Gates · Phase gate
T† gate
The Hermitian conjugate (and inverse) of the non-Clifford T gate, applying a -π/4 phase to |1⟩ and appearing throughout fault-tolerant circuits as the T-count partner of T.
Exact & formalUpdated
A small worked example, computed in your browser. It is not a result from this record's papers.
T†, T turned the other way
One qubit, put into |+⟩ = (|0⟩ + |1⟩)/√2 first, because on |0⟩ or |1⟩ alone this gate changes nothing a measurement or another gate can ever detect.
Step 1 of 2 · H
Probability of each outcome after this step
- 050%
- 150%
Puts the qubit into an equal superposition. Without this there is no second amplitude for the gate's phase to be measured against, and a phase on the whole state is not observable.
Probability spreads from 1 outcome to 2. All 2 are equally likely.
Multiplies the |1⟩ half of the state by a phase of −π/4, printed as 7π/4, and leaves |0⟩ alone. It is what undoes a T applied earlier, and it costs a fault-tolerant circuit exactly as much as a T does. The two outcomes stay equally likely — the difference is carried entirely in the phase between them, which is what the panel beside the bars shows.
The outcome probabilities do not move at all. What this step changes is phase — 2 distinct relative phases across the state now — which no measurement can see until a later step turns it back into probability.
Readout Measuring gives 0 or 1, each half the time — exactly as it did before the gate. A single measurement of this qubit cannot detect this gate at all. Its effect becomes visible only when the phase is turned back into probability by a later interference step, which is what every algorithm that uses it does.
Open in Studio- Steps1
- Wires1
- Qubits1
- Depth1 gate
- Parameternone (fixed -π/4 phase)
- Papers2
T† matters because fault-tolerant compilers count T and T† together as the expensive resource (T-count): the Toffoli decomposition in this catalog, for instance, uses both 4 T and 3 T† gates, and each costs the same magic-state budget.
Takes1 qubitWhat joins here
Takes a 1-qubit register and states no assumption about what is on it. That is the whole condition: anything returning 1 qubits can feed it.
15 entries meet this end. Named below.
Returns1 qubitWhat joins here
Returns a 1-qubit register a next stage can take. This is the end that joins.
13 entries meet this end. Named below.
A unitary. The same register goes in and comes out, and this entry states no assumption about what is on it — which is what lets anything of the same width feed it. See all 29 →
Definition
is the conjugate transpose of , and since is unitary, is also its inverse: .
Action on basis states
Applied to , rotates the equatorial phase by , the mirror image of what does.
Key identities
- and : the phase ladder steps down in increments exactly as steps up.
- up to global phase.
- since , so seven applications of implement one when a compiler's native gate set exposes only .
- is not Clifford: conjugating by does not return a Pauli operator, which is exactly why (or ) is needed for universal fault-tolerant quantum computation.
Takes1 qubitWhat joins here
Takes a 1-qubit register and states no assumption about what is on it. That is the whole condition: anything returning 1 qubits can feed it.
15 entries meet this end. Named below.
Returns1 qubitWhat joins here
Returns a 1-qubit register a next stage can take. This is the end that joins.
13 entries meet this end. Named below.
A unitary. The same register goes in and comes out, and this entry states no assumption about what is on it — which is what lets anything of the same width feed it. See all 29 →
- Qubits
- 1
- Depth
- 1 gate
- Parameter
- none (fixed -π/4 phase)
- Matrix
- diag(1, e^{-iπ/4})
- Inverse of
- T (T·T† = I)
- Power
- (T†)² = S†, (T†)⁸ = I
- Gate family
- Non-Clifford
T†on q[0]
Quantum vs classical
Classical baseline
Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.
Quantum claim
The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.
How to compare
Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.
Nobody has reviewed this record for gaps yet.
OPENQASM 3.0;
include "stdgates.inc";
qubit q;
tdg q;Related entries
- T phase gate
Gates
- S† (inverse phase) gate
Gates
- Magic T state
Operators
- Toffoli (CCX) gate
Gates
References
- Quantum Computation and Quantum Information: 10th Anniversary Edition
Michael A. Nielsen and Isaac L. Chuang · 2010
Standard reference for gate matrices, the Clifford/non-Clifford distinction, and controlled-gate constructions used throughout this record.
- OpenQASM 3: A broader and deeper quantum assembly language
Andrew W. Cross, Ali Javadi-Abhari, Thomas Alexander, Niel de Beaudrap, Lev S. Bishop, Steven Heidel, Colm A. Ryan, Prasahnt Sivarajah, John Smolin, Jay M. Gambetta, Blake R. Johnson · 2021
Defines the standard gate library (id, sdg, tdg, cy, crz, rxx, rzz, ccx, cx…) used natively by the code snippets on this record.