Inner and Outer

 Been looking through the blogs of Andi Sama on Mediuem from 2021. Found interesting 

information and a very nice intro to the math. Below:

Implementation is based on the superconducting qubit in the Josephson junction device, in which a current flows continuously without any voltage applied. It is achieved by having two superconducting materials separated by a thin insulating barrier operating at an extremely low temperature close to absolute zero, at 15 mK (milliKelvin). 

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In this NISQ-era (Noisy Intermediate Scale Quantum), a qubit is not perfect (noisy). The quantum state can only be maintained (useful to do quantum computation) in the range of just tenths of microseconds before experiencing the decaying process called decoherence.

Take a look at the following illustration. The decoherence time (T1) of a 15-qubits IBM Quantum computer on the Cloud “ibmq_16_melbourne” is 54.27 microseconds. The quantum computer is online and operational when accessed on May 18, 2021.

(... turns out a microsecond is one millionth of a second!!)

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A ket |Ψ> is a column vector, with values α and β . a bra <Ψ|is a row vector with values α*and β*. α* and β* are the complex conjugate of α and β. Complex conjugate means that we change the sign of the imaginary part of α and β from plus to minus, or vice versa. Then, we do a transpose operation following the complex conjugate operation — thus, completing the entire complex conjugate transpose operation.

(... Finally!!)

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The inner product of bra and ket is a measure of overlap between quantum states. An inner product of bra and ket, a row vector times the column vector, produces a single value, which is ∈ ℂ².

An outer product of ket and bra, a column vector times the row vector, produces a matrix.

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One can take the inner product of each qubit separately, or one can 

take it for the system as a whole.

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The 0uter product is no different than that of the individual qubits.

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The outer product yields four equally probable outcomes for the system.

https://www.perplexity.ai/search/quantum-computing-with-qiskit-tv0zskdSTtCi_ylvigBChA