# Phase gates

**Phase gates** are single-qubit gates that apply phase shifts to the computational basis states without changing their amplitudes. They are rotations around the z-axis on the Bloch sphere ($R_Z(\theta)$ gates with specific angles). Phase gates are diagonal in the computational basis and leave $|0\rangle$ unchanged while applying phases to $|1\rangle$. They form a family parameterized by angle $\theta$.

## Common phase gates

- **[[quantum-gate-p|P gate]]**: parameterized phase gate; applies angle $\theta$ phase; $P(\theta) = R_Z(\theta)$; special cases include S, T, Z
- **[[quantum-gate-s|S gate]]**: applies $\pi/2$ phase; $S = R_Z(\pi/2)$; $S^4 = I$
- **[[quantum-gate-s-dagger|S† gate]]**: applies $-\pi/2$ phase; $S^\dagger = R_Z(-\pi/2)$; inverse of S
- **[[quantum-gate-t|T gate]]**: applies $\pi/4$ phase; $T = R_Z(\pi/4)$; $T^8 = I$
- **[[quantum-gate-t-dagger|T† gate]]**: applies $-\pi/4$ phase; $T^\dagger = R_Z(-\pi/4)$; inverse of T
- **[[quantum-gate-z|Z gate]]**: applies $\pi$ phase (180°); $Z = R_Z(\pi)$; $Z^2 = I$

## Matrix form

All phase gates are diagonal in the computational basis:

$$\text{Phase gate}(\theta) = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\theta} \end{pmatrix}$$

or equivalently (using the standard rotation convention):

$$R_Z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$$

The difference is a global phase factor $e^{-i\theta/2}$ that doesn't affect measurement.

## Properties

- **Diagonal**: phase gates do not entangle or mix basis states
- **No bit flip**: leave $|0\rangle$ unchanged; only add phase to $|1\rangle$
- **Composition**: $P(\theta_1) P(\theta_2) = P(\theta_1 + \theta_2)$ (phase accumulation)
- **Inverse**: $P(\theta)^\dagger = P(-\theta)$
- **Periodicity**: $P(\theta + 2\pi) = P(\theta)$ (global phase)
- **Clifford structure**: S and S† are [[quantum-gate-clifford|Clifford gates]]; T and T† are non-Clifford

## Relation to rotations

Phase gates are special cases of [[quantum-gate-rz|RZ rotation gates]]:

$$S = R_Z(\pi/2), \quad S^\dagger = R_Z(-\pi/2), \quad T = R_Z(\pi/4), \quad T^\dagger = R_Z(-\pi/4), \quad Z = R_Z(\pi)$$

More generally, $P(\theta) = R_Z(\theta)$ (or $R_Z(2\theta)$ depending on convention). Phase gates are "free" on superconducting qubits when implemented as RZ rotations via reference frame adjustment.

## Relation to Pauli Z

Phase gates form power hierarchies:

**Forward**: $T^2 = S$, $S^2 = Z$, $T^4 = Z$

**Inverse**: $(T^\dagger)^2 = S^\dagger$, $(S^\dagger)^2 = Z$, $(T^\dagger)^4 = Z$

This hierarchical structure extends to arbitrary phase angles: $\cdots \to T \leftrightarrow T^\dagger \to S \leftrightarrow S^\dagger \to Z$

## Conjugation by phase gates

How phase gates conjugate Pauli operators:

- $S X S^\dagger = Y$, $S Y S^\dagger = -X$, $S Z S^\dagger = Z$ (S rotates X → Y)
- $T X T^\dagger = \frac{1}{\sqrt{2}}(X + Y)$ (T breaks Pauli closure)
- $Z X Z^\dagger = -X$, $Z Y Z^\dagger = -Y$, $Z Z Z^\dagger = Z$ (Z phase-flips X and Y)

## Clifford vs non-Clifford

- **Clifford phase gates** (S, S†): preserve Pauli group under conjugation; efficient stabilizer simulation
- **Non-Clifford phase gates** (T, T†): break Pauli closure ($S X S^\dagger \notin \{\pm I, \pm X, \pm Y, \pm Z\}$); enable universal quantum computation

The T gate is the minimal non-Clifford addition to Clifford gates for universality.

## Uses

- **State preparation**: rotate basis states to arbitrary phases on Bloch sphere
- **Circuit synthesis**: decompose arbitrary unitaries into phase and bit-flip layers
- **Variational algorithms**: parameterized phase gates in ansatze (VQE, QAOA)
- **Algorithm implementation**: phase corrections in quantum algorithms
- **Measurement basis rotation**: combined with other gates to measure in different bases
- **Error correction**: phase syndrome measurements in stabilizer codes
- **Magic state distillation**: T gates require resource-expensive distillation in fault-tolerant schemes

## Implementation

- **Superconducting qubits**: RZ($\theta$) is typically virtual (reference frame adjustment); free cost
- **Trapped ions**: detuned laser pulse applies phase without excitation
- **Photonic**: optical phase shifter or wave plate set to angle $\theta$
- **Native vs virtual**: S and T gates often virtual on superconducting platforms, requiring no physical pulse

## Clifford+T decomposition

Quantum circuits decompose into Clifford layers plus T/T† gates. The **T-count** is the primary cost metric in fault-tolerant quantum computing:

- Clifford phase gates (S, S†): nearly free (polynomial overhead)
- Non-Clifford phase gates (T, T†): expensive (magic state distillation, ~100-1000 physical qubits per logical T or T†)

Many quantum algorithms use phase gates extensively; optimizing T-count is central to compiling for FTQC.

## Relations

- [[quantum-gate-p|P gate]]: parameterized phase gate; S, T, and Z are special cases
- [[quantum-gate-s|S gate]] and [[quantum-gate-s-dagger|S† gate]]: fundamental Clifford phase gates
- [[quantum-gate-t|T gate]] and [[quantum-gate-t-dagger|T† gate]]: non-Clifford phase gates; enable universal quantum computation
- [[quantum-gate-z|Z gate]]: Pauli Z is a special case of phase gate at $\theta = \pi$
- [[quantum-gate-rz|RZ gate]]: phase gates are special cases of rotation gates
- [[quantum-gate-clifford|Clifford gates]]: S and S† are Clifford; T and T† are non-Clifford
- [[quantum-gate-rotation|Rotation gates]]: phase gates are z-axis rotations
- Error correction: phase syndrome measurements in stabilizer codes
- Fault tolerance: T-count optimization for fault-tolerant quantum computing