Qubit, An Intuition #1 — First Baby Steps in Exploring the Quantum World

TL;DR; 
- 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.
- Bra-ket notation, bloch sphere, hybrid Classical-Quantum.
- The source code (in Python) for generating illustrated qubit state transitions is available on github.
- This is the first article in the "Qubit, An Intuition" series. Upcoming articles (towards the end of 2021) will discuss "Inner Product, Outer Product, and Tensor Product", "Quantum Measurement", "Unitary Matrices", "Quantum Circuit and Reversible Transformation", as well as "Quantum Algorithms."

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.”

IBM Quantum computers (IBM, 2021d)
The operating status of 26 IBM real Quantum computers. The quantum computers are accessible from the IBM Cloud (as of May 12, 2021).

Classical and Quantum Computing

Classical (based on bits) and Quantum (based on qubits) computations.
Bits in Classical computer and Qubits in Quantum computer.

Quantum bit (Qubit)

A physical qubit realization in IBM Quantum chip, based on superconducting qubit (Zlatko Minev, 2020).
The average T1 time of a qubit is the time it takes for a qubit to decay from the excited state to the ground state. It is important because it limits the duration of meaningful programs we can run on the quantum computer.,” stated in Qiskit Textbook (IBM, 2021b) within the section titled “Calibrating Qubits with Qiskit Pulse.”
The decoherence time (average T1 time) of a 15-qubits IBM Quantum computer on the Cloud “ibmq_16_melbourne”.

Bra-ket Notation

A quantum state in ket notation, |Ψ>.
Bra <Ψ| and ket |Ψ> notation, thus “Bra-ket” notation, 1 of 2.
Bra <Ψ| and ket |Ψ> notation, thus “Bra-ket” notation, 2 of 2.
Bra <Ψ| and ket |Ψ> notation, thus Bra-ket notation. Other possible quantum states are plus (|+>), minus (|->), i (|i>), and -i (|-i>).
When a qubit is in superposition, the probability of measuring 0 or 1 is equal at 50% and 50%, respectively.

Bloch Sphere — Illustrating a 1-qubit state

The Atoms of Computation: The Bloch Sphere of a qubit (IBM, 2021b).
Quantum state transition from |0> to |1>.
Quantum state transition from |1> to |0>.
Quantum state transitions for Superposition. |0> to |+> and |1> to |->.
An illustration of two independent qubits to illustrate Superposition: qubit 0 and qubit 1. The transition for qubit 0 is from state |0> to |+> (by applying H-gate), while qubit 1 from state |1> to |-> (also by applying H-gate). Illustrated by an online tool, Quirk (Strilanc, 2019).
An illustration of two independent qubits to illustrate Superposition: qubit 0 and qubit 1. The transition for qubit 0 is from state |0> to |+> (by applying H-gate), while qubit 1 from state |1> to |-> (also by applying H-gate). Note that, in order to bring qubit 0 to state |1>, an X-gate is applied to the |0> state). Illustrated by an online tool, Quirk (Strilanc, 2019).
Quantum state transitions from |+> to |-> and from |-> to |+>.
An illustration of two independent qubits: qubit 0 and qubit 1. The transition for qubit 0 is from state |+> to |-> (by applying Z-gate, or Y-gate). Illustrated by an online tool, Quirk (Strilanc, 2019).
An illustration of two independent qubits: qubit 0 and qubit 1. The transition for qubit 0 is from state |+> to |-> (by applying Z-gate, or Y-gate), while qubit 1 from state |-> to |+> (also by applying Z-gate, or Y-gate). Note that, in order to bring qubit 0 to state |+> and qubit 1 to |->, an H-gate is applied to both independently, by initially setting the state for qubit 0 = |0> and qubit 1 = |1>. Illustrated by an online tool, Quirk (Strilanc, 2019).
Quantum state transitions from |i> to |-i>.
Quantum state transitions from |-i> to |i>.
An illustration of two independent qubits: qubit 0 and qubit 1. The transition for qubit 0 is from state |i> to -|i> (by applying Z-gate), while qubit 1 is from the state -|i> to |i> (also by applying Z-gate). Illustrated by an online tool, Quirk (Strilanc, 2019).
An illustration of two independent qubits: qubit 0 and qubit 1. The transition for qubit 0 is from state |i> to -|i> (by applying Z-gate), while qubit 1 is from the state -|i> to |i> (also by applying Z-gate). Note that, in order to bring qubit 0 to state |+> and qubit 1 to |->, an H-gate followed by an S-gate (or S-gate inverse) are applied to both independently, by initially setting the states for both qubits = |0>. Illustrated by an online tool, Quirk (Strilanc, 2019).

Measurement — with IBM Quantum

Before execution, the submitted quantum circuit is translated to a series of hardware-specific supported basis gates for the specific Quantum computer. E.g., for the 27 qubits IBM Quantum computer called "ibmq_toronto" (IBM, 2021a), the supported basis gates are CX, ID, RZ, SX, and X. While for for the single qubit IBM Quantum computer called "ibmq_armonk", the supported basis gates are ID, RZ, SX, and X.
A typical workflow in a hybrid Classical-Quantum computation.
A qubit in Superposition, executed in a Quantum simulator.
A qubit in Superposition, executed in a real IBM Quantum computer, a 5-qubits “ibmq_santiago.”
Measurement frequency results for a qubit in Superposition, executed in a real IBM Quantum computer, a 5-qubits “ibmq_santiago.”
Example of Operators. For manipulating quantum state transitions on qubits in a quantum circuit (IBM, 2021c).
Operators are implemented mathematically as matrices operations (IBM, 2021c).

Moving Forward

IBM roadmap to the future of millions of qubits.

References

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