A Controlled-NOT Quantum Logic Gate | Quantum Science Philippines
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A Controlled-NOT Quantum Logic Gate

by Majvell Kay G. Odarve

A quantum computer is device for computation that uses the phenomena of quantum mechanics to perform operations on data. Because of the quantum mechanical phenomena, such as entanglement and superposition of states, quantum computers have great offers in the field of computations and data handling. The distinctive feature of a quantum computer lies on its ability to store and process superposition of numbers.  This potential for parallel computing points out that some problems can be efficiently solved using quantum computers compared to the classical one. Shor’s algorithm, for example, which solves the problem that ‘Given an integer N, find its prime factors’, shows that quantum computer should be able to efficiently factor large numbers. The field appears to be of great interest since most data encryption schemes (in cryptography-science of information security) relies on the inability of the classical computers to factor large numbers.

To have an experimental realization of a quantum computer, there is a requirement of an isolated quantum system which will act as qubits and the presence of controlled unitary interactions between the qubits that allow construction of a controlled-NOT (CN) quantum logic gate. Quantum logic gates are building blocks of quantum circuits which operate on small numbers of quantum bits (qubits).  Quantum logic gates, unlike the classical logic gates, are reversible.  A CN quantum logic gate is one of the commonly used logic gates which operate on two qubits (we label the qubits as [eq]\epsilon_1[/eq] and [eq]\epsilon_2[/eq]).   The CN gate transforms the state of the two qubits from [eq]|\epsilon_1>|\epsilon_2> [/eq] to [eq]|\epsilon_1>|\epsilon_1\oplus\epsilon_2>[/eq] where [eq]\oplus[/eq] is an addition modulo 2. The CN gate represents a computation at the most fundamental level, that is, a certain ‘target’ qubit [eq]\epsilon_2[/eq] is flipped depending on the state of a ‘control’ qubit [eq]\epsilon_1[/eq].

Christopher Monroe and his team from National Institute of Standards and Technology (NIST) Laboratory in Boulder, Colorado, who are working on ion-trapped quantum computers, have been able to demonstrate the operation of a two-bit controlled-NOT quantum logic gate operating on prepared quantum states.  In their experiment, the two qubits comprise two internal (hyperfine) states and two external (quantized motional harmonic oscillator) states of a single trapped atom using a single beryllium ion (Be+). The trapped ions are first laser cooled to zero-point energy for them to stay in the ground state.   In the trapped-ion architecture, the qubits are associated with the internal states of the ions and information is transferred between the qubits through a shared motional degree of freedom. With this configuration, decoherence can be small so it will be easier to extend the idea to large registers and the qubit readout will have a nearly unit efficiency.

To realize the CN gate, three sequential pulses of the Raman beams is applied to the ion, namely the π/2 pulse applied to the carrier transition, a 2π pulse applied on the blue side band transition and a π/2 pulse applied to the carrier transition with a π phase shift relative to the first pulse. The truth table for a CN gate operation is given as:

Input State        [eq]\longrightarrow[/eq]     Output State

[eq]|0>|\downarrow> [/eq]  [eq]\longrightarrow[/eq]  [eq]|0>|\downarrow>[/eq]

[eq]|0>|\uparrow> [/eq]  [eq]\longrightarrow[/eq]  [eq] |0>|\uparrow>[/eq]

[eq]|1>|\downarrow> [/eq] [eq]\longrightarrow[/eq]  [eq]|1>|\uparrow>[/eq]

[eq]|1>|\uparrow> [/eq]  [eq]\longrightarrow[/eq]  [eq]|1>|\downarrow>[/eq]

In their experiment, the key features of the CN gate was demonstrated by verifying that the populations of the register follow the truth table and by demonstrating the conditional quantum dynamics associated with the CN operation. On the results of the experiments of Monroe, et. al., decoherence were still present. The decoherence were caused by instabilities of the laser beam power, the position of the ions relative to the laser beams, the fluctuation of external magnetic fields, instabilities in the trap drive frequency and voltage amplitude, dissipation of the ion motion and some spontaneous emission caused by off-resonant transitions. The single ion quantum register in the experiment comprises only two qubit which is not useful for computations. The next step of the researchers then is to apply their operation techniques to many ions cooled at a state of collective motion for the possibility of implementing computations on larger quantum registers.

Reference:

1. C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, and D.J. Wineland, Physical Review Letters, Vol. 75, Num. 25, December 1995.

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