U3 to Clifford Z rotation alternatives (#386)

U3 is defined by

$$U(\theta,\phi,\lambda) = R_z(\phi) R_y(\theta) R_z(\lambda)$$

When $\theta=0$, we get

$$U(0,\phi,\lambda) = R_z(\phi + \lambda)$$

If we restrict to angles in $[0,2\pi)$, then the Clifford operators are
$R_z(0)=I$, $R_z(\pi)=Z$, $R_z(\pi/2)=S$ $R_z(3\pi/2)=S^\dagger$.

Hence as long as $\phi + \lambda$ is a multiple of $\pi/2$ then we have
a Clifford, this PR ensures that this is the case by fixing the lookup
table and adding tests.

---------

Co-authored-by: Casey Duckering <[email protected]>

Author: ehua7365

src/bloqade/squin/rewrite/U3_to_clifford.py

@@ -126,6 +126,14 @@ def decompose_U3_gates(self, node: op.stmts.U3) -> Tuple[List[ir.Statement], ...
         if theta is None or phi is None or lam is None:
             return ()
 
+        # For U3(2*pi*n, phi, lam) = U3(0, 0, lam + phi) which is a Z rotation.
+        if np.isclose(np.mod(theta, math.tau), 0):
+            lam = lam + phi
+            phi = 0.0
+        elif np.isclose(np.mod(theta + np.pi, math.tau), 0):
+            lam = lam - phi
+            phi = 0.0
+
         theta_half_pi: int | None = self.resolve_angle(theta)
         phi_half_pi: int | None = self.resolve_angle(phi)
         lam_half_pi: int | None = self.resolve_angle(lam)

test/squin/rewrite/test_U3_to_clifford.py

@@ -59,6 +59,18 @@ def test_equiv():
     assert isinstance(get_stmt_at_idx(test_equiv, 5), op.stmts.S)
 
 
+def test_s_alternative():
+
+    @kernel
+    def test():
+        q = qubit.new(4)
+        oper = op.u(theta=0.0, phi=0.25 * math.tau, lam=0.0)
+        qubit.apply(oper, q[0])
+
+    SquinToCliffordTestPass(test.dialects)(test)
+    assert isinstance(get_stmt_at_idx(test, 5), op.stmts.S)
+
+
 def test_z():
 
     @kernel
@@ -81,6 +93,18 @@ def test():
     assert isinstance(get_stmt_at_idx(test, 15), op.stmts.Z)
 
 
+def test_z_alternative():
+
+    @kernel
+    def test():
+        q = qubit.new(4)
+        oper = op.u(theta=0.0, phi=0.5 * math.tau, lam=0.0)
+        qubit.apply(oper, q[0])
+
+    SquinToCliffordTestPass(test.dialects)(test)
+    assert isinstance(get_stmt_at_idx(test, 5), op.stmts.Z)
+
+
 def test_sdag():
 
     @kernel
@@ -104,6 +128,58 @@ def test_equiv():
     assert isinstance(get_stmt_at_idx(test_equiv, 6), op.stmts.Adjoint)
 
 
+def test_sdag_alternative_negative():
+
+    @kernel
+    def test():
+        q = qubit.new(4)
+        oper = op.u(theta=0.0, phi=-0.25 * math.tau, lam=0.0)
+        qubit.apply(oper, q[0])
+
+    SquinToCliffordTestPass(test.dialects)(test)
+    assert isinstance(get_stmt_at_idx(test, 5), op.stmts.S)
+    assert isinstance(get_stmt_at_idx(test, 6), op.stmts.Adjoint)
+
+
+def test_sdag_alternative():
+
+    @kernel
+    def test():
+        q = qubit.new(4)
+        oper = op.u(theta=0.0, phi=0.75 * math.tau, lam=0.0)
+        qubit.apply(oper, q[0])
+
+    SquinToCliffordTestPass(test.dialects)(test)
+    assert isinstance(get_stmt_at_idx(test, 5), op.stmts.S)
+    assert isinstance(get_stmt_at_idx(test, 6), op.stmts.Adjoint)
+
+
+def test_sdag_weird_case():
+
+    @kernel
+    def test():
+        q = qubit.new(4)
+        oper = op.u(theta=2 * math.tau, phi=0.7 * math.tau, lam=0.05 * math.tau)
+        qubit.apply(oper, q[0])
+
+    SquinToCliffordTestPass(test.dialects)(test)
+    assert isinstance(get_stmt_at_idx(test, 5), op.stmts.S)
+    assert isinstance(get_stmt_at_idx(test, 6), op.stmts.Adjoint)
+
+
+def test_sdag_weirder_case():
+
+    @kernel
+    def test():
+        q = qubit.new(4)
+        oper = op.u(theta=0.5 * math.tau, phi=0.05 * math.tau, lam=0.8 * math.tau)
+        qubit.apply(oper, q[0])
+
+    SquinToCliffordTestPass(test.dialects)(test)
+    assert isinstance(get_stmt_at_idx(test, 5), op.stmts.S)
+    assert isinstance(get_stmt_at_idx(test, 6), op.stmts.Adjoint)
+
+
 def test_sqrt_y():
 
     @kernel