Qubit, An Intuition #2 — Inner Product, Outer Product, and Tensor Product in Bra-ket Notation

The beauty of Quantum Mechanics for Quantum Computation, featuring IBM Quantum

Andi Sama — CIO, Sinergi Wahana Gemilang with Cahyati S. Sangaji

TL;DR; 
2 qubits inner product, outer product, and tensor product in bra-ket notation, with examples.
Please refer to the previous article (published in July 12 2021), “Qubit, An Intuition #1 — First Baby Steps in Exploring the Quantum World” for a discussion on a single qubit as a computing unit for quantum computation.

For an introductory helicopter view of the overall six articles in the series, please visit this link “Embarking on a Journey to Quantum Computing — Without Physics Degree.”

In the first article, “Qubit, An Intuition #1 — First Baby Steps in Exploring the Quantum World,” we discussed the intuition of a quantum bit (qubit) as a computing unit in a Quantum computer, in comparison to a binary digit (bit) in a Classical computer. We also discussed Bra-ket notation, Bloch sphere, along with the hybrid classical-quantum approach.

In this follow-on article, we will discuss the inner product, outer product, and tensor product. These are some of the basic mathematical foundations to understand quantum computation.

Quantum Bit

Indeed, a single quantum bit (qubit) is exciting with its nature of being in superposition. Hence |Ψ> = α|0> + β|1>. The quantum state Psi (|Ψ>) is in the linear combination of |0> and |1> with the probability of being in state |0> is |α|², and the probability of being in state|1> is |β|².

α and β are the probability amplitudes while α* and β* are the complex conjugates of α and β, respectively. It fulfills the normalization constraint, such that |α|²+|β|²=1 or α*α+β*β=1, where α, β ∈ ℂ².

Inner Product, Outer Product, and Tensor Product

However, we need more qubits to do meaningful quantum computation. Let’s say that we have the following two qubits, Psi|Ψ> and Phi |Φ>. Each has its respected quantum state.