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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Non-deterministic quantum CNOT gate with double encoding

Amor Gueddana, Moez Attia, Rihab Chatta

Published Sep 26, 2013• 5 citations
Publisher

Abstract

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.

DOI
10.1117/12.2023041

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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.
In this work, we study the error sources standing behind the non-perfect linear optical quantum components composing a non-deterministic quantum CNOT gate model, which performs the CNOT function with a success probability of 4/27 and uses a double encoding technique to represent photonic qubits at the control and the target. We generalize this model to an abstract probabilistic CNOT version and determine the realizability limits depending on a realistic range of the errors. Finally, we discuss physical constraints allowing the implementation of the Asymmetric Partially Polarizing Beam Splitter (APPBS), which is at the heart of correctly realizing the CNOT function.
A new method based on quantum encoders is developed for implementing quantum computational gates, especially the CNOT gate. Photonic qubits are used where ∣0> and ∣1> represent horizontal ∣H> and vertical ∣V> polarized photons, respectively. By the use of polarizing beam splitters (PBSs) and entangled two-photon states in a certain experimental scheme, each input qubit state is copied, with a certain probability of success, into four equivalent input qubit states. The present method concentrates on the special case where the initial two-qubit state is separable. The CNOT gate (multiplied by 2) is decomposed into a summation of four unitary matrices each of dimension 4×4 and where each of these unitary matrices operates on a corresponding copy of the input two-qubit state. By adding the amplitudes of the four two-qubit states, which is obtained in a decoding interference experiment, the CNOT gate is implemented.
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.
Implementation of quantum logic gates with linear optical elements plays a prominent role in quantum computing due to the relatively easier manipulation and realization. We present efficient schemes to implement controlled-NOT (CNOT) gate and controlled-swap (Fredkin) gate by solely using linear optics. We encode the control qubits and target qudits in photonic polarization (two-level) and spatial degrees of freedom (ddd-level), respectively. Based on the hybrid encoding, CNOT and Fredkin gates are constructed in a deterministic way without any borrowed ancillary photons or measurement-induced nonlinearities. Remarkably, the number of linear optics required to implement a CNOT gate has been reduced to one polarization beam splitter (PBS), while only ddd PBSs are necessary to implement a generalized Fredkin gate. The optical depths of all schemes are reduced to one and dimension-independent. Besides, the fidelity of our three-qubit Fredkin gate is higher than 99.7\% under realistic conditions, which is higher than the previous schemes.
We propose a CNOT gate for quantum computation. The CNOT operation is based on existence of triactive molecules, which in one direction have dipole moment and cause rotation of the polarization plane of linearly polarized light and in perpendicular direction have a magnetic moment. The incoming linearly polarized laser beam is divided into two beams by beam splitter. In one beam a control state is prepared and the other beam is a target. The interaction of polarized states of both beams in a solution containing triactive molecules can be described as interaction of two qubits in CNOT.
The Controlled-NOT (CNOT) quantum gate associated with single qubit gates is universal for building all quantum circuits. Although CNOTs based on trapped ions and superconducting qubits have become experimentally feasible, the deterministically functioning quantum CNOT gate with photonic qubits will lead to practical realization of quantum computers in a deterministic manner, and will therefore provide an exponential increase in processing power with lowest decoherence time. Based on a previously proposed CNOT model utilizing a quantum cloner (with a fidelity of 78 %), we present in this work our design of the Compact CNOT gate providing a fidelity of 90.31 %, and give physical realization constraints for building CNOT based decomposition of the Toffoli gate and more generalized photonic CnNOT gates. The photonic qubits are encoded by circularly polarized single photons and the design of the gate is based on copying the polarization degree of freedom by quantum cloners. Our model is based on spin of electron in a quantum dot embedded in a double sided optical microcavity which behaves like a beam splitter. Circularly polarizing beam splitters, beam splitters, delay lines, photonic circulators, single qubit operations and other photonic devices are used for the design and decomposition. We show that the fidelity of the decomposed CnNOT gates is highly dependent on the errors of the non-perfect photonic devices used and more specifically to the theoretical optimal limit of 5/6 of the quantum cloners required for each CNOT in the decomposition.