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arXiv:1110.2998

Equivalent Quantum Circuits

Juan Carlos García-Escartín, Pedro Chamorro‐Posada

Published Oct 13, 2011• 23 citations
arXivHTMLPDF

Abstract

Quantum algorithms and protocols are often presented as quantum circuits for a better understanding. We give a list of equivalence rules which can help in the analysis and design of quantum circuits. As example applications we study quantum teleportation and dense coding protocols in terms of a simple XOR swapping circuit and give an intuitive picture of a basic gate teleportation circuit.

DOI
10.48550/arxiv.1110.2998

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Quantum algorithms and protocols are often presented as quantum circuits for a better understanding. We give a list of equivalence rules which can help in the analysis and design of quantum circuits. As example applications we study quantum teleportation and dense coding protocols in terms of a simple XOR swapping circuit and give an intuitive picture of a basic gate teleportation circuit.
Decomposing a complex system into smaller abstract functional blocks and developing mathematical models to represent their behavior is an important activity towards developing comprehensive system understanding. In this paper, we decompose the ideal Quantum Teleportation protocol into a collection of simple quantum circuit blocks, examine the behavior of each block, and show how collections of blocks operate to create more complex circuits. We believe this approach greatly simplifies the understanding of how the Quantum Teleportation protocol works. This paper is introductory in nature and is intended to help those who are new to modeling, simulating, and analyzing ideal quantum circuits.
The beauty of the quantum circuit lies in the fact that it can perform certain tasks which are impossible in the classical world. For example, consider teleportation and superdense coding. Teleportation is a quantum task in which an unknown quantum state is transmitted from a sender (Alice) to a spatially separated receiver (Bob) via an entangled quantum channel and with the help of some classical communications. Dense coding is another communication scheme in which if Alice and Bob share a prior entanglement then one of them can send n bits of classical information to the other by communicating m-qubits (m < n). Before we go into the technical details of these two schemes let us try to lucidly visualize the uniqueness of teleportation. Suppose Alice and Bob are spatially separated and Alice wants to send an unknown quantum state |ψ〉 = α|0〉 + β|1〉 to Bob. For some reason she is not allowed to send the state directly through a quantum channel. She cannot measure it and send the information to Bob because the moment she will try to measure the state, it will collapse either to |0〉 or to |1〉. But the state can be sent by using a simple quantum circuit (pre-shared entanglement) and some classical communications. In the process the unknown quantum state gets destroyed at Alice’s end and reappears at Bob’s end. The state never exists in the communication channel. This process is called quantum teleportation. The uniqueness of quantum teleportation will be clearer if we note that the Fax is not an example of teleportation because the original copy remains with the sender. Again a letter posted in a letter box is not an example of teleportation because the letter exists (travels) in its original form in the communication channel. In brief, there is no classical analogue of quantum teleportation. The same is true for dense coding. In this chapter we will describe these two unique quantum communication schemes and discuss their significance and applications.
Quantum teleportation is one of the most important protocols in quantum information, which is based on quantum entanglement. This paper provides an overview of the basic theory behind the original quantum teleportation protocol and explores its advanced variations. We validate these protocols' effectiveness by constructing simple quantum circuits. Following this, we delve into deeper theoretical aspects, experimental realizations, and applications of quantum teleportation, including quantum state exchange experiments and weak measurement techniques, as well as environment-assisted measurements. Finally, we briefly discussed its current limitations and the future prospects of quantum teleportation.
Quantum algorithms promise up to a superpolynomial factor speedup in computational tasks compared to their classical counterparts. Quantum computers are now approaching sizes where quantum algorithms can be demonstrated. As a result, the implementation of algorithms as quantum circuits has become an important research topic. In this work, we provide an overview of quantum computation and the design of quantum circuits. Properties unique to quantum computation are outlined. We introduce how a quantum algorithm is translated into a circuit and show measures used to assess the performance of quantum circuits.
Reversible logic is expanding grounds with its applicability to various upcoming technologies and quantum computation. Numerous reversible gates has also been proposed for designing logic circuits apart of basic quantum controlled Toffoli and Fredkin gates. A comparative study and analysis of a range of reversible gates from the literature is presented in this paper to obtain an optimal solution for designing these circuits. The gates are synthesized and implemented to obtain equivalent quantum circuits and operating cost in terms of number of inputs, gate cost, quantum cost, garbage output and ancilla input. The comparison is done at two subsequent levels i.e. gates structural level and circuit design level, providing a better solution for designing logic circuits. It is calculated that, basic gates are 50% efficient at gate level and 33% at design level analysis in terms of gate cost, as compared to the most efficient gates proposed in the literature.