Use basis argument for Decomposition/BasisConversion passes

lib/Optimizer/CodeGen/Passes.cpp

@@ -14,16 +14,13 @@
 
 using namespace mlir;
 
-static SmallVector<std::string> z_disabledPatterns = {"R1ToU3"};
-
 static void addAnyonPPipeline(OpPassManager &pm) {
   using namespace cudaq::opt;
   std::string basis[] = {
       "h", "s", "t", "rx", "ry", "rz", "x", "y", "z", "z(1)",
   };
   BasisConversionOptions options;
   options.basis = basis;
-  options.disabledPatterns = z_disabledPatterns;
   pm.addPass(createBasisConversion(options));
 }
 
@@ -34,7 +31,6 @@ static void addAnyonCPipeline(OpPassManager &pm) {
   };
   BasisConversionOptions options;
   options.basis = basis;
-  options.disabledPatterns = z_disabledPatterns;
   pm.addPass(createBasisConversion(options));
 }
 
@@ -46,7 +42,6 @@ static void addOQCPipeline(OpPassManager &pm) {
   };
   BasisConversionOptions options;
   options.basis = basis;
-  options.disabledPatterns = z_disabledPatterns;
   pm.addPass(createBasisConversion(options));
 }
 
@@ -60,7 +55,6 @@ static void addQCIPipeline(OpPassManager &pm) {
   };
   BasisConversionOptions options;
   options.basis = basis;
-  options.disabledPatterns = z_disabledPatterns;
   pm.addPass(createBasisConversion(options));
 }
 
@@ -71,7 +65,6 @@ static void addQuantinuumPipeline(OpPassManager &pm) {
   };
   BasisConversionOptions options;
   options.basis = basis;
-  options.disabledPatterns = z_disabledPatterns;
   pm.addPass(createBasisConversion(options));
 }
 
@@ -83,7 +76,6 @@ static void addIQMPipeline(OpPassManager &pm) {
   };
   BasisConversionOptions options;
   options.basis = basis;
-  options.disabledPatterns = z_disabledPatterns;
   pm.addPass(createBasisConversion(options));
 }
 
@@ -95,7 +87,6 @@ static void addIonQPipeline(OpPassManager &pm) {
   };
   BasisConversionOptions options;
   options.basis = basis;
-  options.disabledPatterns = z_disabledPatterns;
   pm.addPass(createBasisConversion(options));
 }
 
@@ -106,7 +97,6 @@ static void addFermioniqPipeline(OpPassManager &pm) {
   };
   BasisConversionOptions options;
   options.basis = basis;
-  options.disabledPatterns = z_disabledPatterns;
   pm.addPass(createBasisConversion(options));
 }
 

lib/Optimizer/CodeGen/Pipelines.cpp

@@ -150,10 +150,7 @@ void cudaq::opt::createPipelineTransformsForPythonToOpenQASM(
   pm.addNestedPass<func::FuncOp>(createCSEPass());
   pm.addNestedPass<func::FuncOp>(createMultiControlDecomposition());
   pm.addPass(createDecomposition(
-      {.enabledPatterns = {"SToR1", "TToR1", "R1ToU3", "U3ToRotations",
-                           "CHToCX", "CCZToCX", "CRzToCX", "CRyToCX", "CRxToCX",
-                           "CR1ToCX", "CCZToCX", "RxAdjToRx", "RyAdjToRy",
-                           "RzAdjToRz"}}));
+      {.basis = {"h", "s", "t", "rx", "ry", "rz", "x", "y", "z", "x(1)"}}));
   pm.addPass(createQuakeToCCPrep());
   pm.addNestedPass<func::FuncOp>(createCanonicalizerPass());
   pm.addNestedPass<func::FuncOp>(createExpandControlVeqs());

lib/Optimizer/Transforms/DecompositionPatternSelection.cpp

@@ -38,13 +38,17 @@ namespace {
 struct OperatorInfo {
   StringRef name;
   std::size_t numControls;
+  bool isAdj;
 
-  OperatorInfo(StringRef infoStr) : name(), numControls(0) {
-    auto nameEnd = infoStr.find_first_of('(');
+  OperatorInfo(StringRef infoStr) : name(), numControls(0), isAdj(false) {
+    auto nameEnd = infoStr.find_first_of("(<");
     name = infoStr.take_front(nameEnd);
     if (nameEnd < infoStr.size())
       infoStr = infoStr.drop_front(nameEnd);
 
+    if (infoStr.consume_front("<adj>"))
+      isAdj = true;
+
     if (infoStr.consume_front("(")) {
       infoStr = infoStr.ltrim();
       if (infoStr.consume_front("n"))
@@ -56,7 +60,8 @@ struct OperatorInfo {
   }
 
   bool operator==(const OperatorInfo &other) const {
-    return name == other.name && numControls == other.numControls;
+    return name == other.name && numControls == other.numControls &&
+           isAdj == other.isAdj;
   }
 };
 
@@ -111,7 +116,7 @@ namespace std {
 template <>
 struct hash<OperatorInfo> {
   std::size_t operator()(const OperatorInfo &info) const {
-    return llvm::hash_combine(info.name, info.numControls);
+    return llvm::hash_combine(info.name, info.numControls, info.isAdj);
   }
 };
 } // namespace std

lib/Optimizer/Transforms/DecompositionPatterns.cpp

@@ -645,7 +645,7 @@ struct R1AdjToR1
     return success();
   }
 };
-REGISTER_DECOMPOSITION_PATTERN(R1AdjToR1, "r1", "r1");
+REGISTER_DECOMPOSITION_PATTERN(R1AdjToR1, "r1<adj>", "r1");
 
 // quake.swap a, b
 // ───────────────────────────────────
@@ -717,6 +717,8 @@ struct CHToCX : public cudaq::DecompositionPattern<CHToCXType, quake::HOp> {
     return success();
   }
 };
+// TODO: Technically, this pattern also produces s<adj> and t<adj> ops, but we
+// currently don't treat them as distinct from their non-adjoint counterparts.
 REGISTER_DECOMPOSITION_PATTERN(CHToCX, "h(1)", "s", "h", "t", "x(1)");
 
 //===----------------------------------------------------------------------===//
@@ -1075,6 +1077,8 @@ struct CYToCX : public cudaq::DecompositionPattern<CYToCXType, quake::YOp> {
     return success();
   }
 };
+// TODO: Technically, this pattern also produces s<adj> ops, but we currently
+// don't treat it as distinct from their non-adjoint counterparts.
 REGISTER_DECOMPOSITION_PATTERN(CYToCX, "y(1)", "s", "x(1)");
 
 //===----------------------------------------------------------------------===//
@@ -1152,6 +1156,8 @@ struct CCZToCX : public cudaq::DecompositionPattern<CCZToCXType, quake::ZOp> {
     return success();
   }
 };
+// TODO: Technically, this pattern also produces t<adj> ops, but we currently
+// don't treat it as distinct from their non-adjoint counterparts.
 REGISTER_DECOMPOSITION_PATTERN(CCZToCX, "z(2)", "t", "x(1)");
 
 // quake.z [control] target
@@ -1488,7 +1494,7 @@ struct RxAdjToRx
     return success();
   }
 };
-REGISTER_DECOMPOSITION_PATTERN(RxAdjToRx, "rx", "rx");
+REGISTER_DECOMPOSITION_PATTERN(RxAdjToRx, "rx<adj>", "rx");
 
 //===----------------------------------------------------------------------===//
 // RyOp decompositions
@@ -1617,7 +1623,7 @@ struct RyAdjToRy
     return success();
   }
 };
-REGISTER_DECOMPOSITION_PATTERN(RyAdjToRy, "ry", "ry");
+REGISTER_DECOMPOSITION_PATTERN(RyAdjToRy, "ry<adj>", "ry");
 
 //===----------------------------------------------------------------------===//
 // RzOp decompositions
@@ -1757,7 +1763,7 @@ struct RzAdjToRz
     return success();
   }
 };
-REGISTER_DECOMPOSITION_PATTERN(RzAdjToRz, "rz", "rz");
+REGISTER_DECOMPOSITION_PATTERN(RzAdjToRz, "rz<adj>", "rz");
 
 //===----------------------------------------------------------------------===//
 // U3Op decompositions

python/tests/mlir/test_output_translate_openqasm.py

@@ -81,84 +81,28 @@ def gates():
 # CHECK-NEXT: y var0[0];
 # CHECK-NEXT: z var0[0];
 # CHECK-NEXT: rz(1.570796e+00) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: rz(7.853982e-01) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: rz(1.570796e+00) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: h var0[0];
 # CHECK-NEXT: rz(7.853982e-01) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
 # CHECK-NEXT: rz(-7.853982e-01) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: h var0[0];
 # CHECK-NEXT: rz(-1.570796e+00) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
 # CHECK-NEXT: rz(-7.853982e-01) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
 # CHECK-NEXT: rz(7.853982e-01) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
 # CHECK-NEXT: rz(-7.853982e-01) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
 # CHECK-NEXT: rz(7.853982e-01) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0{{\[[12]\]}};
 # CHECK-NEXT: rz(-7.853982e-01) var0{{\[[12]\]}};
-# CHECK-NEXT: rx(1.570796e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rz(0.000000e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rx(-1.570796e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rz(0.000000e+00) var0{{\[[12]\]}};
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0{{\[[12]\]}};
 # CHECK-NEXT: rz(7.853982e-01) var0{{\[[12]\]}};
-# CHECK-NEXT: rx(1.570796e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rz(0.000000e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rx(-1.570796e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rz(0.000000e+00) var0{{\[[12]\]}};
 # CHECK-NEXT: rz(7.853982e-01) var0{{\[[12]\]}};
-# CHECK-NEXT: rx(1.570796e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rz(0.000000e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rx(-1.570796e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rz(0.000000e+00) var0{{\[[12]\]}};
 # CHECK-NEXT: rz(1.570796e+00) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
 # CHECK-NEXT: ry(-1.570796e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
@@ -173,27 +117,11 @@ def gates():
 # CHECK-NEXT: rz(-1.570796e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
 # CHECK-NEXT: rz(1.570796e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rx(1.570796e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rz(0.000000e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rx(-1.570796e+00) var0{{\[[12]\]}};
-# CHECK-NEXT: rz(0.000000e+00) var0{{\[[12]\]}};
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
 # CHECK-NEXT: rz(-1.570796e+00) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: cx var0{{\[[12]\]}}, var0[0];
 # CHECK-NEXT: rz(1.570796e+00) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: rz(3.141593e+00) var0[0];
-# CHECK-NEXT: rx(1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
-# CHECK-NEXT: rx(-1.570796e+00) var0[0];
-# CHECK-NEXT: rz(0.000000e+00) var0[0];
 # CHECK-NEXT: rx(3.141593e+00) var0[0];
 # CHECK-NEXT: rx(-3.141593e+00) var0[0];
 # CHECK-NEXT: ry(3.141593e+00) var0[0];

runtime/cudaq/platform/default/rest/helpers/braket/braket.yml

@@ -19,7 +19,7 @@ config:
   # Add preprocessor defines to compilation
   preprocessor-defines: ["-D CUDAQ_QUANTUM_DEVICE"]
   # Define the JIT lowering pipeline
-  jit-mid-level-pipeline: "lower-to-cfg,decomposition{enable-patterns=SToR1,TToR1,R1ToU3,U3ToRotations,CHToCX,CCZToCX,CRzToCX,CRyToCX,CRxToCX,CR1ToCX,RxAdjToRx,RyAdjToRy,RzAdjToRz,ExpPauliDecomposition},quake-to-cc-prep,func.func(expand-control-veqs,combine-quantum-alloc,canonicalize,combine-measurements)"
+  jit-mid-level-pipeline: "lower-to-cfg,decomposition{basis=h,s,t,rx,ry,rz,x,y,z,x(1)},quake-to-cc-prep,func.func(expand-control-veqs,combine-quantum-alloc,canonicalize,combine-measurements)"
   # Tell the rest-qpu that we are generating OpenQASM 2.0.
   codegen-emission: qasm2
   # Library mode is only for simulators, physical backends must turn this off

runtime/cudaq/platform/default/rest/helpers/infleqtion/infleqtion.yml

@@ -19,7 +19,7 @@ config:
   # Add preprocessor defines to compilation
   preprocessor-defines: ["-D CUDAQ_QUANTUM_DEVICE"]
   # Define the JIT lowering pipeline
-  jit-mid-level-pipeline: "lower-to-cfg,decomposition{enable-patterns=SToR1,TToR1,CCZToCX,CRyToCX,CRxToCX,R1AdjToR1,RxAdjToRx,RyAdjToRy,RzAdjToRz,ExpPauliDecomposition},quake-to-cc-prep,func.func(memtoreg{quantum=0})"
+  jit-mid-level-pipeline: "lower-to-cfg,decomposition{basis=h,s,t,r1,rx,ry,rz,x,y,z,x(1)},quake-to-cc-prep,func.func(memtoreg{quantum=0})"
   # Tell the rest-qpu that we are generating OpenQASM 2.0.
   codegen-emission: qasm2
   # Library mode is only for simulators, physical backends must turn this off

runtime/cudaq/platform/default/rest/helpers/quantum_machines/quantum_machines.yml

@@ -19,7 +19,7 @@ config:
   # Add preprocessor defines to compilation
   preprocessor-defines: ["-D CUDAQ_QUANTUM_DEVICE"]
   # Define the JIT lowering pipeline
-  jit-mid-level-pipeline: "lower-to-cfg,decomposition{enable-patterns=CHToCX,RzAdjToRz,CCZToCX,CR1ToCX,SwapToCX,CRxToCX,CRyToCX,CRzToCX},quake-to-cc-prep,func.func(expand-control-veqs,combine-quantum-alloc,canonicalize,combine-measurements)"
+  jit-mid-level-pipeline: "lower-to-cfg,decomposition{basis=h,s,t,r1,rx,ry,rz,x,y,z,x(1)},quake-to-cc-prep,func.func(expand-control-veqs,combine-quantum-alloc,canonicalize,combine-measurements)"
   # Tell the rest-qpu that we are generating OpenQASM 2.0.
   codegen-emission: qasm2
   # Library mode is only for simulators, physical backends must turn this off

runtime/cudaq/platform/default/rest/helpers/scaleway/scaleway.yml

@@ -19,7 +19,7 @@ config:
   # Add the rest-qpu library to the link list
   link-libs: ["-lcudaq-rest-qpu"]
   # Define the JIT lowering pipeline
-  jit-mid-level-pipeline: "lower-to-cfg,decomposition{enable-patterns=SToR1,TToR1,R1ToU3,U3ToRotations,CHToCX,CCXToCCZ,CCZToCX,CRzToCX,CRyToCX,CRxToCX,CR1ToCX,RxAdjToRx,RyAdjToRy,RzAdjToRz,ExpPauliDecomposition},quake-to-cc-prep,func.func(expand-control-veqs,combine-quantum-alloc,canonicalize,combine-measurements)"
+  jit-mid-level-pipeline: "lower-to-cfg,decomposition{basis=h,s,t,rx,ry,rz,x,y,z,x(1)},quake-to-cc-prep,func.func(expand-control-veqs,combine-quantum-alloc,canonicalize,combine-measurements)"
   # Tell the rest-qpu that we are generating QIR.
   codegen-emission: qasm2
   # Library mode is only for simulators, physical backends must turn this off

runtime/cudaq/platform/default/rest/helpers/tii/tii.yml

@@ -19,7 +19,7 @@ config:
   # Add preprocessor defines to compilation
   preprocessor-defines: ["-D CUDAQ_QUANTUM_DEVICE"]
   # Define the JIT lowering pipelines
-  jit-mid-level-pipeline: "lower-to-cfg,func.func(canonicalize,multicontrol-decomposition),decomposition{enable-patterns=U3ToRotations,SToR1,TToR1,CHToCX,CCZToCX,CRzToCX,CRyToCX,CRxToCX,CR1ToCX,R1AdjToR1,RxAdjToRx,RyAdjToRy,RzAdjToRz,ExpPauliDecomposition},quake-to-cc-prep,func.func(expand-control-veqs,canonicalize),symbol-dce"
+  jit-mid-level-pipeline: "lower-to-cfg,func.func(canonicalize,multicontrol-decomposition),decomposition{basis=h,s,t,r1,rx,ry,rz,x,y,z,x(1)},quake-to-cc-prep,func.func(expand-control-veqs,canonicalize),symbol-dce"
   codegen-emission: qasm2
   # Library mode is only for simulators, physical backends must turn this off
   library-mode: false

test/Transforms/BasisConversion/all-qir-gates.qke

@@ -6,7 +6,7 @@
 // the terms of the Apache License 2.0 which accompanies this distribution.   //
 // ========================================================================== //
 
-// RUN: cudaq-opt --basis-conversion="basis=h,s,t,rx,ry,rz,x,y,z,x(1) disable-patterns=R1ToU3" %s | FileCheck %s
+// RUN: cudaq-opt --basis-conversion="basis=h,s,t,rx,ry,rz,x,y,z,x(1)" %s | FileCheck %s
 
 /// This test covers all the supported CUDA-Q gates along with control and
 /// adjoint modifiers, similar to the test in `python/tests/backends/test_Infleqtion.py::test_all_gates`
@@ -304,7 +304,8 @@ module {
 // CHECK:           %[[VAL_87:.*]] = cc.cast unsigned %[[VAL_85]] : (i1) -> i8
 // CHECK:           cc.store %[[VAL_87]], %[[VAL_86]] : !cc.ptr<i8>
 // CHECK:           %[[VAL_88:.*]] = quake.alloca !quake.ref
-// CHECK:           quake.rz<adj> (%[[VAL_4]]) %[[VAL_88]] : (f64, !quake.ref) -> ()
+// CHECK:           %[[NEG_PI:.*]] = arith.negf %[[VAL_4]] : f64
+// CHECK:           quake.rz (%[[NEG_PI]]) %[[VAL_88]] : (f64, !quake.ref) -> ()
 // CHECK:           quake.rx<adj> (%[[VAL_0]]) %[[VAL_88]] : (f64, !quake.ref) -> ()
 // CHECK:           quake.ry<adj> (%[[VAL_3]]) %[[VAL_88]] : (f64, !quake.ref) -> ()
 // CHECK:           quake.rz<adj> (%[[VAL_2]]) %[[VAL_88]] : (f64, !quake.ref) -> ()

test/Transforms/BasisConversion/phased_rx-cz.qke

@@ -6,8 +6,8 @@
 // the terms of the Apache License 2.0 which accompanies this distribution.   //
 // ========================================================================== //
 
-// RUN: cudaq-opt --basis-conversion="basis=z(1),phased_rx disable-patterns=R1ToU3" %s | FileCheck %s
-// RUN: cudaq-opt --basis-conversion="basis=z(1),phased_rx disable-patterns=R1ToU3" %s | CircuitCheck %s --up-to-global-phase
+// RUN: cudaq-opt --basis-conversion="basis=z(1),phased_rx" %s | FileCheck %s
+// RUN: cudaq-opt --basis-conversion="basis=z(1),phased_rx" %s | CircuitCheck %s --up-to-global-phase
 
 
 // CHECK-LABEL: func.func @cx