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Journal of the Optical Society of America B

Effect of loss on linear optical quantum logic gates

James E. Davis, Durdu Ö. Güney

Published Jul 12, 2021• 5 citations
Publisher

Abstract

Linear optical quantum gates have been proposed as a possible implementation for quantum computers. Most experimental linear optical quantum gates are constructed with free-space optical components with negligible loss. In this work, we analyze symmetric and asymmetric partially polarizing lossy beam splitters. Using the generalized beam splitter equations, we study the effects of loss on two linear optical quantum gates: the first is a commonly used CNOT gate, and the second is a W state expansion gate. Envisioning inherent loss in plasmonics and metamaterials as a new degree of freedom and those materials systems as a route for miniaturization, we reconsider the requirements of the lossy CNOT gate and show it is possible to simplify the three-beam-splitter design to a single beam splitter without sacrificing success probability.

DOI
10.1364/josab.430603

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Linear optical quantum gates have been proposed as a possible implementation for quantum computers. Most experimental linear optical quantum gates are constructed with free-space optical components with negligible loss. In this work, we analyze symmetric and asymmetric partially polarizing lossy beam splitters. Using the generalized beam splitter equations, we study the effects of loss on two linear optical quantum gates: the first is a commonly used CNOT gate, and the second is a W state expansion gate. Envisioning inherent loss in plasmonics and metamaterials as a new degree of freedom and those materials systems as a route for miniaturization, we reconsider the requirements of the lossy CNOT gate and show it is possible to simplify the three-beam-splitter design to a single beam splitter without sacrificing success probability.
It has previously been shown that probabilistic quantum logic operations may be performed using linear optical elements, additional photons (ancilla), and post-selection based on the output of single-photon detectors. Here we describe the operation of several quantum logic operations of an elementary nature, including a quantum parity check and a quantum encoder, and we show how they may be combined to implement a controlled-NOT (CNOT) gate. All of these gates may be constructed using polarizing beam splitters that completely transmit one state of polarization and totally reflect the orthogonal state of polarization, which allows a simple explanation of each operation. We also describe a polarizing beam splitter implementation of a CNOT gate that is closely analogous to the quantum teleportation technique previously suggested by Gottesman and Chuang [Nature 402, 390 (1999)]. Finally, our approach has the interesting feature that it makes practical use of a quantum-eraser technique.
Photonic quantum gates are fundamental building blocks for a wide range of optical quantum information processing tasks. We propose an efficient linear-optical scheme for implementing a post-selected three-qubit fan-out gate (also known as a controlled-NOT-NOT gate) using only two ancillary single photons and linear optical elements. The scheme can be generalized to an nnn-qubit fan-out gate, requiring (n−1)(n-1)(n−1) ancillary single photons and (3n−3)(3n-3)(3n−3) polarizing beam splitters (PBSs), with a success probability of (14)n−1\left(\frac{1}{4}\right)^{n-1}(41​)n−1. Compared to the standard gate decomposition approach, which requires (2n−2)(2n-2)(2n−2) ancillary single photons, (5n−5)(5n-5)(5n−5) PBSs, and yields a success probability of (18)n−1\left(\frac{1}{8}\right)^{n-1}(81​)n−1. Our scheme significantly reduces the resource overhead and improves the success probability. We further evaluate the gate performance and demonstrate improved robustness compared to gate decomposition-based methods.
Linear optical quantum computing in the dual-rail representation offers the practical advantage of room-temperature operation, with quantum gates implemented through beam splitters and phase shifters. Despite its appeal, this approach is fundamentally constrained by the nondeterministic nature of entangling gates, which limits scalability within the circuit-based paradigm. Measurement-based quantum computation (MBQC) addresses this issue by employing pre-generated cluster states and single-qubit projective measurements, but both frameworks remain highly susceptible to photon loss, the dominant error in photonic systems. In this work, we investigate the effects of photon loss on single-qubit gates, heralded two-qubit entangling gates, the two-qubit Grover algorithm, and the preparation of a three-qubit cluster state. By employing simplified photon-loss models, we quantify both the success probability and fidelity degradation of the resulting states. These results provide practical insight into the limitations of current photonic architectures and highlight pathways toward efficient error mitigation in near-term quantum technologies.
The first section of the work investigates light modes as a means of implementing optical qubits and qudits. The modes considered are polarization mode, path mode, transverse spatial mode, frequency mode, temporal bin-mode and temporal mode. Subsequently, mathematical model of linear optical elements like beam splitters (BS) and phase shifters are deduced and their capability of representing any single qubit optical gate is exposed. Finally Knill, Laflamme and Milburn (KLM) method of using linear optical elements to promote nonlinear operations based on nonlinear (nondeterministic) sign-flip gate (NS) is explained, and designs of two qubits conditional sign flip gate (c-z1/16) and CNOT gate, both based on KLM method, are demonstrated. So, universal set of quantum gates based on linear optics is possible.
One of the main problems that optical quantum computing has to overcome is the efficient construction of two-photon gates. Theoretically these gates can be realized using Kerr-nonlinearities, but the techniques involved are experimentally very difficult. We therefore employ linear optics with projective measurements to generate these non-linearities. The downside is that the measurement-induced nonlinearities achieved with linear optics are less versatile and the success rate can be quite low. This project is mainly the result of a literature study but also a theoretical work on the physics behind quantum optical multiports which is essential for realizing two-photon gates. By applying different postcorrection techniques we increase the probability of success in a modifed non-linear sign shift gate which is foundational for the two photon controlled-NOT gate. We prove that it's not possible to correct the states by only using a single beam splitter. We show that it might be possible to increase the probability of success using a more complex setup with at least two error-correcting beam splitters.
We define an Asymmetric Partially Polarizing Beam Splitter (APPBS) to be a linear optical component having different reflectivity (transmittance) coefficients, on the upper and the lower arms, for horizontally and vertically Polarized incident photons. Our CNOT model is composed by two APPBSs, one Half Wave Plate (HWP), two Polarizing Beam Splitters (PBSs), a Beam Splitter (BS) and a -phase rotator for specific wavelength. Control qubit operates with dual rail encoding while target qubit is based on polarization encoding. To perform CNOT operation in 4/27 of the cases, input and target incoming photons are injected with different wavelengths.