# T† gate (inverse T gate)

**T† gate** (also written T-dagger or T inverse) applies a $-45°$ phase to the $|1\rangle$ state. It is the inverse of the [[quantum-gate-t|T gate]], with $T^\dagger T = I$ and $T^\dagger = T^7$ (since $T^8 = I$). Like the T gate, T† is non-Clifford and essential for universal quantum computation.

Matrix:

$$T^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & e^{-i\pi/4} \end{pmatrix}$$

Action: $T^\dagger|0\rangle = |0\rangle$, $T^\dagger|1\rangle = e^{-i\pi/4}|1\rangle$ (adds $-\pi/4$ phase to $|1\rangle$).

## Properties

- **Inverse of T**: $T^\dagger T = T T^\dagger = I$
- **Self-adjoint modulo phase**: $(T^\dagger)^\dagger = T$
- **$(T^\dagger)^8 = I$**: applying T† eight times returns to identity
- **$(T^\dagger)^4 = Z$**: four T† gates equal one Z gate
- **$(T^\dagger)^2 = S^\dagger$**: two T† gates equal one S† gate
- **Non-Clifford**: T† is not in the Clifford group; enables universal computation

## Relation to T and phase gates

$$T^\dagger = T^{-1} = T^7 = R_Z(-\pi/4) = e^{i\pi Z/8}$$

$$(T^\dagger)^2 = S^\dagger = R_Z(-\pi/2)$$

$$(T^\dagger)^4 = Z = R_Z(\pi)$$

$$T T^\dagger = I$$

The hierarchy: $T^\dagger$ ↔ $S^\dagger$ ↔ $Z$ mirrors the $T$ → $S$ → $Z$ progression.

## Expense and magic state distillation

- **Non-Clifford**: T† cannot be simulated classically, like T
- **Magic state distillation**: fault-tolerant systems require distillation; one logical T† costs ~1000 physical qubits (same overhead as T)
- **T-count**: optimization includes both T and T† gates; total count is primary cost metric
- **Interchangeability**: T and T† have identical cost and resource requirements

## Uses

- **Universal computation**: T† + Clifford gates form universal set (as does T + Clifford)
- **Circuit synthesis**: undo T gate phases added earlier; phase corrections
- **Variational algorithms**: T† gates appear in some ansatze but should be minimized
- **Solovay-Kitaev approximation**: T† gates required for arbitrary unitary approximation

## Implementation

- **Superconducting qubits**: RZ($-\pi/4$) via microwave pulse or virtual rotation; ~20 ns
- **Trapped ions**: controlled phase via detuned laser (opposite phase from T)
- **Photonic**: phase shifter set to $-\pi/4$
- **Cost note**: typically ~20 ns on NISQ hardware, but bottleneck for fault-tolerant systems (same as T)

## Relations

- [[quantum-gate-phase|Phase gates]]: category containing T† and other phase gates
- [[quantum-gate-t|T gate]]: inverse relationship; $T^\dagger = T^{-1}$
- [[quantum-gate-s-dagger|S† gate]]: $(T^\dagger)^2 = S^\dagger$; intermediate in phase gate hierarchy
- [[quantum-gate-z|Z gate]]: $(T^\dagger)^4 = Z$; related via phase accumulation
- [[quantum-gate-rz|RZ gate]]: $T^\dagger = R_Z(-\pi/4)$
- [[quantum-gate-clifford|Clifford gates]]: T† is non-Clifford, unlike S and S†
- [[quantum-gate-single-qubit|Single-qubit gates]]: fundamental building block