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 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 $|1\rangle$).

• 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

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

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

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

$$T T^\dagger = I$$

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

• 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

• 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

• 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)

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