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Physical Review A

Design and stability of NOR and NAND logic gates constructed with three quantum states

Nicolas Renaud, Christian Joachim

Published Dec 8, 2008• 17 citations
Publisher

Abstract

By encoding the digital input of a classical logic gate on the Hamiltonian of a quantum system and driving the logic operation by preparing it in the same nonstationary state whatever the input, NOR and NAND classical logic gates are designed with a minimum of three quantum states. The outputs of one gate are obtained either by measuring the distance between the $\ensuremath{\rho}(t)$ periodic quantum trajectory and the output target state ${\ensuremath{\rho}}_{b}$, or by measuring the secular frequency of the $\ensuremath{\rho}(t)$ almost periodic trajectory in the ${\ensuremath{\rho}}_{b}$ direction. A comparison of the stability to noise between the two approaches demonstrates that the frequency approach is more immune to random fluctuations in the Hamiltonian than a distance control approach, opening the way to determine the logic output using a tunneling current passing through the gate.

DOI
10.1103/physreva.78.062316

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By encoding the digital input of a classical logic gate on the Hamiltonian of a quantum system and driving the logic operation by preparing it in the same nonstationary state whatever the input, NOR and NAND classical logic gates are designed with a minimum of three quantum states. The outputs of one gate are obtained either by measuring the distance between the \ensuremathρ(t)\ensuremath{\rho}(t)\ensuremathρ(t) periodic quantum trajectory and the output target state \ensuremathρb{\ensuremath{\rho}}_{b}\ensuremathρb​, or by measuring the secular frequency of the \ensuremathρ(t)\ensuremath{\rho}(t)\ensuremathρ(t) almost periodic trajectory in the \ensuremathρb{\ensuremath{\rho}}_{b}\ensuremathρb​ direction. A comparison of the stability to noise between the two approaches demonstrates that the frequency approach is more immune to random fluctuations in the Hamiltonian than a distance control approach, opening the way to determine the logic output using a tunneling current passing through the gate.
An analytical method is proposed to implement any classical Boolean function in a small quantum system by taking the advantage of its electronic transport properties. The logical input, α = {α1, ..., αN}, is used to control well-identified parameters of the Hamiltonian of the system noted . The logical output is encoded in the tunneling current intensity passing through the quantum system when connected to conducting electrodes. It is demonstrated how to implement the six symmetric two-input/one-output Boolean functions in a quantum system. This system can be switched from one logic function to another by changing its structural parameters. The stability of the logic gates is discussed, perturbing the Hamiltonian with noise sources and studying the effect of decoherence.
We present how basic logic gates including nand, nor, and xor gates can be implemented counterfactually. The two inputs (Bob and Charlie) and the output (Alice) of the proposed counterfactual logic gate are not within the same station but rather separated in three different locations. We show that there is no need to prearrange entanglement for the gate, and, more importantly, there are no real physical particles traveling among Alice, Bob, and Charlie during the information processing. Bob and Charlie only need to independently control the blocking and unblocking of the transmission channels that connect them to Alice. In this way, they can completely determine the state of a real photon at Alice's end, thereby leading to implemention of a counterfactual logic gate. The functionality of a particular counterfactual logic gate is determined only by an appropriate design of Alice's optical device. Furthermore, by utilizing the proposed counterfactual logic gates, we demonstrate how to counterfactually prepare the Greenberger-Horne-Zeilinger state and W state with three remote quantum objects, which are in superposition states of blocking and unblocking the transmission channel.
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.