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new 598a9ad36 docs: improve website math format (#1131)
598a9ad36 is described below
commit 598a9ad36886b3d010245a04eff13fea6ab59950
Author: Tim Hsiung <[email protected]>
AuthorDate: Sat Mar 7 10:02:28 2026 +0800
docs: improve website math format (#1131)
---
docs/qumat/basic-gates.md | 30 ++++++++++++++----------------
1 file changed, 14 insertions(+), 16 deletions(-)
diff --git a/docs/qumat/basic-gates.md b/docs/qumat/basic-gates.md
index b39aa4760..60dfd8819 100644
--- a/docs/qumat/basic-gates.md
+++ b/docs/qumat/basic-gates.md
@@ -10,8 +10,6 @@ The Hadamard gate, denoted as the H-gate, is used to create
superposition states
$$H|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$$
-
-
This gate is crucial in quantum algorithms like Grover's and quantum
teleportation.
## CNOT Gate (Controlled-X Gate)
@@ -26,24 +24,24 @@ The SWAP gate exchanges the states of two qubits. When
applied, it swaps the sta
## Pauli Y Gate (Y-Gate)
The Pauli Y gate introduces complex phase shifts along with a bit-flip
operation. It can be thought of as a combination of bit-flip and phase-flip
gates. Mathematically:
-$$Y|0\rangle = i|1\rangle$$
-$$Y|1\rangle = -i|0\rangle$$
+- $$Y|0\rangle = i|1\rangle$$
+- $$Y|1\rangle = -i|0\rangle$$
It's essential in quantum error correction and quantum algorithms.
## Pauli Z Gate (Z-Gate)
The Pauli Z gate introduces a phase flip without changing the qubit's state.
It leaves $|0\rangle$ unchanged and transforms $|1\rangle$ to $-|1\rangle$.
Mathematically:
-$$Z|0\rangle = |0\rangle$$
-$$Z|1\rangle = -|1\rangle$$
+- $$Z|0\rangle = |0\rangle$$
+- $$Z|1\rangle = -|1\rangle$$
It's used for measuring the phase of a qubit.
## T-Gate (π/8 Gate)
The T-Gate applies a **π/4 phase shift** to the qubit. It is essential for
quantum computing because it, along with the Hadamard and CNOT gates, allows
for **universal quantum computation**. Mathematically:
-$$T|0\rangle = |0\rangle$$
-$$T|1\rangle = e^{i\pi/4} |1\rangle$$
+- $$T|0\rangle = |0\rangle$$
+- $$T|1\rangle = e^{i\pi/4} |1\rangle$$
## CSWAP Gate (Controlled-SWAP / Fredkin Gate)
The CSWAP gate, also known as the **Fredkin gate**, is a three-qubit gate that
conditionally swaps the states of two target qubits based on the state of a
control qubit. If the control qubit is in the $|1\rangle$ state, it swaps the
states of the two target qubits; otherwise, it leaves them unchanged.
@@ -92,17 +90,17 @@ The U gate can be decomposed into rotations around the Z,
Y, and Z axes:
$$U(\theta, \phi, \lambda) = R_z(\phi) \cdot R_y(\theta) \cdot R_z(\lambda)$$
This decomposition shows that the U gate applies:
-1. A rotation by λ around the Z-axis
-2. A rotation by θ around the Y-axis
-3. A rotation by φ around the Z-axis
+1. A rotation by $\lambda$ around the Z-axis
+2. A rotation by $\theta$ around the Y-axis
+3. A rotation by $\phi$ around the Z-axis
### Special Cases
-- **Identity**: U(0, 0, 0) = I
-- **Pauli X**: U(π, 0, π) = X
-- **Pauli Y**: U(π, π/2, π/2) = Y
-- **Pauli Z**: U(0, 0, π) = Z
-- **Hadamard**: U(π/2, 0, π) = H
+- **Identity**: $U(0, 0, 0) = I$
+- **Pauli X**: $U(\pi, 0, \pi) = X$
+- **Pauli Y**: $U(\pi, \pi/2, \pi/2) = Y$
+- **Pauli Z**: $U(0, 0, \pi) = Z$
+- **Hadamard**: $U(\pi/2, 0, \pi) = H$
This gate is particularly useful in parameterized quantum circuits and
variational quantum algorithms where you need to optimize over all possible
single-qubit operations.
