Circuit Identities

Circuit identities replace a sequence of ideal operations by another with the same action. This lookup fixes exact matrix representatives so that a discarded phase does not silently become an observable relative phase. Derivations and compilation strategies belong to Circuit Optimization, Gate Decomposition, and Controlled Operations.

Pauli and Clifford conjugations

Use the Gate Library conventions. Products act rightmost first. All equalities below are exact; $R_j(\theta)=e^{-i\theta\sigma_j/2}$.

Identity	Use and qualification
$HXH=Z$, $HZH=X$, $HYH=-Y$	Exchange computational and Hadamard bases. The Y sign matters for observables and controlled operations.
$SXS^\dagger=Y$, $SYS^\dagger=-X$, $SZS^\dagger=Z$	Propagate Pauli frames through S. Reversing S and its adjoint reverses the rotation.
$XY=iZ$, $YZ=iX$, $ZX=iY$	Reduce Pauli products. Reversing distinct factors changes the sign.
$HR_z(\theta)H=R_x(\theta)$	Convert an axis rotation. The angle sign follows the declared exponential convention.
$XR_z(\theta)X=R_z(-\theta)$	Reverse a Z rotation. The Pauli factors reverse the angle, preserving the operator order.
$R_j(\alpha)R_j(\beta)=R_j(\alpha+\beta)$	Fuse same-axis rotations. Different axes do not obey this additive rule.

These relations follow from $Ue^AU^\dagger=e^{UAU^\dagger}$. They apply to the ideal gates; replacing two calibrated pulses by one may change the noise and scheduling even when the ideal unitary agrees.

CNOT propagation and swaps

Let $C=\operatorname{CNOT}_{1\to2}$, $X_1=X\otimes I$, and similarly for other Pauli factors. Since $C=C^\dagger$, the following table is also an error-propagation lookup.

Input Pauli P	Conjugated Pauli CPC	Interpretation
$X_1$	$X_1X_2$	A control bit flip propagates to the target.
$Z_1$	$Z_1$	A control phase flip remains on the control.
$X_2$	$X_2$	A target bit flip remains on the target.
$Z_2$	$Z_1Z_2$	A target phase flip propagates to the control.

For example, $CX_1=X_1X_2C$. The extra target X cannot be omitted when commuting the control X through the gate. The table supplies generators; products determine the remaining Pauli cases with their phases.

Write $C_{\mathrm{rev}}=\operatorname{CNOT}_{2\to1}$ for the reverse control direction. Useful exact rewrites are

$$\begin{aligned} \operatorname{CZ}&=(I\otimes H)\,C\,(I\otimes H),\\ C_{\mathrm{rev}}&=(H\otimes H)\,C\,(H\otimes H),\\ \operatorname{SWAP}&=C\,C_{\mathrm{rev}}\,C. \end{aligned}$$

CNOT gates with the same control and distinct targets commute, as do CNOTs with distinct controls and the same target. Adjacent CNOTs in a chain generally do not commute. Disjoint support is sufficient for commutation but is not necessary.

Controlled phase decomposition

For $P(\phi)=\operatorname{diag}(1,e^{i\phi})$, an exact identity using two CNOTs is

$$\operatorname{CP}(\phi) ={}[P(\phi/2)\otimes P(\phi/2)] \times C\,[I\otimes P(-\phi/2)]\,C.$$

On a basis state $\lvert a,b\rangle$, the accumulated phase is

$$\frac{\phi}{2}\,[a+b-(a\oplus b)]=\phi ab.$$

Checking four basis states therefore verifies both the action and its exact phase. Substituting $R_z$ for every P without retaining the corresponding global factors changes exact equality, even though a separately executed channel may remain equivalent.

Teleportation as an identity channel

Prepare a Bell pair on qubits 2 and 3, apply CNOT from input 1 to 2 and H on 1, and measure qubits 1 and 2 in the computational basis. For outcomes $(a,b)$, the unnormalized state on 3 is

$$\frac12 X^bZ^a\lvert\psi\rangle.$$

The exact inverse correction is $Z^aX^b$. The alternative order $X^bZ^a$ differs by the branch-global sign $(-1)^{ab}$ and yields the same corrected density operator when the outcomes are classical. Forgetting the outcomes instead produces the maximally mixed state for an arbitrary input qubit. The communication and resource accounting are developed in Quantum Teleportation.

References

• Nielsen, Michael A., and Isaac L. Chuang, Quantum Computation and Quantum Information, 10th anniversary edition, Cambridge University Press, 2010, doi:10.1017/CBO9780511976667 — circuit identities, Clifford propagation, and teleportation.