Optical photons present many interesting aspects in the context of the research for suitable physical representations of quantum bits. Photons are chargeless particles, and do not interact strongly with each other, or even with most matter. They can be guided along long distances with low loss in optical fibers, delayed efficiently using phase shifters, and combined easily using beamsplitters. Photons exhibit typical quantum phenomena, such as the interference produced in two-slit experiments. Furthermore, in principle, photons can be made interact with each other if carefully handled. There are many practical problems with this approach; nevertheless it presents, beyond what has already been noted, an appealing conceptual simplicity.The purpose of this work is to present - at least, in its fundamental principles - a set of optical photon quantum gates that is universal for quantum computation. To do so, we begin by gathering the necessary mathematical and physical tools. We discuss canonical quantization for the electromagnetic field, and we formalize in an abstract context the characteristic properties of every realization of the Fock canonical commutation relationships (CCR). We analize the class of the spaces where it is possible to give such a realization, and that naturally lend themselves to host the physical model of our interest. Once acquired these bases, we turn ourselves to a particular optic system, the Mach-Zehnder interferometer, and we study the general properties of this system and of its essential constituents. At this point we choose a particular encoding of quantum bit and we look at the optical devices just discussed as elements of a quantum circuit. We then discuss the fundamental ideas that may guide the implementation of some important quantum gates: Hadamard gate, $\pi/8$ gate and controlled-NOT. We conclude by showing that these three quantum gates form a universal set of gates for quantum computation.
Optical photons present many interesting aspects in the context of the research for suitable physical representations of quantum bits. Photons are chargeless particles, and do not interact strongly with each other, or even with most matter. They can be guided along long distances with low loss in optical fibers, delayed efficiently using phase shifters, and combined easily using beamsplitters. Photons exhibit typical quantum phenomena, such as the interference produced in two-slit experiments. Furthermore, in principle, photons can be made interact with each other if carefully handled. There are many practical problems with this approach; nevertheless it presents, beyond what has already been noted, an appealing conceptual simplicity.The purpose of this work is to present - at least, in its fundamental principles - a set of optical photon quantum gates that is universal for quantum computation. To do so, we begin by gathering the necessary mathematical and physical tools. We discuss canonical quantization for the electromagnetic field, and we formalize in an abstract context the characteristic properties of every realization of the Fock canonical commutation relationships (CCR). We analize the class of the spaces where it is possible to give such a realization, and that naturally lend themselves to host the physical model of our interest. Once acquired these bases, we turn ourselves to a particular optic system, the Mach-Zehnder interferometer, and we study the general properties of this system and of its essential constituents. At this point we choose a particular encoding of quantum bit and we look at the optical devices just discussed as elements of a quantum circuit. We then discuss the fundamental ideas that may guide the implementation of some important quantum gates: Hadamard gate, π/8 gate and controlled-NOT. We conclude by showing that these three quantum gates form a universal set of gates for quantum computation.
Optical photons present many interesting aspects in the context of the research for suitable physical representations of quantum bits. Photons are chargeless particles, and do not interact strongly with each other, or even with most matter. They can be guided along long distances with low loss in optical fibers, delayed efficiently using phase shifters, and combined easily using beamsplitters. Photons exhibit typical quantum phenomena, such as the interference produced in two-slit experiments. Furthermore, in principle, photons can be made interact with each other if carefully handled. There are many practical problems with this approach; nevertheless it presents, beyond what has already been noted, an appealing conceptual simplicity.The purpose of this work is to present - at least, in its fundamental principles - a set of optical photon quantum gates that is universal for quantum computation. To do so, we begin by gathering the necessary mathematical and physical tools. We discuss canonical quantization for the electromagnetic field, and we formalize in an abstract context the characteristic properties of every realization of the Fock canonical commutation relationships (CCR). We analize the class of the spaces where it is possible to give such a realization, and that naturally lend themselves to host the physical model of our interest. Once acquired these bases, we turn ourselves to a particular optic system, the Mach-Zehnder interferometer, and we study the general properties of this system and of its essential constituents. At this point we choose a particular encoding of quantum bit and we look at the optical devices just discussed as elements of a quantum circuit. We then discuss the fundamental ideas that may guide the implementation of some important quantum gates: Hadamard gate, π/8 gate and controlled-NOT. We conclude by showing that these three quantum gates form a universal set of gates for quantum computation.
This Spotlight aims to provide a general introduction to linear and nonlinear optical components that are frequently used to implement the protocols for quantum computation and communication. The role of each optical element is described briefly, and it is shown that these optical elements can be combined to constitute quantum circuits for performing various quantum computing and communication tasks. Present challenges and future scopes are also mentioned briefly. Specifically, after introducing the qubit, it is stated that a quantum gate is a device that transforms a quantum state into another quantum state. Thus, most of the optical components (e.g., a beamsplitter, a wave plate, a mirror, a polarizing beamsplitter) can be viewed as quantum gates because each of these optical elements can transform a quantum state into another quantum state. This fact is the basis for constructing useful quantum circuits using optical elements. Finally, a few proposals for optical realizations of schemes for quantum communication and computation are described.
In classical computation and communication, we use classical gates and bits. Similarly, in quantum computation and communication, we use their quantum analogs, i.e., quantum gates and qubit(s) [quantum bit(s)]. We have already mentioned that a bit can be realized in various ways. In the same way, a qubit can also be realized in various ways, and accordingly, corresponding quantum gates that would operate on qubits can also be realized in various ways. Here, we focus on a specific implementation: optical implementation, where qubits are realized using photons and quantum gates, and circuits are realized using various optical components. Specific merits of this realization lie in the following facts: (i) in quantum communication we need to transmit qubits from one location to the other, and transmission of photons is easier than any other realization of qubits; (ii) the effect of the environment (e.g., external electric and magnetic fields) on photons is less compared to any other implementations of qubits; and (iii) integrated optics are already a well-developed field. Keeping these points in mind, this Spotlight describes basic optical elements and how to use those elements to implement various schemes of quantum computation and communication. We will mostly describe realizations using bulk optics. However, in most cases, it is not difficult to transform these bulk-optics-based realizations into integrated optics. To begin with, in Section 1.1, we formally define a qubit and describe two alternative ways of realizing a qubit using a photon.
Quantum information science aims to harness uniquely quantum mechanical properties to enhance measurement and information technologies, and to explore fundamental aspects of quantum physics. Of the various approaches to quantum computing [1], photons are particularly appealing for their low-noise properties and ease of manipulation at the single qubit level [2,3]. Encoding quantum information in photons is also an appealing approach to quantum communication, metrology (eg. [4]), measurement (eg. [5]) and other quantum technologies [6]. However, the implementation of optical quantum circuits with bulk optics has reached practical limits. We have developed an integrated waveguide approach to photonic quantum circuits for high performance, miniaturization and scalability [7]. Here we report high-fidelity silica-on-silicon integrated optical realizations of key quantum photonic circuits, including two-photon quantum interference and a controlled-NOT logic gate [8]. We have demonstrated controlled manipulation of up to four photons on-chip, including highfidelity single qubit operations, using a lithographically patterned resistive phase shifter [9]. We have used this architecture to implement a small-scale compiled version of Shor's quantum factoring algorithm [10], demonstrated heralded generation of tunable four photon entangled states from a six photon input [11], a reconfigurable two-qubit circuit [12], and combined waveguide photonic circuits with superconducting single photon detectors [13]. We describe complex quantum interference behavior in multi-mode interference devices with up to eight inputs and outputs [14], and quantum walks of correlated particles in arrays of coupled waveguides [15]. Finally, we give an overview of our recent work on fundamental aspects of quantum measurement [16,17] and diamond [18,19] and nonlinear [20,21] photon sources.
At the beginning of the last century, the theory of quantum optics arose and led to a revolution in physics, since it allowed the interpretation of many unknown phenomena and the development of numerous powerful, cutting edge engineering applications, \nsuch as high precision laser technology. The analysis and verification of such applications and systems, however, are very complicated. Moreover, traditional analysis tools, e.g., simulation, numerical methods, computer algebra systems, and paper-and-pencil approaches are not well suited for quantum systems. In the last decade, a new emerging verification technique, called formal methods, became common among engineering domains, and has proven to be effective as an analysis tool. Formal methods \nconsist in the development of mathematical models of the system subject for analysis, and deriving computer-aided mathematical proofs. In this thesis, we propose a framework for the analysis of quantum optics based on formal methods, in particular theorem proving. The framework aims at implementing necessary quantum mechanics and optics concepts and theorems that facilitate the modelling of quantum optical devices and circuits, and then reason about them formally. To this end, the framework \nconsists of three major libraries: 1) Mathematical foundations, which mainly contain the theory of complex-valued-function linear spaces, 2) Quantum mechanics, which develops the general rules of quantum physics, and 3) Quantum Optics, which \nspecializes these rules for light beams and implements all related concepts, e.g., light coherence which is typically emitted by laser sources. On top of these theoretical foundations, we build a library of formal models of a number of optical devices commonly \nused in quantum circuits, including, beam splitters, light displacers, and light phase shifters. Using the proposed framework, we have been able to formally verify common quantum optical computing circuits, namely the Flip gate, CNOT gate, and Mach-Zehnder interferometer.
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.
Under the label of optical quantum computing, there are a variety of protocols and experiments that use the quantum properties of light to achieve a computational advantage over classical computing machines. In this review, we describe some of the main implementations, which differ in the type of encoding and in how the computation is performed, whether using gates or cluster states. For each protocol, we describe advantages and challenges with an overview of the experimental results obtained, summarized in tables. Because of the great relevance achieved in this field, there is a section dedicated to non-universal quantum computation with photons, where boson sampling, variational quantum eigensolvers, and quantum machine learning applications are described. The aim is to give the reader the broadest overview of the applications where photons and their quantum properties play a key role in computation.