This note reviews the model of computation generated by photonic circuits, comprising edges that are traversed by photons in a single time-bin and vertices given by idealised lossless beam splitters and phase shifters. The circuit model is abstracted as a representation of the symmetric monoidal category of unitary matrices on the bosonic Fock space of multimode photons. A diagrammatic language, designed to aid with the understanding and development of photonic algorithms, is presented that encapsulates the category properties of this representation. As demonstrations of the formalism, the boson sampling scheme and the protocol of Knill, Laflamme and Milburn are developed on the single-rail photonic computer, and a parity-based model for the qudit is investigated.
This note reviews the model of computation generated by photonic circuits, comprising edges that are traversed by photons in a single time-bin and vertices given by idealised lossless beam splitters and phase shifters. The circuit model is abstracted as a representation of the symmetric monoidal category of unitary matrices on the bosonic Fock space of multimode photons. A diagrammatic language, designed to aid with the understanding and development of photonic algorithms, is presented that encapsulates the category properties of this representation. As demonstrations of the formalism, the boson sampling scheme and the protocol of Knill, Laflamme and Milburn are developed on the single-rail photonic computer, and a parity-based model for the qudit is investigated.
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.
This thesis covers several topics in optical quantum computation. A quantum computer is a computational device which is able to manipulate information by performing unitary operations on some physical system whose state can be described as a vector (or mixture of vectors) in a Hilbert space. The basic unit of information, called the qubit, is considered to be a system with two orthogonal states, which are assigned logical values of 0 and 1. Photons make excellent candidates to serve as qubits. They have little interactions with the environment. Many operations can be performed using very simple linear optical devices such as beam splitters and phase shifters. Photons can easily be processed through circuit-like networks. Operations can be performed in very short times. Photons are ideally suited for the long-distance communication of quantum information. The great difficulty in constructing an optical quantum computer is that photons naturally interact weakly with one another. This thesis first gives a brief review of two early approaches to optical quantum computation. It will describe how any discrete unitary operation can be performed using a single photon and a network of beam splitters, and how the Kerr effect can be used to construct a two photon logic gate. Second, this work provides a thorough introduction to the linear optical quantum computer developed by Knill, Laflamme, and Milburn. It then presents this author's results on the reliability of this scheme when implemented using imperfect photon detectors. This author finds that quantum computers of this sort cannot be built using current technology. Third, this dissertation describes a method for constructing a linear optical quantum computer using nearly orthogonal coherent states of light as the qubits. It shows how a universal set of logic operations can be performed, including calculations of the fidelity with which these operations may be accomplished. It discusses methods for reducing and correcting errors and recovering from failed operations. Lastly it describes an analysis of the long distance transmission of the coherent state qubits and shows how transmission errors can be corrected.
Boson sampling arising from multiphoton interference in linear-optical networks is a prominent non-universal model for quantum computation. Here, by encoding the multi-qubit state to bosonic Fock state, we present a scalable quantum-circuit framework for simulating boson sampling on a universal quantum computing platform. Beginning with balanced beam-splitter transformations on the single- and two-photon sectors, we derive equivalent quantum-circuit implementations and unify them within a common Hilbert-space representation using an ancilla-assisted encoding. This construction is then generalized to arbitrary interferometers by replacing each optical beam splitter with a repeating quantum-circuit unit that selectively acts only within the relevant local interference subspace, requiring N+1 qubits for a two-photon N-mode interferometer and a linear-overhead subspace-identification procedure. Using this framework, gate-based quantum circuit for a four-mode boson-sampling circuit is developed and experimentally implemented on a four-qubit gate-based photonic qubit system. The qubit framework for emulating boson sampling of n−photons in m−mode will be useful to solve a broad class of sampling complexity problem on a gate-based quantum computers.
This work presents a developed Wolfram Mathematica software tool based on the Quantum extension package. This tool is able to simulate the output state for any linear optical circuit based on Beam Splitters (BSs) and Phase Sifters (PSs) by using as input, quantum states of light. The input of the software is described via the use of photon number states capable of emulating Fock states or coherent states, whereas the output provides the probability of detecting a photon at each output port. This photonic circuit may include any number of linear optical elements, including beam splitters with variable splitting ratios, as well as phase shifters. By using the unitary transformation of any photonic circuit (e.g., photonic chip), the output states can be computed, and detection probabilities of the chip can be obtained. The unitary transformation of the Controlled NOT (CNOT) quantum gate through a linear optical layout is presented, validating the functionality of the proposed software tool. In conclusion, the proposed software aims to enable testing of simple photonic circuits in the quantum regime.
ABSTRACT Unitary operations lie at the heart of quantum computing, enabling the manipulation of quantum states in a reversible and coherent manner. In photonic integrated circuits, these operations are realized using photonic waveguide architectures composed of passive beam splitters and reconfigurable phase shifters. This review provides an overview of photonic circuit architectures designed to implement unitary transformations. We examine various design strategies, material platforms, and the current state‐of‐the‐art in realizing high‐fidelity photonic unitaries. In addition, we explore the theoretical underpinnings of photon transport in such circuits, shedding light on the quantum behavior of light in unitary photonic systems. Finally, we introduce a simulation framework for modeling and analyzing these photonic networks, offering a practical tool for research and development in this rapidly growing field.
Graphical languages are powerful and useful to represent, rewrite and simplify different kinds of processes. In particular, they have been widely used for quantum processes, improving the state of the art for compilation, simulation and verification. In this work, we focus on one of the main carrier of quantum information and computation: linear optical circuits. We introduce the LO_fi-calculus, the first graphical language to reason on the infinite-dimensional photonic space with circuits only composed of the four core elements of linear optics: the phase shifter, the beam splitter, and auxiliary sources and detectors with bounded photon number. First, we study the subfragment of circuits composed of phase shifters and beam splitters, for which we provide the first minimal equational theory. Next, we introduce a rewriting procedure on those LO_fi-circuits that converge to normal forms. We prove those forms to be unique, establishing both a novel and unique representation of linear optical processes. Finally, we complement the language with an equational theory that we prove to be complete: two LO_fi-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LO_fi-calculus.