Quantum Programming with High School Math: Part 2

Recap on Previous Article(s)

This article is part of a series. The earlier article is Demystify Quantum Programming with High School Math Series: Part 1 - The Qubit.

Syllabus Overview

In this article, I will introduce the Bloch sphere, which provides an intuitive visual representation of the quantum state of a qubit. This visual representation also makes it easier to appreciate how a quantum gate operation transforms the qubit state.

Although complex numbers are used in quantum mechanics (QM) linear algebra, I will limit to using real numbers in this article. As such, we will cover some gates that operate only on one qubit (also known as single-qubit gates), namely Identity (I) gate, Pauli-X (X) gate, and Hadamard (H) gate.

Bloch Sphere, A First Visit

The Bloch sphere, invented by physicist Felix Bloch, is a visual presentation of the mathematical representation of the quantum state of a quit. In technical terms, the Bloch sphere is a geometric representation of a two-level quantum mechanical system aka a qubit.

Northern and Southern Hemispheres

Diagram 1. |0⟩ and |1⟩ basis states representation in a Bloch sphere.

Recall, in Part 1, that the quantum state of the qubit |Ψ⟩ can be described as a linear combination of two complex numbers α and β, where α describes the amplitude that the qubit is in the |0⟩ state and β describes the amplitude that the qubit is in the |1⟩ state.

In Diagram 1 above, the northern hemisphere is the graphical representation of α; the southern hemisphere is the graphical representation of β. Together, both hemispheres form the Bloch sphere, which graphically represents the linear combination of α and β of one qubit.

The Three Axes

Diagram 2. x-axis, y-axis and z-axis of a Bloch sphere. The solid lines represent positive values along the axes; the dotted lines represent negative values along the axes.

Here is a quick refresher on complex numbers. A complex number consists of two parts: a real number component and an imaginary number component. Specifically, a complex number, z = a + bi, where a & b are real numbers and i is the square root of -1 . Intuitively, this implies that there are four real numbers to describe a quantum state in a qubit.

However, in the context of quantum mechanics, the quantum state of a qubit can be described using only three real numbers (by factoring out the global phase from α and β), namely the amplitude of |0⟩, the amplitude of |1⟩ and the relative phase between |0⟩ and |1⟩. These three real numbers are commonly represented in α and β as follows:

α contains the amplitude of |0⟩. It is a real number that is more than or equals to zero.
β contains both the amplitude of |1⟩ and the relative phase between |0⟩ and |1⟩. It is a complex number.

As a result, the quantum state of a qubit can be visually represented as a state vector in a 3-dimension Bloch sphere shown in Diagram 2 above. In case you were wondering, the x-axis represents the real part of the state vector; the y-axis represents the imaginary part of the state vector; the z-axis represents the probabilities of a qubit being observed as 0 or 1 upon measurement.

State Vector for a Coherent Qubit

Diagram 3. State vector |Ψ⟩, representing a pure state or coherent state, pointing to the surface of the Bloch sphere.

Recall, in Part 1, we briefly mentioned that when a qubit maintains quantum coherence, it possesses a valid quantum state. A valid quantum state is visually represented as a state vector pointing to the surface of the Bloch sphere with a vector length of 1 (also known as a unit vector) as shown in Diagram 3 above.

Conversely, when a qubit experiences quantum decoherence, the invalid quantum state (also known as a mixed state) is visually represented as a state vector pointing within the volume of the Bloch sphere.

Now, let us take a look at how the two basis states are visually represented in the Bloch sphere.

State Vector for Basis States of a Qubit

Diagram 4. State vector |0⟩ in a Bloch sphere.

When the qubit |Ψ⟩ is in the |0⟩ state, the state vector points to the surface of the Bloch sphere vertically upwards along the z-axis, as shown in Diagram 4 above. This state vector indicates that there is a 100% chance of observing a 0 upon measuring the qubit.

Diagram 5. State vector |1⟩ in a Bloch sphere.

When the qubit |Ψ⟩ is in the |1⟩ state, the state vector points to the surface of the Bloch sphere vertically downwards along the z-axis, as shown in Diagram 5 above. This state vector indicates that there is a 100% chance of observing a 1 upon measuring the qubit.

State Vector for Superposition States of a Qubit

Recall, in Part 1, when a qubit |Ψ⟩ is in a superposition of its basis states, both α and β have non-zero values. Visually, the state vector representing this superposition state can point to any point on the surface of the Bloch sphere except for |0⟩ and |1⟩.

If the state vector points to any point along the equator (the horizontal ellipse with dashes) on the the surface of the Bloch sphere, the qubit will have a 50% chance of observing a 0 or 1 upon measuring the qubit.

Math for Single-qubit Quantum Gate

Recall, in Part 1, that the state of a qubit can be represented as a 2x1 vector. From now on, I will refer to this as the state vector.

A single-qubit quantum gate operation on a quantum state can be represented as a 2x2 matrix multiplying with a state vector. To facilitate visualization of this concept, I will use some quantum gates as examples.

Math for Identity (I) Gate

Diagram 6. Circuit form of I gate. The I gate is represented in a circuit diagram as a square with an "I" in it (left). Since the I gate does nothing to the input state, it is usually omitted and drawn as a straight line to the next gate (right).

The I gate takes in one qubit input, and outputs the same quantum state without any modifications. The matrix for the I gate is defined as

I = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

The effect of the I gate is shown below in both Dirac and vector notations.

I|0⟩ = |0⟩, I|0⟩ = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}]

I|1⟩ = |1⟩, I|1⟩ = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) [\begin{matrix} 0 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 1 \end{matrix}]

Math for Pauli-X (X) Gate

Diagram 7. Circuit forms of X gate. The X gate is represented in a circuit diagram either as a square with an "X" in it (left) or as a circled plus (right).

The X gate takes in one qubit input, flips the input from |0⟩ to |1⟩, and vice versa from |1⟩ to |0⟩. Due to this bit-flip operation, the X gate is also commonly known as a "bit-flip operator", "NOT gate" or "addition mod 2". In the Bloch sphere, the X gate performs a rotation on the quantum state vector by π radians around the x-axis.

The matrix for the X gate is defined as

X = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})

The effect of the X gate is shown below in both Dirac and vector notations.

X|0⟩ = |1⟩, X|0⟩ = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 0 \\ 1 \end{matrix}]

X|1⟩ = |0⟩, X|1⟩ = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) [\begin{matrix} 0 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}]

If you compare the effect of the X gate against the state vector direction in Diagrams 4 and 5, you may have noticed that the X gate actually flips the state vector from up to down, and vice versa. In mathematical terms, the X gate rotates the state vector around the x-axis by π radians.

Math for Hadamard (H) Gate

Diagram 8. Circuit form of H gate. The H gate is represented in a circuit diagram as a square with an "H" in it.

The H gate takes in one qubit input, transforms the input state into a superposition state with 50% chance of observing a 0 or 1 upon measurement.

The matrix for the H gate is defined as

H = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix})

The effect of the H gate is shown below in both Dirac and vector notations.

H|0⟩ = \frac{1}{\sqrt{2}} (|0⟩ + |1⟩) = |+⟩, H|0⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}) [\begin{matrix} 1 \\ 0 \end{matrix}] = \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ 1 \end{matrix}]

H|1⟩ = \frac{1}{\sqrt{2}} (|0⟩ - |1⟩) = |-⟩, H|1⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}) [\begin{matrix} 0 \\ 1 \end{matrix}] = \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ -1 \end{matrix}]

In the equations above, |+⟩ and |-⟩ are popular labels used to represent commonly used superposition states transformed by the Hadamard gate. We can call |+⟩ the plus state and |-⟩ the minus state.

Diagram 9. State vector |+⟩ in a Bloch sphere. Upon measurement, there is a 50% chance to observe a |0⟩ or |1⟩.

Diagram 10. State vector |-⟩ in a Bloch sphere. Upon measurement, there is a 50% chance to observe a |0⟩ or |1⟩.

The |+⟩ state is visually represented in Diagram 9 above; the |-⟩ state is visually represented in Diagram 10 above. The state vectors for both |+⟩ and |-⟩ point in opposite directions along the x-axis, but have the same 50% chance to be observed as a 0 or 1 upon measurement. In simpler terms, the state vectors for both |+⟩ and |-⟩ have the same amplitudes in both basis states but different relative phase.

Review

In this article, we learnt how to visually represent the quantum state of a qubit in a Bloch sphere, in particular, for the four quantum states: |0⟩, |1⟩, |+⟩, and |-⟩. We also learnt the mathematical representation for the operations of the I gate, X gate and H gate.

In the next article, I will expand to using the full scope of complex numbers used to represent the quantum state of a qubit and additional quantum gate operations.

Bonus Question

Using our deeper understanding of the H gate, given a quantum circuit with two H gates connected in series along one line, what is the output state of the circuit for the following two input states? (Hint: You can use Quantum-in-a-Browser to check against your answer.)

Input qubit is in |0⟩ state. In QuICScript, the code is "H,H" for 1 qubit.
Input qubit is in |1⟩ state. In QuICScript, the code is "X,H,H" for 1 qubit.

See Answer

Using the matrix definition for H gate, we can apply the H gate twice on the input state vector. The resulting effect is two H gate operations will cancel out and return an output state that was the same as the input state. Thus HH|0⟩ = |0⟩, and HH|1⟩ = |1⟩. The math (in both notations) for this is as follows

HH|0⟩ = H|+⟩ = |0⟩, HH|0⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}) \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}) [\begin{matrix} 1 \\ 0 \end{matrix}] = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}) \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ 1 \end{matrix}] = \frac{1}{2} [\begin{matrix} 2 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}]

HH|1⟩ = H|-⟩ = |1⟩, HH|1⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}) \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}) [\begin{matrix} 0 \\ 1 \end{matrix}] = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}) \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ -1 \end{matrix}] = \frac{1}{2} [\begin{matrix} 0 \\ 2 \end{matrix}] = [\begin{matrix} 0 \\ 1 \end{matrix}]

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