![ALT](../images/gemm-hierarchy-with-epilogue-no-labels.png "CUTLASS Functionality")

[README](../../README.md#documentation) > **Functionality**

# Functionality

Note : CUTLASS-3 requires users to use CUDA 11.4 or newer, and SM70 or newer, for the target toolkit and architecture, respectively.
Please refer to the [Compatibility](/README.md#Compatibility) section for more details.

- N - Column Major Matrix
- T - Row Major matrix
- {N,T} x {N,T} - All combinations, i.e., NN, NT, TN, TT
- [NHWC](/include/cutlass/layout/tensor.h#L63-206) - 4 dimension tensor used for convolution
- [NCxHWx](/include/cutlass/layout/tensor.h#L290-395) - Interleaved 4 dimension tensor used for convolution
- f - floating point
- s - signed int
- b - bit
- cf - complex float
- bf16 - bfloat16
- tf32 - tfloat32
- Simt - Use Simt CUDA Core MMA
- TensorOp - Use Tensor Core MMA
- SpTensorOp - Use Sparse Tensor Core MMA
- WmmaTensorOp - Use WMMA abstraction to use Tensor Core MMA

## Device-level GEMM

The following tables summarize device-level GEMM kernels in CUTLASS, organized by opcode class, data type, and layout.
Hyperlinks to relevant unit tests demonstrate how specific template instances may be defined.

### CUTLASS 3.x Kernels

|**Opcode Class** | **Compute Capability** | **CUDA Toolkit** | **Data Type**                  | **Layouts**            | **Unit Test**    |
|-----------------|------------------------|------------------|--------------------------------|------------------------|------------------|
| **TensorOp**        | 90a                 |  12.0+           | `f16 * f16 + { f16, f32 } => { f16, f32 }`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/sm90_gemm_f16_f16_f16_tensor_op_f32_cluster_warpspecialized.cu) |
| **TensorOp**        | 90a                 |  12.0+           | `bf16 * bf16 + { f16, f32 } => { bf16, f32 }`| {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/sm90_gemm_bf16_bf16_bf16_tensor_op_f32.cu) |
| **TensorOp**        | 90a                 |  12.0+           | `{f32, tf32} * {f32, tf32} + f32 => f32`| { T } x { N } => {N,T} |  [example](/test/unit/gemm/device/sm90_gemm_f32_f32_f32_tensor_op_f32.cu) |
| **TensorOp**        | 90a                 |  12.0+           | `s8 * s8 + s32 => {s32, s8}`   | { T } x { N } => {N,T} |  [example](/test/unit/gemm/device/sm90_gemm_s8_s8_s8_tensor_op_s32.cu) |
 

### CUTLASS 2.x Kernels

|**Opcode Class** | **Compute Capability** | **CUDA Toolkit** | **Data Type**                  | **Layouts**            | **Unit Test**    |
|-----------------|------------------------|------------------|--------------------------------|------------------------|------------------|
| **Simt**        | 50+                    |  11.4+            | `f32 * f32 + f32 => f32`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/simt_sgemm_nt_sm50.cu)                |
| **Simt**        | 50+                    |  11.4+            | `f64 * f64 + f64 => f64`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/simt_dgemm_nt_sm50.cu)                |
| **Simt**        | 60+                    |  11.4+            | `f16 * f16 + f16 => f16`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/simt_hgemm_nt_sm50.cu)                |
| **Simt**        | 61+                    |  11.4+            | `s8 * s8 + s32 => {s32,s8}`    | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/simt_igemm_nt_sm50.cu)              |
| **WmmaTensorOp**    | 70+                |  11.4+            | `f16 * f16 + f16 => f16`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f16t_f16t_f16n_wmma_tensor_op_f16_sm70.cu)     |
| **WmmaTensorOp**    | 70+                |  11.4+            | `f16 * f16 + f32 => {f16, f32}`| {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f16t_f16t_f16n_wmma_tensor_op_f32_sm70.cu)                |
| **WmmaTensorOp**    | 75+                |  11.4+           | `s8 * s8 + s32 => {s32, s8}`   | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_s8t_s8n_s8t_wmma_tensor_op_s32_sm72.cu) |
| **WmmaTensorOp**    | 75+                |  11.4+           | `s4 * s4 + s32 => {s32, s4}`   | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_s4t_s4n_s4t_wmma_tensor_op_s32_sm75.cu)                |
| **WmmaTensorOp**    | 75+                |  11.4+           | `b1 ^ b1 + s32 => {s32, b1}`   | { T } x { N } => {N,T} |  [example](/test/unit/gemm/device/gemm_b1t_b1n_b1t_wmma_tensor_op_s32_sm75.cu)                |
| **TensorOp**        | 70+                |  11.4+           | `f16 * f16 + f16 => f16`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f16n_f16t_f16t_volta_tensor_op_f16_sm70.cu)                |
| **TensorOp**        | 70+                |  11.4+           | `f16 * f16 + f32 => {f16, f32}`| {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f16n_f16t_f16t_volta_tensor_op_f32_sm70.cu)                |
| **TensorOp**        | 75+                |  11.4+           | `f16 * f16 + f16 => f16`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f16n_f16t_f16t_tensor_op_f16_sm75.cu) |
| **TensorOp**        | 75+                |  11.4+           | `f16 * f16 + f32 => {f16, f32}`| {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f16n_f16t_f16t_tensor_op_f32_sm75.cu) |
| **TensorOp**        | 75+                |  11.4+           | `s8 * s8 + s32 => {s32, s8}`   | { T } x { N } => {N,T} |  [example](/test/unit/gemm/device/gemm_s8t_s8n_s32n_tensor_op_s32_sm75.cu) |
| **TensorOp**        | 75+                |  11.4+           | `s4 * s4 + s32 => {s32, s4}`   | { T } x { N } => {N,T} |  [example](/test/unit/gemm/device/gemm_s4t_s4n_s32n_tensor_op_s32_sm75.cu) |
| **TensorOp**        | 75+                |  11.4+           | `b1 ^ b1 + s32 => {s32, b1}`   | { T } x { N } => {N,T} |  [example](/test/unit/gemm/device/gemm_b1t_b1n_s32n_tensor_op_s32_sm75.cu) |
| **TensorOp**        | 80+                |  11.4+           | `f16 * f16 + f16 => f16`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f16n_f16t_f16t_tensor_op_f16_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `f16 * f16 + f32 => {f16, f32}`| {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f16n_f16t_f16t_tensor_op_f32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `bf16 * bf16 + f32 => {bf16, f32}`| {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_bf16n_bf16t_bf16t_tensor_op_f32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `tf32 * tf32 + f32 => f32`| {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f32n_f32t_f32t_tensor_op_f32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `s8 * s8 + s32 => {s32, s8}`   | { T } x { N } => {N,T} |  [example](/test/unit/gemm/device/gemm_s8t_s8n_s32n_tensor_op_s32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `s4 * s4 + s32 => {s32, s4}`   | { T } x { N } => {N,T} |  [example](/test/unit/gemm/device/gemm_s4t_s4n_s32n_tensor_op_s32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `b1 ^ b1 + s32 => {s32, b1}`   | { T } x { N } => {N,T} |  [example](/test/unit/gemm/device/gemm_b1t_b1n_s32n_tensor_op_s32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `f64 * f64 + f64 => f64`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f64n_f64t_f64t_tensor_op_f64_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `cf32 * cf32 + cf32 => cf32`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_cf32n_cf32t_cf32t_tensor_op_tf32_f32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `cf64 * cf64 + cf64 => cf64`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_cf64n_cf64t_cf64t_tensor_op_f64_sm80.cu), [Gaussian 3m](/test/unit/gemm/device/gemm_cf64n_cf64t_cf64t_tensor_op_f64_gaussian_sm80.cu) |
| **SpTensorOp**      | 80+                |  11.4+           | `f16 * f16 + f32 => {f16, f32}`    | {N,T} x {N,T} => {N,T} | [example](/test/unit/gemm/device/gemm_f16n_f16n_f32t_tensor_op_f32_sparse_sm80.cu) |
| **SpTensorOp**      | 80+                |  11.4+           | `bf16 * bf16 + f32 => {bf16, f32}` | {N,T} x {N,T} => {N,T} | [example](/test/unit/gemm/device/gemm_f16n_f16n_f32t_tensor_op_f32_sparse_sm80.cu) |
| **SpTensorOp**      | 80+                |  11.4+           | `tf32 * tf32 + f32 => f32`         | {N,T} x {N,T} => {N,T} | [example](/test/unit/gemm/device/gemm_f32n_f32n_f32t_tensor_op_f32_sparse_sm80.cu) |
| **SpTensorOp**      | 80+                |  11.4+           | `s8 * s8 + s32 => {s8, s32}`       | {N,T} x {N,T} => {N,T} | [example](/test/unit/gemm/device/gemm_s8t_s8n_s32t_tensor_op_s32_sparse_sm80.cu) |
| **SpTensorOp**      | 80+                |  11.4+           | `s4 * s4 + s32 => {s4, s32}`       | {N,T} x {N,T} => {N,T} | [example](/test/unit/gemm/device/gemm_s4t_s4n_s32t_tensor_op_s32_sparse_sm80.cu) |
| **TensorOp**        | 90+                |  11.8+           | `f64 * f64 + f64 => f64`       | {N,T} x {N,T} => {N,T} |  [example](/test/unit/gemm/device/gemm_f64n_f64t_f64t_tensor_op_f64_sm90.cu) |


## Device-level Implicit GEMM convolution

The following table summarizes device-level implicit GEMM convolution kernels in CUTLASS, organized by opcode class, data type, and layout.
Hyperlinks to relevant conv2d fprop unit tests demonstrate how specific template instances may be defined. 
One can find and/or create equivalent dgrad and wgrad convolutional operators.

|**Opcode Class** | **Compute Capability** | **CUDA Toolkit** | **Data Type**                  | **Layouts**      | **Unit Test**    |
|-----------------|------------------------|------------------|--------------------------------|------------------|------------------|
| **Simt**            | 50+                |  11.4+            | `f32 * f32 + f32 => f32`       | NHWC             |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_f32nhwc_f32nhwc_f32nhwc_simt_f32_sm50.cu)                |
| **Simt**            | 50+                |  11.4+            | `cf32 * cf32 + cf32 => cf32`   | NHWC             |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_cf32nhwc_cf32nhwc_cf32nhwc_simt_f32_sm50.cu)                |
| **TensorOp**        | 70+                |  11.4+           | `f16 * f16 + f32 => {f16, f32}`| NHWC             |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_f16nhwc_f16nhwc_f32nhwc_tensor_op_f32_sm70.cu) |
| **TensorOp**        | 75+                |  11.4+           | `f16 * f16 + f32 => {f16, f32}`| NHWC             |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_f16nhwc_f16nhwc_f32nhwc_tensor_op_f32_sm75.cu) |
| **TensorOp**        | 75+                |  11.4+           | `s8 * s8 + s32 => {s32, s8}`   | NHWC, NCxHWx     |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_s8nhwc_s8nhwc_s32nhwc_tensor_op_s32_sm75.cu), [ncxhwx](/test/unit/conv/device/conv2d_fprop_implicit_gemm_s8ncxhwx_s8cxrskx_s8ncxhwx_tensor_op_s32_sm75.cu) |
| **TensorOp**        | 75+                |  11.4+           | `s4 * s4 + s32 => {s32, s4}`   | NHWC, NCxHWx     |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_s4nhwc_s4nhwc_s32nhwc_tensor_op_s32_sm75.cu), [ncxhwx](/test/unit/conv/device/conv2d_fprop_implicit_gemm_s4ncxhwx_s4cxrskx_s4ncxhwx_tensor_op_s32_sm75.cu) |
| **Simt**            | 80+                |  11.4+           | `f32 * f32 + f32 => f32`       | NHWC             |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_f32nhwc_f32nhwc_f32nhwc_simt_f32_sm80.cu)                |
| **Simt**            | 80+                |  11.4+           | `cf32 * cf32 + cf32 => cf32`   | NHWC             |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_cf32nhwc_cf32nhwc_cf32nhwc_simt_f32_sm80.cu)                |
| **TensorOp**        | 80+                |  11.4+           | `f16 * f16 + f32 => {f16, f32}`| NHWC             |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_f16nhwc_f16nhwc_f32nhwc_tensor_op_f32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `f16 * f16 + f16 => f16`       | NHWC             |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_f16nhwc_f16nhwc_f32nhwc_tensor_op_f32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `tf32 * tf32 + f32 => f32`     | NHWC             |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_tf32nhwc_tf32nhwc_f32nhwc_tensor_op_f32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `s8 * s8 + s32 => {s32, s8}`   | NHWC, NCxHWx     |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_s8nhwc_s8nhwc_s32nhwc_tensor_op_s32_sm80.cu), [ncxhwx](/test/unit/conv/device/conv2d_fprop_implicit_gemm_s8ncxhwx_s8cxrskx_s8ncxhwx_tensor_op_s32_sm80.cu) |
| **TensorOp**        | 80+                |  11.4+           | `s4 * s4 + s32 => {s32, s4}`   | NHWC, NCxHWx     |  [example](/test/unit/conv/device/conv2d_fprop_implicit_gemm_s4nhwc_s4nhwc_s32nhwc_tensor_op_s32_sm80.cu), [ncxhwx](/test/unit/conv/device/conv2d_fprop_implicit_gemm_s4ncxhwx_s4cxrskx_s4ncxhwx_tensor_op_s32_sm80.cu) |



## Warp-level Matrix Multiply with Tensor Cores

The following table summarizes supported warp level shapes for each TensorOp instruction.

|**Opcode Class** | **Instruction Shape** | **Warp Shapes**                            |
|-----------------|-----------------------|--------------------------------------------|
| **TensorOp**    | 8-by-8-by-4           | 32x32x4, 32x64x4, 64x32x4, 64x64x4         |
| **TensorOp**    | 16-by-8-by-8          | 32x32x8, 32x64x8, 64x32x8, 64x64x8         |
| **TensorOp**    | 16-by-8-by-16         | 32x32x16, 32x64x16, 64x32x16, 64x64x16     |
| **TensorOp**    | 8-by-8-by-16          | 32x32x16, 32x64x16, 64x32x16, 64x64x16     |
| **TensorOp**    | 8-by-8-by-32          | 32x32x32, 32x64x32, 64x32x32, 64x64x32     |
| **TensorOp**    | 16-by-8-by-32         | 32x32x32, 32x64x32, 64x32x32, 64x64x32     |
| **TensorOp**    | 16-by-8-by-64         | 32x32x64, 32x64x64, 64x32x64, 64x64x64     |
| **TensorOp**    | 8-by-8-by-128         | 32x32x128, 32x64x128, 64x32x128, 64x64x128 |
| **TensorOp**    | 16-by-8-by-256        | 32x32x256, 32x64x256, 64x32x256, 64x64x256 |
| **SpTensorOp**  | 16-by-8-by-16         | 64x64x16, 64x32x16, 32x64x16, 32x32x16     |
| **SpTensorOp**  | 16-by-8-by-32         | 64x64x32, 64x32x32, 32x64x32, 32x32x32     |
| **SpTensorOp**  | 16-by-8-by-64         | 64x64x64, 64x32x64, 32x64x64, 32x32x64     |
| **SpTensorOp**  | 16-by-8-by-128        | 64x64x128, 64x32x128, 32x64x128, 32x32x128 |


TensorOp instructions depend on a permuted shared memory layout that can be efficiently
loaded from. The following tables summarize the destination shared memory layout that
can be targeted by matrix operands. It is assumed that each thread loads 128b vectors
from global memory with layout specified in the column "GMEM Layout."

**TensorOp 8-by-8-by-4.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                         |
|-----------|--------------|-----------------|-----------------------------------------|
|  **A**    | `half_t`     | `ColumnMajor`   | `ColumnMajorVoltaTensorOpCongruous<16>` |
|  **A**    | `half_t`     | `RowMajor`      | `RowMajorVoltaTensorOpCrosswise<16>`    |
|  **B**    | `half_t`     | `ColumnMajor`   | `ColumnMajorVoltaTensorOpCrosswise<16>` |
|  **B**    | `half_t`     | `RowMajor`      | `RowMajorVoltaTensorOpCongruous<16>`    |
|  **C**    | `half_t`     | `RowMajor`      | `RowMajor`                              |
|  **C**    | `float`      | `RowMajor`      | `RowMajor`                              |

**TensorOp 16-by-8-by-8.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `half_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCongruous<16>` |
|  **A**    | `half_t`     | `RowMajor`      | `RowMajorTensorOpCrosswise<16>`    |
|  **B**    | `half_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCrosswise<16>` |
|  **B**    | `half_t`     | `RowMajor`      | `RowMajorTensorOpCongruous<16>`    |
|  **C**    | `half_t`     | `RowMajor`      | `RowMajor`                         |
|  **C**    | `float`      | `RowMajor`      | `RowMajor`                         |

**TensorOp 16-by-8-by-8.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `tfloat32_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCongruous<32>` |
|  **A**    | `tfloat32_t`     | `RowMajor`      | `RowMajorTensorOpCrosswise<32>`    |
|  **B**    | `tfloat32_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCrosswise<32>` |
|  **B**    | `tfloat32_t`     | `RowMajor`      | `RowMajorTensorOpCongruous<32>`    |
|  **C**    | `float`          | `RowMajor`      | `RowMajor`                         |


**TensorOp 16-by-8-by-16.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `half_t`, `bfloat16_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCongruous<16>` |
|  **A**    | `half_t`, `bfloat16_t`     | `RowMajor`      | `RowMajorTensorOpCrosswise<16>`    |
|  **B**    | `half_t`, `bfloat16_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCrosswise<16>` |
|  **B**    | `half_t`, `bfloat16_t`     | `RowMajor`      | `RowMajorTensorOpCongruous<16>`    |
|  **C**    | `half_t`     | `RowMajor`      | `RowMajor`                         |
|  **C**    | `float`      | `RowMajor`      | `RowMajor`                         |

**TensorOp 8-by-8-by-4.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `double`     | `ColumnMajor`   | `ColumnMajorTensorOpCongruous<64>` |
|  **A**    | `double`     | `RowMajor`      | `RowMajorTensorOpCrosswise<64>`    |
|  **B**    | `double`     | `ColumnMajor`   | `ColumnMajorTensorOpCrosswise<64>` |
|  **B**    | `double`     | `RowMajor`      | `RowMajorTensorOpCongruous<64>`    |
|  **C**    | `double`     | `RowMajor`      | `RowMajor`                         |

**TensorOp 8-by-8-by-16.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `int8_t`     | `RowMajor`      | `RowMajorTensorOpCrosswise<8>`     |
|  **B**    | `int8_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCongruous<8>`  |
|  **C**    | `int32_t`    | `RowMajor`      | `RowMajor`                         |

**TensorOp 16-by-8-by-32.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `int8_t`     | `RowMajor`      | `RowMajorTensorOpCrosswise<8>`     |
|  **B**    | `int8_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCongruous<8>`  |
|  **C**    | `int32_t`    | `RowMajor`      | `RowMajor`                         |

**TensorOp 8-by-8-by-32.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `int4b_t`    | `RowMajor`      | `RowMajorTensorOpCrosswise<4>`     |
|  **B**    | `int4b_t`    | `ColumnMajor`   | `ColumnMajorTensorOpCongruous<4>`  |
|  **C**    | `int32_t`    | `RowMajor`      | `RowMajor`                         |

**TensorOp 16-by-8-by-64.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `int4b_t`    | `RowMajor`      | `RowMajorTensorOpCrosswise<4>`     |
|  **B**    | `int4b_t`    | `ColumnMajor`   | `ColumnMajorTensorOpCongruous<4>`  |
|  **C**    | `int32_t`    | `RowMajor`      | `RowMajor`                         |

**TensorOp 8-by-8-by-128.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `bin1_t`     | `RowMajor`      | `RowMajorTensorOpCrosswise<4>`     |
|  **B**    | `bin1_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCongruous<4>`  |
|  **C**    | `int32_t`    | `RowMajor`      | `RowMajor`                         |


**SpTensorOp 16-by-8-by-16.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `tfloat32_t` | `RowMajor`      | `RowMajorTensorOpCrosswise<32, 32>`   |
|  **B**    | `tfloat32_t` | `ColumnMajor`   | `ColumnMajorTensorOpCrosswise<32, 32>`|
|  **C**    | `float`      | `RowMajor`      | `RowMajor`                            |

**SpTensorOp 16-by-8-by-32.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                       |
|-----------|--------------|-----------------|---------------------------------------|
|  **A**    | `half_t`     | `RowMajor`      | `RowMajorTensorOpCrosswise<16, 64>`   |
|  **B**    | `half_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCrosswise<16, 64>`|
|  **C**    | `float`      | `RowMajor`      | `RowMajor`                            |

**SpTensorOp 16-by-8-by-64.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                       |
|-----------|--------------|-----------------|---------------------------------------|
|  **A**    | `int8_t`     | `RowMajor`      | `RowMajorTensorOpCrosswise<8, 128>`   |
|  **B**    | `int8_t`     | `ColumnMajor`   | `ColumnMajorTensorOpCrosswise<8, 128>`|
|  **C**    | `int32_t`    | `RowMajor`      | `RowMajor`                            |

**SpTensorOp 16-by-8-by-128.**

|**Operand**|**Element**   | **GMEM Layout** | **SMEM Layout**                    |
|-----------|--------------|-----------------|------------------------------------|
|  **A**    | `int4b_t`    | `RowMajor`      | `RowMajorTensorOpCrosswise<4, 256>`   |
|  **B**    | `int4b_t`    | `ColumnMajor`   | `ColumnMajorTensorOpCrosswise<4, 256>`|
|  **C**    | `int32_t`    | `RowMajor`      | `RowMajor`                           |



## Warp-level Matrix Multiply with CUDA WMMA API

The following table summarizes supported warp level shapes for each WmmaTensorOp instruction.

|**Opcode Class**     | **Instruction Shape** | **Warp Shapes**                            |
|---------------------|-----------------------|--------------------------------------------|
| **WmmaTensorOp**    | 16-by-16-by-16        | 32x32x16, 32x64x16, 64x32x16               |
| **WmmaTensorOp**    | 8-by-32-by-16         | 32x32x16, 32x64x16, 64x32x16               |
| **WmmaTensorOp**    | 32-by-8-by-16         | 32x32x16, 32x64x16, 64x32x16               |
| **WmmaTensorOp**    | 8-by-8-by-32          | 32x32x32, 32x64x32, 64x32x32, 64x64x32     |
| **WmmaTensorOp**    | 8-by-8-by-128         | 32x32x128, 32x64x128, 64x32x128, 64x64x128 |


CUDA exposes warp-level matrix operations in the CUDA C++ WMMA API. The CUDA C++ WMMA API exposes Tensor Cores via a set of functions and types in the `nvcuda::wmma` namespace. The functions and types in `nvcuda::wmma` provide target-independent APIs and implement architecture-specific tensor operation using TensorOp instruction underneath. CUTLASS exposes WMMA API through WmmaTensorOp. The WmmaTensorOp supports canonical shared memory layouts. The following table summarizes the destination shared memory layout that can be targeted by matrix operands. The WMMA API expects that matrices in shared memory loaded by `nvcuda::wmma::load_matrix_sync()` satisfy 128 bit alignment.


**WmmaTensorOp (all matrix sizes and data types).**

|**Operand** |       **GMEM Layout**      |       **SMEM Layout**        |
|------------|----------------------------|------------------------------|
|  **A**     | `RowMajor`, `ColumnMajor`  | `RowMajor`, `ColumnMajor`    |
|  **B**     | `RowMajor`, `ColumnMajor`  | `RowMajor`, `ColumnMajor`    |
|  **C**     | `RowMajor`, `ColumnMajor`  | `RowMajor`, `ColumnMajor`    |

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