Optical quantum devices are promoting the development not only of communication but also of quantum computation using optical quantum gates. Here, we will explain how to replace circuits consisting of gates such as NOT, AND, etc. used in classical logic calculations with quantum calculation gates. These classical gates can be expressed as NOT and CNOT of quantum computing gates. We show how to express these quantum gates as optical devices using the behavior of the polarization of discrete photons. Next, we describe the backtracking method of quantum circuits utilizing the reversibility of quantum gates. Since classical gates are not unitary, we use probabilistic methods that rely on measurements. In a typical classical gate, there are multiple inputs corresponding to a certain output, so backtracking requires tracing multiple states. The quantum bit representation, which can represent superposition states, is suitable for this purpose. As an application example, we explain how to backtrack integer multipliers and perform factorization. In configuration using quantum gates, one measurement quantum bit is required for each element, resulting in a large number of state bits. However, by reusing bits that are no longer needed, it was possible to construct a circuit with a number of state bits proportional to the number of bits for the integer multiplicand. We simulated an integer multiplier circuit by combining classical gates, and backtrack the circuit to derive two factors from a composite number represented by quantum bits. We performed simulations on a PC and succeeded in factorizing a 100-bit composite number.
Optical quantum devices are promoting the development not only of communication but also of quantum computation using optical quantum gates. Here, we will explain how to replace circuits consisting of gates such as NOT, AND, etc. used in classical logic calculations with quantum calculation gates. These classical gates can be expressed as NOT and CNOT of quantum computing gates. We show how to express these quantum gates as optical devices using the behavior of the polarization of discrete photons. Next, we describe the backtracking method of quantum circuits utilizing the reversibility of quantum gates. Since classical gates are not unitary, we use probabilistic methods that rely on measurements. In a typical classical gate, there are multiple inputs corresponding to a certain output, so backtracking requires tracing multiple states. The quantum bit representation, which can represent superposition states, is suitable for this purpose. As an application example, we explain how to backtrack integer multipliers and perform factorization. In configuration using quantum gates, one measurement quantum bit is required for each element, resulting in a large number of state bits. However, by reusing bits that are no longer needed, it was possible to construct a circuit with a number of state bits proportional to the number of bits for the integer multiplicand. We simulated an integer multiplier circuit by combining classical gates, and backtrack the circuit to derive two factors from a composite number represented by quantum bits. We performed simulations on a PC and succeeded in factorizing a 100-bit composite number.
Optical quantum devices are promoting the development not only of communication but also of quantum computation using optical quantum gates. We will explain how to replace circuits consisting of gates such as NOT and AND used in classical logic calculations with quantum calculation gates. These classical gates can be expressed as NOT and CNOT of quantum computing gates. We show how to express these quantum gates as optical devices using the behavior of the polarization of discrete photons. Next, we describe the backtracking method of quantum circuits utilizing the reversibility of quantum gates. As classical gates are not unitary, we use probabilistic methods that rely on measurements. In a typical classical gate, there are multiple inputs corresponding to a certain output, so backtracking requires tracing multiple states. The quantum bit representation, which can represent superposition states, is suitable for this purpose. As an application example, we explain how to backtrack integer multipliers and perform factorization. In a configuration using quantum gates, one measurement quantum bit is required for each element, resulting in a large number of state bits. However, by reusing bits that are no longer needed, it was possible to construct a circuit with a number of state bits proportional to the number of bits for the integer multiplicand. We simulated an integer multiplier circuit by combining classical gates and backtracked the circuit to derive two factors from a composite number represented by quantum bits. By performing a simulation on a PC, we succeeded in factorizing a 100-bit composite number, subject to the condition that it is a product of two prime numbers close to less than 2m1−1 and 2m2−1, where m1 and m2 are integers. We compare our method with the calculation using satisfiability (SAT) for the factoring problem.
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.
A quantum computer is a device for computation that makes direct use of distinctively quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. In a classical (or conventional) computer, the amount of data is measured by bits, in a quantum computer; it is measured by quantum of bits (qubits). This work describes, and implements the universal quantum logic gates, the terminology is introduced with two well known quantum gates, the quantum NOT, and the quantum XOR gates. The NOT and XOR gates have already been described in classical reversible logic. This work gives the implementation of the quantum circuits, such as the Quantum Half-Adder Circuits, which consists of quantum control control not gate (CCNot) and quantum control not gate (CNot). Also, it describes and implements the Quantum Full-Adder Circuits, which consists of two Quantum Half-Adder Circuits and one control not gate (CNot). Depending on the quantum circuits, one can implement the quantum basic arithmetic operations such as (addition, subtraction, multiplication and division). After each implementation, the computational complexity of each step is calculated. Key words:
Quantum computing is an emerging technology that has the potential to change the perspectives and applications of computing in general. A wide range of applications are enabled: from faster algorithmic solutions of classically still difficult problems to theoretically more secure communication protocols. A quantum computer uses the quantum mechanical effects of particles or particle-like systems, and a major similarity between quantum and classical computers consists of both being abstracted as information processing machines. Whereas a classical computer operates on classical digital information, the quantum computer processes quantum information, which shares similarities with analog signals. One of the central differences between the two types of information is that classical information is more fault-tolerant when compared to its quantum counterpart. Faults are the result of the quantum systems being interfered by external noise, but during the last decades quantum error correction codes (QECC) were proposed as methods to reduce the effect of noise. Reliable quantum circuits are the result of designing circuits that operate directly on encoded quantum information, but the circuit’s reliability is also increased by supplemental redundancies, such as sub-circuit repetitions. Reliable quantum circuits have not been widely used, and one of the major obstacles is their vast associated resource overhead, but recent quantum computing architectures show promising scalabilities. Consequently the number of particles used for computing can be more easily increased, and that the classical control hardware (inherent for quantum computation) is also more reliable. Reliable quantum circuits haev been investigated for almost as long as general quantum computing, but their limited adoption (until recently) has not generated enough interest into their systematic design. The continuously increasing practical relevance of reliability motivates the present thesis to investigate some of the first answers to questions related to the background and the methods forming a reliable quantum circuit design stack. The specifics of quantum circuits are analysed from two perspectives: their probabilistic behaviour and their topological properties when a particular class of QECCs are used. The quantum phenomena, such as entanglement and superposition, are the computational resources used for designing quantum circuits. The discrete nature of classical information is missing for quantum information. An arbitrary quantum system can be in an infinite number of states, which are linear combinations of an exponential number of basis states. Any nontrivial linear combination of more than one basis states is called a state superposition. The effect of superpositions becomes evident when the state of the system is inferred (measured), as measurements are probabilistic with respect to their output: a nontrivial state superposition will collapse to one of the component basis states, and the measurement result is known exactly only after the measurement. A quantum system is, in general, composed from identical subsystems, meaning that a quantum computer (the complete system) operates on multiple similar particles (subsystems). Entanglement expresses the impossibility of separating the state of the subsystems from the state of the complete system: the nontrivial interactions between the subsystems result into a single indivisible state. Entanglement is an additional source of probabilistic behaviour: by measuring the state of a subsystem, the states of the unmeasured subsystems will probabilistically collapse to states from a well defined set of possible states. Superposition and entanglement are the building blocks of quantum information teleportation protocols, which in turn are used in state-of-the-art fault-tolerant quantum computing architectures. Information teleportation implies that the state of a subsystem is moved to a second subsystem without copying any information during the process. The probabilistic approach towards the design of quantum circuits is initiated by the extension of classical test and diagnosis methods. Quantum circuits are modelled similarly to classical circuits by defining gate-lists, and missing quantum gates are modelled by the single missing gate fault. The probabilistic approaches towards quantum circuits are facilitated by comparing these to stochastic circuits, which are a particular type of classical digital circuits. Stochastic circuits can be considered an emulation of analogue computing using digital components. A first proposed design method, based on the direct comparison, is the simulation of quantum circuits using stochastic circuits by mapping each quantum gate to a stochastic computing sub-circuit. The resulting stochastic circuit is compiled and simulated on FPGAs. The obtained results are encouraging and illustrate the capabilities of the proposed simulation technique. However, the exponential number of possible quantum basis states was translated into an exponential number of stochastic computing elements. A second contribution of the thesis is the proposal of test and diagnosis methods for both stochastic and quantum circuits. Existing verification (tomographic) methods of quantum circuits were targeting the reconstruction of the gate-lists. The repeated execution of the quantum circuit was followed by different but specific measurement at the circuit outputs. The similarities between stochastic and quantum circuits motivated the proposal of test and diagnosis methods that use a restricted set of measurement types, which minimise the number of circuit executions. The obtained simulation results show that the proposed validation methods improve the feasibility of quantum circuit tomography for small and medium size circuits. A third contribution of the thesis is the algorithmic formalisation of a problem encountered in teleportation-based quantum computing architectures. The teleportation results are probabilistic and require corrections represented as quantum gates from a particular set. However, there are known commutation properties of these gates with the gates used in the circuit. The corrections are not applied as dynamic gate insertions (during the circuit’s execution) into the gate-lists, but their effect is tracked through the circuit, and the corrections are applied only at circuit outputs. The simulation results show that the algorithmic solution is applicable for very large quantum circuits. Topological quantum computing (TQC) is based on a class of fault-tolerant quantum circuits that use the surface code as the underlying QECC. Quantum information is encoded in lattice-like structures and error protection is enabled by the topological properties of the lattice. The 3D structure of the lattice allows TQC computations to be visualised similarly to knot diagrams. Logical information is abstracted as strands and strand interactions (braids) represent logical quantum gates. Therefore, TQC circuits are abstracted using a geometrical description, which allows circuit input-output transformations (correlations) to be represented as geometric sub-structures. TQC design methods were not investigated prior to this work, and the thesis introduces the topological computational model by first analysing the necessary concepts. The proposed TQC design stack follows a top-down approach: an arbitrary quantum circuit is decomposed into the TQC supported gate set; the resulting circuit is mapped to a lattice of appropriate dimensions; relevant resulting topological properties are extracted and expressed using graphs and Boolean formulas. Both circuit representations are novel and applicable to TQC circuit synthesis and validation. Moreover, the Boolean formalism is broadened into a formal mechanism for proving circuit correctness. The thesis introduces TQC circuit synthesis, which is based on a novel logical gate geometric description, whose formal correctness is demonstrated. Two synthesis methods are designed, and both use a general planar representation of the circuit. Initial simulation results demonstrate the practicality and performance of the methods. An additional group of proposed design methods solves the problem of automatic correlation construction. The methods use validity criteria which were introduced and analysed beforehand in the thesis. Input-output correlations existing in the circuit are inferred using both the graph and the Boolean representation. The thesis extends the TQC state-of-the-art by recognising the importance of correlations in the validation process: correlation construction is used as a sub-routine for TQC circuit validation. The presented cross-layer validation procedure is useful when investigating both the QECC and the circuit, while a second proposed method is QECC-independent. Both methods are scalable and applicable even to very large circuits. The thesis completes with the analysis of TQC circuit identities, where the developed Boolean formalism is used. The proofs of former known circuit identities were either missing or complex, and the presented approach reduces the length of the proofs and represents a first step towards standardising them. A new identity is developed and detailed during the process of illustrating the known circuit identities. Reliable quantum circuits are a necessity for quantum computing to become reality, and specialised design methods are required to support the quest for scalable quantum computers. This thesis used a twofold approach towards this target: firstly by focusing on the probabilistic behaviour of quantum circuits, and secondly by considering the requirements of a promising quantum computing architecture, namely TQC. Both approaches resulted in a set of design methods enabling the investigation of reliable quantum circuits. The thesis contributes with the proposal of a new quantum simulation technique, no
A quantum gate is the quantum analog of the classical logic gate of digital computers. It is a basic structural unit of a quantum circuit which operates on a small number of qubits to deliver a predefined output. A quantum circuit is a graphical representation of the sequence of the quantum gates used and the measurements performed in a quantum computation. Various quantum logic gates, namely, the Hadamard, T phase, NOT, Pauli-X, controlled NOT (CNOT), controlled-Z, SWAP and controlled phase gates as well as higher-level Toffoli and Fredkin gates are described. Conventions of reading quantum circuit diagrams are explained. Operation of a quantum full adder consisting of registers and gates is discussed, elaborating the role of the NOT gate, the sum calculation with CNOTgates and the carry C-out calculation with Toffoli gates. A primary tool used for performing arithmetic operations on a quantum computer is the quantum Fourier transform (QFT). The QFT adder and multiplier circuits are presented.
Except for the NOT gate, all the other familiar classical gates are irreversible in the sense that we cannot uniquely reconstruct the input states from the output states. For example, OR, AND, NAND, NOR, etc. are irreversible. They all map a two-bit input state into a single bit output state. Thus one bit is erased during operation of each of these gates, and according to Landauer’s principle that requires dissipation of a minimum amount of energy. Now a simple question arises in our curious mind: Is it possible to avoid this loss of energy? The answer is yes. If we don’t erase any bit then we can circumvent this energy loss. Thus we need to map n bit input states to n bit output states. In addition, if a one-toone correspondence exists between the input states and the output states then only we will be able to uniquely reconstruct the input states from the output states. In such a case the gate is called reversible. For example, if we have f(00) = 00, f(01) = 10, f(10) = 01, f(11) = 11 then f represents a reversible gate. Quantum evolution operators are unitary so for every operator U we have an inverse operator U−1 = U†. Therefore, quantum evolution operators are the natural choice for the construction of energy efficient reversible gates. However, it is not the only choice. We can have classical reversible gates, too. Usually by reversible gates we refer to classical reversible gates, and quantum gates are specifically referred to as quantum gates. Reversible gates and quantum gates are similar but there exists a fundamental difference that reversible gates cannot accept superposition states (e.g. α|0〉 + β|1〉) as input states, whereas quantum gates can. Thus all quantum gates are essentially reversible but the converse is not true.