harnessed as sources of linear photonic cluster states. This significantly reduces the required resource overhead to reach fault-tolerant optical quantum computing. Here, we develop an algorithm that analytically tracks the global density matrix through the process of the protocol for generating linear-cluster states by Lindner and Rudolph. From this we derive a model to calculate the entangling gate fidelity and the state fidelity of the generated linear optical cluster states. Our model factors in various sources of error, such as spin decoherence and the finite excited state lifetime. Additionally, we highlight the presence of partial reinitialization of spin coherence with each photon emission, eliminating the hard limitation of coherence time. Our framework provides valuable insight into the cost-to-improvement trade-offs for device design parameters as well as the identification of optimal working points. For a combined state-of-the-art quantum dot with a spin coherence time of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mmultiscripts><mml:mi>T</mml:mi> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> <mml:mrow><mml:mrow></mml:mrow> <mml:mo>∗</mml:mo></mml:mrow> </mml:mmultiscripts> <mml:mo>=</mml:mo> <mml:mn>535</mml:mn></mml:mrow> </mml:math> ns and an excited state lifetime of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>23</mml:mn></mml:mrow> </mml:math> ps, we show that a near-unity entangling gate fidelity as well as near-unity state fidelity for 3-photon and 7-photon linear cluster states can be reached.