Table of Contents

1. Introduction
2. Basics of Controlled Gates
3. The Concept of Control and Target Qubits
4. Controlled Gates as Conditional Operations
5. Matrix Representation of CNOT
6. Action of CNOT on Basis States
7. Creating Entanglement with CNOT
8. CNOT in Bell State Preparation
9. Reversibility and Unitarity
10. Controlled-Z and Its Relation to CNOT
11. Circuit Diagram Notation for Controlled Gates
12. Multi-Controlled Gates
13. The Toffoli Gate (CCNOT)
14. Matrix Representation of Toffoli
15. Toffoli as Universal Classical Gate
16. Toffoli in Reversible Computation
17. Use of Toffoli in Error Correction
18. Quantum Circuits with Multiple CNOTs
19. Gate Decomposition of Toffoli
20. Physical Implementation Challenges
21. Gate Cost and Depth Considerations
22. CNOT in Quantum Teleportation
23. Role in Quantum Fourier Transform
24. Controlled Gates and Quantum Universality
25. Conclusion

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1. Introduction

Controlled gates are essential in quantum computing because they allow operations to be conditioned on the state of other qubits. Two of the most important controlled gates are the CNOT (Controlled-NOT) and the Toffoli (Controlled-Controlled-NOT) gate. These gates are critical for entanglement, logical operations, and universal quantum computation.

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2. Basics of Controlled Gates

A controlled gate applies a quantum operation to a target qubit only if a control qubit is in a particular state, typically \( |1\rangle \). This allows for conditional logic in quantum circuits.

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3. The Concept of Control and Target Qubits

In a controlled gate:

• Control qubit: Determines whether the operation is applied.
• Target qubit: Receives the operation (e.g., a bit flip).

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4. Controlled Gates as Conditional Operations

For the CNOT gate, if the control qubit is \( |1\rangle \), the target qubit is flipped:

\[
\text{CNOT}|00\rangle = |00\rangle,\quad \text{CNOT}|10\rangle = |11\rangle
\]

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5. Matrix Representation of CNOT

The 4×4 matrix representation in the basis \( \{|00\rangle, |01\rangle, |10\rangle, |11\rangle\} \) is:

\[
\text{CNOT} =
\begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 0 & 1 \
0 & 0 & 1 & 0 \
\end{bmatrix}
\]

---

6. Action of CNOT on Basis States

\[
\text{CNOT}|00\rangle = |00\rangle \
\text{CNOT}|01\rangle = |01\rangle \
\text{CNOT}|10\rangle = |11\rangle \
\text{CNOT}|11\rangle = |10\rangle
\]

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7. Creating Entanglement with CNOT

Apply CNOT after a Hadamard:

\[
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
\]

Then:

\[
\text{CNOT}(H \otimes I)|00\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)
\]

This is the Bell state \( |\Phi^+\rangle \), an entangled state.

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8. CNOT in Bell State Preparation

Bell states are key resources in quantum communication and are created using Hadamard + CNOT:

\[
|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)
\]

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9. Reversibility and Unitarity

CNOT and Toffoli are reversible and unitary, making them ideal for reversible logic and quantum algorithms.

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10. Controlled-Z and Its Relation to CNOT

Controlled-Z is another 2-qubit gate:

\[
\text{CZ}|11\rangle = -|11\rangle
\]

It is related to CNOT by conjugation with Hadamard on the target:

\[
\text{CNOT} = (I \otimes H) \cdot \text{CZ} \cdot (I \otimes H)
\]

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11. Circuit Diagram Notation for Controlled Gates

• CNOT: dot on control, ⊕ on target
• Toffoli: two dots on control, ⊕ on target

These are standard in quantum circuit diagrams.

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12. Multi-Controlled Gates

Controlled gates can have multiple control qubits:

• CCNOT (Toffoli): flips target if both controls are \( |1\rangle \)
• General multi-controlled-U gates are built from elementary gates

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13. The Toffoli Gate (CCNOT)

Applies NOT to the target qubit only if both control qubits are \( |1\rangle \):

\[
\text{Toffoli}|110\rangle = |111\rangle, \quad \text{Toffoli}|100\rangle = |100\rangle
\]

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14. Matrix Representation of Toffoli

An 8×8 matrix, acts trivially on 6 states, flips last two:

\[
\text{Toffoli}|110\rangle = |111\rangle, \quad \text{Toffoli}|111\rangle = |110\rangle
\]

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15. Toffoli as Universal Classical Gate

Toffoli can simulate all classical logic gates:

• AND, OR, NOT
• NAND and NOR (universal gates)
  Thus, it forms the basis for reversible classical computation.

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16. Toffoli in Reversible Computation

Reversible computing uses Toffoli gates to minimize energy dissipation (Landauer’s principle). It also aids in designing quantum algorithms with classical logic layers.

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17. Use of Toffoli in Error Correction

Toffoli is used in error correction circuits:

• Steane code
• Shor code
• Syndrome extraction

It helps with conditional logic on ancilla bits.

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18. Quantum Circuits with Multiple CNOTs

CNOTs are used extensively in:

• Quantum Fourier Transform
• Grover’s algorithm
• Quantum addition and multiplication

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19. Gate Decomposition of Toffoli

Toffoli can be decomposed into:

• 6 CNOT gates
• Single-qubit gates (T, H, S)
  This is required for implementation on hardware with only 1- and 2-qubit gates.

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20. Physical Implementation Challenges

• CNOT is native in many platforms
• Toffoli requires decompositions
• Fidelity drops with more control lines

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21. Gate Cost and Depth Considerations

Toffoli increases circuit depth and error. Optimizing:

• T-count
• Gate depth
• Entangling gate count

is crucial in NISQ-era devices.

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22. CNOT in Quantum Teleportation

CNOT plays a vital role in:

• Entanglement preparation
• Bell measurements
• Teleportation protocol

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23. Role in Quantum Fourier Transform

Controlled phase gates and CNOTs are used to entangle qubits for frequency-domain information processing in QFT.

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24. Controlled Gates and Quantum Universality

CNOT, together with single-qubit gates (H, T), forms a universal gate set for quantum computing. Toffoli adds classical universality.

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25. Conclusion

Controlled gates, especially CNOT and Toffoli, are indispensable for both classical logic simulation and quantum logic construction. They enable entanglement, conditional operations, error correction, and serve as the backbone of many quantum algorithms. Understanding their structure and function is fundamental to mastering quantum circuit design.

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Table of Contents

1. Introduction
2. Phase and T Gates in Quantum Circuits
3. Importance of Phase in Quantum Computation
4. Matrix Representations
5. The Phase (S) Gate
6. The T Gate (π/8 Gate)
7. Action on Computational Basis States
8. Geometric Interpretation on the Bloch Sphere
9. Phase Accumulation and Relative Phase
10. Phase Gate as Square Root of Z
11. T Gate as Fourth Root of Z
12. Powering and Composition
13. Unitary and Hermitian Properties
14. Eigenvalues and Eigenvectors
15. Use in Superposition and Interference
16. Phase Kickback Effect
17. Clifford Group and Phase Gate
18. Non-Clifford Nature of T Gate
19. Role in Quantum Universality
20. Solovay-Kitaev Theorem and Gate Approximations
21. T Gate in Fault-Tolerant Quantum Computing
22. T-Gate Distillation
23. Native Implementations and Hardware Considerations
24. Use in Quantum Algorithms and QFT
25. Conclusion

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1. Introduction

In quantum computing, gates that manipulate the phase of qubits are as important as those that flip their state. Among these, the Phase gate (S) and the T gate are essential single-qubit gates, influencing interference and contributing to quantum circuit universality.

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2. Phase and T Gates in Quantum Circuits

These gates don’t alter the probability amplitudes’ magnitudes but change their phases, which affects how quantum states interfere.

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3. Importance of Phase in Quantum Computation

Unlike classical computing, where global phase is irrelevant, relative phase in quantum mechanics determines interference patterns, which are essential for quantum algorithms like Grover’s and Shor’s.

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4. Matrix Representations

• Phase (S) gate:
  \[
  S = \begin{bmatrix} 1 & 0 \ 0 & i \end{bmatrix}
  \]
• T gate:
  \[
  T = \begin{bmatrix} 1 & 0 \ 0 & e^{i\pi/4} \end{bmatrix}
  \]

---

5. The Phase (S) Gate

Applies a phase of \( \frac{\pi}{2} \) to the \( |1\rangle \) state:

\[
S|0\rangle = |0\rangle, \quad S|1\rangle = i|1\rangle
\]

This shifts the qubit’s phase without changing its magnitude.

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6. The T Gate (π/8 Gate)

Applies a \( \frac{\pi}{4} \) phase to \( |1\rangle \):

\[
T|0\rangle = |0\rangle, \quad T|1\rangle = e^{i\pi/4}|1\rangle
\]

It is also called the π/8 gate and is crucial for universality.

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7. Action on Computational Basis States

Both gates leave \( |0\rangle \) unchanged and modify only the phase of \( |1\rangle \), making them diagonal in the computational basis.

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8. Geometric Interpretation on the Bloch Sphere

• The S and T gates rotate the state around the Z-axis.
• They shift phase without changing probability amplitudes.
• The effect becomes visually noticeable in superpositions.

---

9. Phase Accumulation and Relative Phase

In a superposition:

\[
|\psi\rangle = \alpha|0\rangle + \beta|1\rangle
\]

After applying T:

\[
T|\psi\rangle = \alpha|0\rangle + \beta e^{i\pi/4}|1\rangle
\]

Relative phase changes interference and measurement outcomes.

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10. Phase Gate as Square Root of Z

The Z gate:

\[
Z = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}
\]

The phase gate satisfies:

\[
S^2 = Z
\]

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11. T Gate as Fourth Root of Z

The T gate satisfies:

\[
T^4 = Z
\]

and

\[
T^2 = S
\]

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12. Powering and Composition

Successive applications rotate the qubit further:

\[
T^n = \begin{bmatrix} 1 & 0 \ 0 & e^{in\pi/4} \end{bmatrix}
\]

This allows fine control over relative phases in quantum circuits.

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13. Unitary and Hermitian Properties

• S gate: Unitary but not Hermitian
• T gate: Unitary, not Hermitian
• Inverse of S: \( S^\dagger = S^3 \)
• Inverse of T: \( T^\dagger = T^7 \)

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14. Eigenvalues and Eigenvectors

Both S and T have eigenvalues on the unit circle (complex modulus 1). They act trivially on eigenstates but induce critical changes in superpositions.

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15. Use in Superposition and Interference

In states like:

\[
|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
\]

S and T introduce relative phase, changing measurement probabilities and enabling quantum speedups.

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16. Phase Kickback Effect

Used in controlled gates and phase estimation, phase kickback transfers phase information from target to control qubits — crucial in algorithms like Shor’s.

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17. Clifford Group and Phase Gate

S is part of the Clifford group, which includes gates like:

• H, X, Z, S, CNOT

Clifford circuits alone can be efficiently simulated classically.

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18. Non-Clifford Nature of T Gate

The T gate is non-Clifford, meaning:

• It enables universality when added to Clifford gates.
• Necessary for fault-tolerant universal quantum computing.

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19. Role in Quantum Universality

The set:

\[
\{H, T, \text{CNOT}\}
\]

is universal — it can approximate any unitary operation on any number of qubits to arbitrary precision.

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20. Solovay-Kitaev Theorem and Gate Approximations

This theorem guarantees that any quantum gate can be approximated efficiently with sequences of universal gates like H, T, and CNOT, with high fidelity.

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21. T Gate in Fault-Tolerant Quantum Computing

T gates are more expensive than Clifford gates in error-corrected computation. Thus, optimizing T-count is a major goal in quantum compiler design.

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22. T-Gate Distillation

A method to produce high-fidelity T gates from noisy ones. Used in:

• Magic state distillation
• Fault-tolerant circuits based on surface codes

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23. Native Implementations and Hardware Considerations

Different quantum hardware platforms implement S and T using:

• Pulse sequences
• Microwave control
• Optical phase shifts

Fidelity and gate time depend on technology.

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24. Use in Quantum Algorithms and QFT

S and T gates are crucial in:

• Quantum Fourier Transform (QFT): for precise phase rotations
• Phase estimation algorithms
• Grover’s and Shor’s algorithms

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25. Conclusion

The Phase (S) and T gates may seem simple — applying phase shifts to \( |1\rangle \) — but they are fundamental to quantum computing. They influence interference, enable universality, and are indispensable in real-world algorithms and fault-tolerant designs. Mastery of these gates is key to understanding the power of quantum logic.

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