Euler Angle Decomposition of Single-Qubit Unitaries

Hello! Welcome to your fourth lesson in our course on Quantum Computing Foundations.

In our last two lessons, we built up our understanding of the fundamental quantum gates. We started with single-qubit gates, visualizing them as rotations on the Bloch sphere, and then moved to two-qubit gates like CNOT, which are essential for creating entanglement and performing conditional logic.

Today, we return to single-qubit operations to prove a crucial result about their structure and universality. The question we'll answer is: can any possible single-qubit operation be constructed from a simple, standard set of gates?

The learning outcome for this lesson is to derive the decomposition of an arbitrary single-qubit unitary into a sequence of rotation gates using the Euler angle representation.

To achieve this, we will:

1. Review the geometric picture of single-qubit gates as rotations.
2. Examine the general mathematical form of a 2x2 unitary matrix, separating the global phase from the SU(2) component.
3. Derive the Z-Y-Z Euler angle decomposition, which shows that any SU(2) operation can be expressed as three successive rotations.
4. Briefly discuss other Euler decompositions and how they relate to the Z-Y-Z form.

This is a cornerstone result in quantum computation, as it proves that we don't need an infinite library of distinct physical gates to implement any desired single-qubit transformation. A small set of continuously parameterised rotation gates is sufficient.

1. Gates as Rotations: A Recap

As we saw in Lesson 2, single-qubit gates correspond to rotations of the state vector on the Bloch sphere. The Pauli gates $X,Y,Z$ are rotations by $\pi$ radians around their respective axes, and the Hadamard gate is a rotation by $\pi$ around an axis halfway between the x and z axes.

More generally, we can define rotation gates $R_x(\theta ),R_y(\theta ),R_z(\theta )$ that rotate a qubit state by an arbitrary angle $\theta$ around the specified axis. These are our fundamental building blocks.

For a quick visual refresher on this concept, please watch the following clip.

2. The General Form of a Single-Qubit Unitary

A single-qubit gate is mathematically described by a $2\times 2$ unitary matrix, $U$. A key property of any unitary matrix is that its determinant has a magnitude of 1, i.e., $|\det (U)|=1$. This means the determinant must be a pure phase factor, $\det (U)=e^{i\varphi }$ for some real $\varphi$.

This allows us to factor any unitary matrix $U$ into two parts:

$$U=e^{i\alpha }V$$

where:

• $e^{i\alpha }$ is a global phase. This phase has no observable physical consequence, as it multiplies all basis states equally.
• $V$ is a special unitary matrix, which means its determinant is exactly 1, $\det (V)=1$. The set of all such $2\times 2$ matrices forms the special unitary group SU(2).

This separation is useful because the SU(2) matrix $V$ captures all the relative phase changes and rotations that are physically meaningful. The resource below, from Gavin E. Crooks, provides the details for how to perform this separation.

The key insight is that since $\det (U)=\det (e^{i\alpha }V)=e^{i2\alpha }\det (V)=e^{i2\alpha }$, we can find the global phase angle $\alpha$ by taking the argument of the determinant.

3. The Z-Y-Z Euler Angle Decomposition

Now we come to the main result. Any special unitary operation $V\in \text{SU(2)}$ can be decomposed into a sequence of three rotations. A standard choice, analogous to Euler angles in classical mechanics, is the Z-Y-Z decomposition.

The theorem states that for any single-qubit unitary $U$, we can find four angles $\alpha ,{\theta }_0,{\theta }_1,{\theta }_2$ such that:

$$U=e^{i\alpha }{R}_z({\theta }_2)R_y({\theta }_1)R_z({\theta }_0)$$

Note the ordering of the rotation angles, which corresponds to the reverse order of application in operator notation.

The derivation involves writing out the matrix for the product of rotations and equating its elements to the elements of the target SU(2) matrix $V=e^{-i\alpha }U$. This creates a system of equations that can be solved for the angles ${\theta }_0,{\theta }_1,{\theta }_2$.

The following reading walks through this derivation in detail.

Summary of the Derivation Strategy

Let's recap the logic.

1. Start with an arbitrary unitary $U$.
2. Calculate $\alpha =\frac{1}{2}\arg (\det (U))$ and define the special unitary matrix $V=e^{-i\alpha }U$. Let its elements be $V=\left(\begin{array}{cc}{V}_{00}& V_{01}\\ V_{10}& V_{11}\end{array}\right)$.
3. The matrix for the rotation product is $R_z({\theta }_2)R_y({\theta }_1)R_z({\theta }_0)$. By multiplying out the rotation matrices $R_z(\theta )=e^{-i\theta Z/2}$ and $R_y(\theta )=e^{-i\theta Y/2}$, we get the form shown in Eq. (37) of the resource.
4. By comparing the matrix elements of $V$ with the elements of the rotation product, we can solve for the angles.
   • The angle ${\theta }_1$ is found from the magnitudes of the matrix elements, e.g., $|V_{00}|=|\cos ({\theta }_1/2)|$.
   • The angles ${\theta }_0$ and ${\theta }_2$ are found from the phases of the matrix elements, which give expressions for the sum ${\theta }_0+{\theta }_2$ and difference ${\theta }_0-{\theta }_2$.

This procedure provides a constructive algorithm for finding the rotation angles for any given unitary.

4. A Concrete Example: Decomposing the Hadamard Gate

Let's apply this procedure to the Hadamard gate, $H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$.

1. Global Phase:
   $\det (H)=-1=e^{i\pi }$. From $\det (U)=e^{i2\alpha }$, we have $2\alpha =\pi$, so $\alpha =\pi /2$.

2. Special Unitary Part:
   $V=e^{-i\pi /2}H=-iH=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-i& -i\\ -i& i\end{array}\right)$.
   The matrix elements are $V_{00}=-i/\sqrt{2}$, $V_{01}=-i/\sqrt{2}$, $V_{10}=-i/\sqrt{2}$, and $V_{11}=i/\sqrt{2}$.

3. Find Rotation Angles (using formulas from Crooks):

   • ${\theta }_1=2\arccos (|V_{00}|)=2\arccos (1/\sqrt{2})=2(\pi /4)=\pi /2$.
   • ${\theta }_0+{\theta }_2=2\arctan 2(\text{Im}(V_{11}/\cos ({\theta }_1/2)),\text{Re}(V_{11}/\cos ({\theta }_1/2)))$.
     Since $\cos ({\theta }_1/2)=\cos (\pi /4)=1/\sqrt{2}$, the argument of $\arctan 2$ is $V_{11}/\cos ({\theta }_1/2)=(i/\sqrt{2})/(1/\sqrt{2})=i$.
     So, ${\theta }_0+{\theta }_2=2\arctan 2(1,0)=2(\pi /2)=\pi$.
   • ${\theta }_0-{\theta }_2=2\arctan 2(\text{Im}(V_{10}/\sin ({\theta }_1/2)),\text{Re}(V_{10}/\sin ({\theta }_1/2)))$.
     Since $\sin ({\theta }_1/2)=\sin (\pi /4)=1/\sqrt{2}$, the argument of $\arctan 2$ is $V_{10}/\sin ({\theta }_1/2)=(-i/\sqrt{2})/(1/\sqrt{2})=-i$.
     So, ${\theta }_0-{\theta }_2=2\arctan 2(-1,0)=2(-\pi /2)=-\pi$.

4. Solve for Angles:
   We have a system of two linear equations:
   ${\theta }_0+{\theta }_2=\pi$
   ${\theta }_0-{\theta }_2=-\pi$
   Adding them gives $2{\theta }_0=0⟹{\theta }_0=0$. Substituting back gives ${\theta }_2=\pi$.

5. Final Decomposition:

   $$H=e^{i\pi /2}{R}_z(\pi )R_y(\pi /2)R_z(0)$$

   Since $R_z(0)=I$, this simplifies to $H=e^{i\pi /2}{R}_z(\pi )R_y(\pi /2)$. This is a standard decomposition of the Hadamard gate.

5. General Euler Decompositions

The Z-Y-Z decomposition is not unique. Any sequence of three rotations about two non-parallel axes is sufficient. For example, X-Y-X or Z-X-Z are also valid Euler decompositions. These different forms can be useful depending on the native gate set of a particular quantum hardware.

The relationship between these different decompositions is a simple basis change, which can be implemented by a similarity transform. For instance, since $R_x(\theta )=H{R}_z(\theta )H$, we can convert a Z-Y-Z decomposition into a X-Y-X decomposition.

Conclusion

This lesson established a powerful and fundamental result in quantum computing.

Key Takeaways:

• Any single-qubit unitary operation $U$ can be decomposed into a global phase and a sequence of three rotation gates, for example: $U=e^{i\alpha }{R}_z({\theta }_2)R_y({\theta }_1)R_z({\theta }_0)$.
• This is a universality theorem for single-qubit gates. It means that if we can physically implement rotations around two different axes, we can construct any possible single-qubit gate.
• The decomposition is found via a constructive procedure: first, extract the global phase using the matrix determinant, then solve for the three Euler angles by equating the elements of the target SU(2) matrix with the rotation product matrix.

Preview of the Next Lesson:

We have now thoroughly covered the mathematical description of single- and two-qubit gates. We know what they are and how to build complex ones from simpler ones. The next logical step is to understand how we extract information from a quantum system after these operations have been applied.

In the next lesson, we will learn to calculate measurement outcomes and probabilities for quantum states in computational and arbitrary bases. This will bridge the gap between the abstract quantum state vector and the concrete, probabilistic outcomes observed in a real experiment.