diff --git a/mlir/include/mlir/Dialect/Complex/IR/ComplexOps.td b/mlir/include/mlir/Dialect/Complex/IR/ComplexOps.td
--- a/mlir/include/mlir/Dialect/Complex/IR/ComplexOps.td
+++ b/mlir/include/mlir/Dialect/Complex/IR/ComplexOps.td
@@ -351,6 +351,8 @@
     %a = complex.mul %b, %c : complex<f32>
     ```
   }];
+
+  let hasFolder = 1;
 }
 
 //===----------------------------------------------------------------------===//
diff --git a/mlir/lib/Dialect/Complex/IR/ComplexOps.cpp b/mlir/lib/Dialect/Complex/IR/ComplexOps.cpp
--- a/mlir/lib/Dialect/Complex/IR/ComplexOps.cpp
+++ b/mlir/lib/Dialect/Complex/IR/ComplexOps.cpp
@@ -248,6 +248,41 @@
   return {};
 }
 
+//===----------------------------------------------------------------------===//
+// MulOp
+//===----------------------------------------------------------------------===//
+
+OpFoldResult MulOp::fold(FoldAdaptor adaptor) {
+  if (auto constant = getRhs().getDefiningOp<ConstantOp>()) {
+    auto arrayAttr = constant.getValue();
+    auto real = llvm::cast<FloatAttr>(arrayAttr[0]).getValue();
+    auto imag = llvm::cast<FloatAttr>(arrayAttr[1]).getValue();
+
+    if (imag.isZero()) {
+      // complex.mul(a, complex.constant<0.0, 0.0>)
+      //   -> complex.constant<0.0, 0.0>
+      if (real.isZero())
+        return getRhs();
+
+      // complex.mul(a, complex.constant<1.0, 0.0>) -> a
+      switch (real.getSizeInBits(real.getSemantics())) {
+      case 64:
+        if (real == APFloat(1.0))
+          return getLhs();
+        return {};
+      case 32:
+        if (real == APFloat(1.0f))
+          return getLhs();
+        return {};
+      default:
+        return {};
+      }
+    }
+  }
+
+  return {};
+}
+
 //===----------------------------------------------------------------------===//
 // TableGen'd op method definitions
 //===----------------------------------------------------------------------===//
diff --git a/mlir/test/Dialect/Complex/canonicalize.mlir b/mlir/test/Dialect/Complex/canonicalize.mlir
--- a/mlir/test/Dialect/Complex/canonicalize.mlir
+++ b/mlir/test/Dialect/Complex/canonicalize.mlir
@@ -177,3 +177,36 @@
   // CHECK-NEXT: return %[[NEG]]
   return %im : f32
 }
+
+// CHECK-LABEL: func @mul_zero
+//  CHECK-SAME: (%[[ARG0:.*]]: f32, %[[ARG1:.*]]: f32) -> complex<f32>
+func.func @mul_zero(%arg0: f32, %arg1: f32) -> complex<f32> {
+  %create = complex.create %arg0, %arg1: complex<f32>  
+  %zero = complex.constant [0.0 : f32, 0.0 : f32] : complex<f32>
+  %mul = complex.mul %create, %zero : complex<f32>
+  // CHECK: %[[ZERO:.*]] = complex.constant [0.000000e+00 : f32, 0.000000e+00 : f32] : complex<f32>
+  // CHECK-NEXT: return %[[ZERO]]
+  return %mul : complex<f32>
+}
+
+// CHECK-LABEL: func @mul_one_f32
+//  CHECK-SAME: (%[[ARG0:.*]]: f32, %[[ARG1:.*]]: f32) -> complex<f32>
+func.func @mul_one_f32(%arg0: f32, %arg1: f32) -> complex<f32> {
+  %create = complex.create %arg0, %arg1: complex<f32>  
+  %one = complex.constant [1.0 : f32, 0.0 : f32] : complex<f32>
+  %mul = complex.mul %create, %one : complex<f32>
+  // CHECK: %[[CREATE:.*]] = complex.create %[[ARG0]], %[[ARG1]] : complex<f32>
+  // CHECK-NEXT: return %[[CREATE]]
+  return %mul : complex<f32>
+}
+
+// CHECK-LABEL: func @mul_one_f64
+//  CHECK-SAME: (%[[ARG0:.*]]: f64, %[[ARG1:.*]]: f64) -> complex<f64>
+func.func @mul_one_f64(%arg0: f64, %arg1: f64) -> complex<f64> {
+  %create = complex.create %arg0, %arg1: complex<f64>  
+  %one = complex.constant [1.0 : f64, 0.0 : f64] : complex<f64>
+  %mul = complex.mul %create, %one : complex<f64>
+  // CHECK: %[[CREATE:.*]] = complex.create %[[ARG0]], %[[ARG1]] : complex<f64>
+  // CHECK-NEXT: return %[[CREATE]]
+  return %mul : complex<f64>
+}