Phase and T Gates

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

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.

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.

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.

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.

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.

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.

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.

10. Phase Gate as Square Root of Z

The Z gate:

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

The phase gate satisfies:

\[
S^2 = Z
\]

11. T Gate as Fourth Root of Z

The T gate satisfies:

\[
T^4 = Z
\]

and

\[
T^2 = S
\]

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.

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: \( T^\dagger = T^7 \)

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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

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.