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数理解析研究所講究録

On quantum logical gate based on ESR (Analytical Study of Quantum Information and Related Fields)

Masanori Ohya, Igor Vasil'evich Volovich, Noboru Watanabe

Published Apr 1, 2001• 0 citations
OpenAlex

Abstract

In classical computer, there exist inevitable demerits for discussing log- ical gates.One of the demerits is an irreversibility of logical gates, that is the AND and the OR gates.This property causes to the restriction of computtional speed for the classical computer.$\mathrm{T}\mathrm{h}\dot{\mathrm{e}}$ re are several kind of ap- proaches for avoiding these demetits.One of these approaches is proposed by Feynman [1].He proved that every logical gates can be contructed by combining with only two reversible gates, i.e., the NOT and the Controlled NOT (CNOT) gates.By the way, there are several approaches for realizing quantum logical gates.One of those approaches is the study by means of a nuclear magnetic 1

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In classical computer, there exist inevitable demerits for discussing log- ical gates.One of the demerits is an irreversibility of logical gates, that is the AND and the OR gates.This property causes to the restriction of computtional speed for the classical computer.The˙\mathrm{T}\mathrm{h}\dot{\mathrm{e}}The˙ re are several kind of ap- proaches for avoiding these demetits.One of these approaches is proposed by Feynman [1].He proved that every logical gates can be contructed by combining with only two reversible gates, i.e., the NOT and the Controlled NOT (CNOT) gates.By the way, there are several approaches for realizing quantum logical gates.One of those approaches is the study by means of a nuclear magnetic 1
In usual computer, there is an upper bound of computational speed because of irreversibility of logical gate.In order to avoid this demerit, Fredkin and Tof- foli [3] proposed a conservative logical gate.After that, Milburn [4] constructed a physical model of reversible quantum gate with beam splittings and a Kerr medium.This model is called FTM (Fredkin −-− Toffoli −-− Milburn gate) in this paper.This FTM gate was described by the quantum channel and the efficiency of information transmission of the FTM gate was discussed in [10].FTM gate is using a photon number state as an input state for control gate.The photon number state might be difficult to realize physically.In this paper, we introduced a new device on symmetric Fock space in order to avoid this difficulty.In Section 1, we briefly review noisy quantum channels and generalized beam splittings.In Section 2, we explain the quantum channel for FTM gate based on the notation of Section 1.In Section 3, we intruduced a new operator on symmetric Fock space and discuss the truth table for our new gate.
In the last chapter, we saw that it is theoretically possible to switch a logically reversible gate without dissipating any energy at all (recall the proof of the Landauer-Shannon result). If there is no dissipation, then no information is ever being discarded; therefore, if we know the state of the system completely at any instant of time, we should be able to infer its state at any previous instant. This is consistent with the equations of quantum mechanics (e.g., the Schro¨dinger equation, or the Pauli Equation) which are reversible in time. This property of quantum mechanical systems, known as ‘time reversal symmetry’, guarantees that if we know the state of a system at a time t0, we can tell what the state was at any time t < t0. Because of this fundamental connection between physical reversibility, logical reversibility, and the equations of quantum mechanics, we would expect logically reversible gates to behave as non-classical (quantum mechanical) computing machinery∗. This will bring us to the topic of quantum computing and quantum logic gates, but before we get there, let us explore reversible logic in a little more detail.
Quantum is an emerging technology in future computers. Reversibility is the main advantage of quantum computers. In conventional computers, the computation is irreversible i.e. the input bits are lost once the logic block generates the output and input bits cannot be restored but it can be done in reversible computation because in reversible computation the inputs and outputs have a one-to-one correspondence. Therefore, a reversible gate input could even be uniquely determined from their output which leads to less power consumption. Hence the complexity of the digital circuits can be reduced by using reversible computing. In quantum computer to perform reversible operations, we need to implement the reversible gates using quantum gates. In this paper, we discussed various reversible logic gates like Feynman, Toffoli, R, Peres and TR gates using basic quantum gates like CNOT, Pauli, Swap gates and their implementation using IBM quantum experience.
In the last chapter, we saw that it is theoretically possible to switch a logically reversible gate without dissipating any energy at all (recall the proof of the Landauer-Shannon result). If there is no dissipation, then no information is ever being discarded; therefore, if we know the state of the system completely at any instant of time, we should be able to infer its state at any previous instant. This is consistent with the equations of quantum mechanics (e.g., the Schro¨dinger equation, or the Pauli equation) which are reversible in time. This property of quantum mechanical systems, known as “time reversal symmetry”, guarantees that if we know the state of a system at a time t0, we can tell what the state was at any time t < t0. Because of this fundamental connection between physical reversibility, logical reversibility, and the equations of quantum mechanics, we would expect logically reversible gates to behave as non-classical (quantum mechanical) computing machinery.∗ This will bring us to the topic of quantum computing and quantum logic gates, but before we get there, let us explore reversible logic in a little more detail.