Lena Aly (New York University Abu Dhabi, UAE)
LinkedIn: @Lena Aly
Abstract: In this work, we develop a MATLAB-based simulation model for NMR pulse protocols in weakly coupled spin-½ systems, with a focus on implementing quantum logic gates such as the Controlled-NOT (CNOT) gate. The model connects the unitary matrix representation of quantum gates to experimentally realizable pulse sequences using angular momentum operator formalism. It allows direct evaluation of pulse performance and provides quantitative error estimates for various sequences. The model enables informed comparisons between alternative pulse schemes, and lays the groundwork for future extensions to include advanced techniques such as composite pulses for NMR based quantum information science.
-
Hi, what is the classical analogy of the CNOT gate, and what is the main difference between the two?
-
Hi!
The action of the CNOT gate is analogous to the classical XOR operation, and the output of the XOR gate can be thought of as the target qubit in a CNOT. Unlike the XOR, however, the CNOT does not reduce the two qubits into one qubit: the control qubit remains as it is, and the target undergoes the XOR operation. This makes the operation of the CNOT reversible and can be represented by a unitary matrix, as required for all quantum gates.
It can be more challenging to implement the CNOT gate, though, because it is important to only flip the state of the target qubit without changing the coherence of the states.
Also, the CNOT is an essential component in implementing quantum entanglement, which is necessary to give quantum computers an advantage over their classical counterparts.
-
-
Thank you for the presentation. How easily can this expanded to investigate e.g., the Toffoli gate?
-
Such an interesting question!
The current work lays the framework for transforming the desired projection operators in the form of idempotents into realizable pulse sequences and testing them. Since any desired expansions would be unitary, they can also be modeled as idempotents and follow the same logic with a minor modification in the defined Hamiltonian.
-
-
Hi Lena, nice presentation! I am curious, what are some examples of those small mistakes that could occur when trying to go from the matrices to the pulse sequences?
-
Thank you so much!
Well, since the process of going from the matrices to pulse sequences includes identifying the desired projection operators, factorizing them to include complex numbers, using idempotent identities to rewrite them as exponents, and then trying to separate the exponents such that each exponent applies only to one axis, many mistakes could happen, especially with signs.
The one I identified in Price et al 1999 (https://doi.org/10.1006/jmre.1999.1851) using the program was in their application of the identity highlighted in the presentation. It was inconsistent with their choice of axis, which did not end well in the simulation as shown (they should have used U^-1 where they have used U). This also makes sense why their result was different from the one by Volkov and Salikhov 2011 (10.1007/s00723-011-0297-2), although they both used the same mathematical foundation to derive the pulse sequence for the CNOT gate.
-
Thank you! That is very interesting–it turns out sign mistakes are crucial to avoid for these kinds of calculations!
-
Leave a Reply