Skip to content

vllm.model_executor.layers.mamba.ops.gdn_chunk_cutedsl.kernel_kkt_inv_uw

Classes:

Sm100ChunkUWKernel

Compute per-chunk KKT inverse preprocessing and U/W tiles.

Gamma[i,j] = exp(g_cu[i] - g_cu[j]) A = strictLower(beta * (K @ K.T) * Gamma) Ai = inverse(I + A) U = (Ai * beta) @ V W = (Ai * beta * exp(g_cu)) @ K

Source code in vllm/model_executor/layers/mamba/ops/gdn_chunk_cutedsl/kernel_kkt_inv_uw.py
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
class Sm100ChunkUWKernel:
    """Compute per-chunk KKT inverse preprocessing and U/W tiles.

    Gamma[i,j] = exp(g_cu[i] - g_cu[j])
    A = strictLower(beta * (K @ K.T) * Gamma)
    Ai = inverse(I + A)
    U = (Ai * beta) @ V
    W = (Ai * beta * exp(g_cu)) @ K
    """

    def __init__(
        self,
        H: int,
        Hv: int,
        K_dim: int,
        V_dim: int,
        num_stages: int = 2,
    ) -> None:
        assert Hv % H == 0
        assert K_dim == V_dim == 128
        self.H = H
        self.Hv = Hv
        self.K_dim = K_dim
        self.V_dim = V_dim
        self.num_stages = num_stages

        # hard-code
        self.BT = 64
        self.num_warps = 2 + 4 + 4

    @cute.jit
    def _make_tma_args(
        self,
        tensor: cute.Tensor,
        dim: cutlass.Constexpr[int],
        num_stages: int,
        op: cpasync.TmaCopyOp,
    ):
        # logical layout: [BT, dim]
        # permute for TMA: [dim/64, BT, 64] with swizzling
        swizzle_128B = cute.make_swizzle(3, 4, 3)
        slayout = cute.make_layout(
            (self.BT, 1, (64, dim // 64), num_stages),
            stride=(64, 0, (1, self.BT * 64), self.BT * dim),
        )
        slayout = cute.make_composed_layout(swizzle_128B, 0, slayout)

        # we need to convert gmem layout to (T, H, (64, D/64)) for make_tiled_tma_atom()
        # to emit a single 4D TMA. otherwise, it will emit (D/64)x 3D TMA.
        atom, tma_tensor = cpasync.make_tiled_tma_atom(
            op,
            cute.logical_divide(tensor, (None, None, 64)),
            slayout,
            cta_tiler=(self.BT, 1, dim),
        )
        return atom, tma_tensor, slayout

    @cute.jit
    def __call__(
        self,
        K: cute.Tensor,
        V: cute.Tensor,
        U: cute.Tensor,
        W: cute.Tensor,
        g: cute.Tensor,
        beta: cute.Tensor,
        g_cu: cute.Tensor,
        cu_seqlens: cute.Tensor,
        chunk_indices: cute.Tensor,
        total_chunks: cute.Tensor,
        num_sms: Int32,
        stream: CUstream,
    ):
        tma_g2s = cpasync.CopyBulkTensorTileG2SOp()
        tma_s2g = cpasync.CopyBulkTensorTileS2GOp()

        K_args = self._make_tma_args(K, self.K_dim, self.num_stages, tma_g2s)
        V_args = self._make_tma_args(V, self.V_dim, self.num_stages, tma_g2s)
        U_args = self._make_tma_args(U, self.V_dim, 1, tma_s2g)
        W_args = self._make_tma_args(W, self.K_dim, 1, tma_s2g)

        grid = (num_sms // self.Hv, self.Hv, 1)
        block = (self.num_warps * 32, 1, 1)
        self.kernel(
            K_args,
            V_args,
            U_args,
            W_args,
            g,
            beta,
            g_cu,
            cu_seqlens,
            chunk_indices,
            total_chunks,
        ).launch(grid=grid, block=block, stream=stream)

    @cute.kernel
    def kernel(
        self,
        K_args: tuple[cute.CopyAtom, cute.Tensor, cute.ComposedLayout],
        V_args: tuple[cute.CopyAtom, cute.Tensor, cute.ComposedLayout],
        U_args: tuple[cute.CopyAtom, cute.Tensor, cute.ComposedLayout],
        W_args: tuple[cute.CopyAtom, cute.Tensor, cute.ComposedLayout],
        g: cute.Tensor,
        beta: cute.Tensor,
        g_cu: cute.Tensor,
        cu_seqlens: cute.Tensor,
        chunk_indices: cute.Tensor,
        total_chunks: cute.Tensor,
    ):
        tid, _, _ = cute.arch.thread_idx()
        bid, head_id, _ = cute.arch.block_idx()
        grid_x, _, _ = cute.arch.grid_dim()

        warp_id = cute.arch.make_warp_uniform(tid // 32)
        lane_id = tid % 32
        k_head_id = head_id // (self.Hv // self.H)

        BT = self.BT
        K_dim = self.K_dim
        V_dim = self.V_dim
        num_stages = self.num_stages

        K_tma_atom, tmaK, sK_layout = K_args
        V_tma_atom, tmaV, sV_layout = V_args
        U_tma_atom, tmaU, sU_layout = U_args
        W_tma_atom, tmaW, sW_layout = W_args

        def allocate_tensor(smem, dtype, layout):
            return smem.allocate_tensor(
                dtype, layout.outer, byte_alignment=128, swizzle=layout.inner
            )

        smem = cutlass.utils.SmemAllocator()
        sK = allocate_tensor(smem, BFloat16, sK_layout)[None, 0, None, None]
        sV = allocate_tensor(smem, BFloat16, sV_layout)[None, 0, None, None]
        sU = allocate_tensor(smem, BFloat16, sU_layout)[None, 0, None, 0]
        sW = allocate_tensor(smem, BFloat16, sW_layout)[None, 0, None, 0]

        swizzle_128B = cute.make_swizzle(3, 4, 3)
        sA_layout = cute.make_layout((BT, (64, 1)), stride=(64, (1, BT * 64)))
        sA_layout = cute.make_composed_layout(swizzle_128B, 0, sA_layout)
        sA = allocate_tensor(smem, BFloat16, sA_layout)
        sAi = allocate_tensor(smem, BFloat16, sA_layout)

        s_beta = smem.allocate_array(Float32, BT)
        s_g_cu_exp = smem.allocate_array(Float32, BT)
        s_g_cu = smem.allocate_array(Float32, BT)

        tma_mbar = smem.allocate_array(Int64, num_stages)
        mma_kkt_mbar = smem.allocate_array(Int64, num_stages)
        inv_mbar = smem.allocate_array(Int64, num_stages)
        mma_u_mbar = smem.allocate_array(Int64, num_stages)
        mma_w_mbar = smem.allocate_array(Int64, num_stages)
        epi_mbar = smem.allocate_array(Int64, num_stages)
        taddr = smem.allocate(Int32, 4)

        kkt_tmem = 0
        U_tmem_base = kkt_tmem + BT
        Ab_tmem_base = U_tmem_base + V_dim * num_stages
        assert Ab_tmem_base + (BT // 2) * num_stages <= 512

        # prepare ldmatrix/stmatrix ops
        ldsm_op = warp.LdMatrix8x8x16bOp(num_matrices=4)
        stsm_op = warp.StMatrix8x8x16bOp(num_matrices=4)
        ldsm_trans_op = warp.LdMatrix8x8x16bOp(num_matrices=4, transpose=True)
        ldsm_atom = cute.make_copy_atom(ldsm_op, BFloat16)
        stsm_atom = cute.make_copy_atom(stsm_op, BFloat16)
        ldsm_trans_atom = cute.make_copy_atom(ldsm_trans_op, BFloat16)

        if warp_id == 0:
            with cute.arch.elect_one():
                for i in cutlass.range_constexpr(num_stages):
                    cute.arch.mbarrier_init(tma_mbar + i, 1)
                    cute.arch.mbarrier_init(mma_kkt_mbar + i, 1)
                    cute.arch.mbarrier_init(inv_mbar + i, 128)
                    cute.arch.mbarrier_init(mma_u_mbar + i, 1)
                    cute.arch.mbarrier_init(mma_w_mbar + i, 1)
                    cute.arch.mbarrier_init(epi_mbar + i, 128)
                cute.arch.mbarrier_init_fence()
        elif warp_id == 1:
            cpasync.prefetch_descriptor(K_tma_atom)
            cpasync.prefetch_descriptor(V_tma_atom)
            cpasync.prefetch_descriptor(U_tma_atom)
            cpasync.prefetch_descriptor(W_tma_atom)
        cute.arch.sync_threads()

        num_global_chunks = total_chunks[0]
        if warp_id == 9:
            # TMA warp
            stage_id = 0
            parity = 1

            for global_chunk_id in range(bid, num_global_chunks, grid_x):
                seq_id = chunk_indices[global_chunk_id, 0]
                chunk_id = chunk_indices[global_chunk_id, 1]
                bos = cu_seqlens[seq_id]

                # since off_t is not a multiple of BT, we need to use
                # domain_offset() to shift the pointer first.
                mbar = tma_mbar + stage_id
                gK = cute.local_tile(
                    cute.domain_offset((bos, 0), tmaK[None, k_head_id, None]),
                    tiler=(BT, K_dim),
                    coord=(chunk_id, 0),
                )
                gV = cute.local_tile(
                    cute.domain_offset((bos, 0), tmaV[None, head_id, None]),
                    tiler=(BT, V_dim),
                    coord=(chunk_id, 0),
                )

                # when UW MMA is done, K and V TMA buffers are released
                cute.arch.mbarrier_wait(mma_u_mbar + stage_id, parity)

                with cute.arch.elect_one():
                    STAGE_SIZE = BT * (K_dim + V_dim) * 2
                    cute.arch.mbarrier_arrive_and_expect_tx(mbar, STAGE_SIZE)
                simple_tma_copy(K_tma_atom, gK, sK[None, None, stage_id], mbar)
                simple_tma_copy(
                    V_tma_atom, gV, sV[None, None, stage_id], mbar, EVICT_FIRST
                )

                stage_id = (stage_id + 1) % num_stages
                if stage_id == 0:
                    parity ^= 1

        elif warp_id == 8:
            # MMA warp
            _tcgen05.alloc(taddr)

            stage_id = 0
            parity = 0

            kkt_idesc = _tcgen05.make_bf16_idesc(BT, BT)
            u_idesc = _tcgen05.make_bf16_idesc(BT, V_dim, transpose_B=True)
            w_idesc = _tcgen05.make_bf16_idesc(BT, K_dim, transpose_B=True)

            # LBO=BT*128 is ignored for K-major
            sdesc_template = _tcgen05.make_sdesc_128B_swizzle(BT * 128)

            for global_chunk_id in range(bid, num_global_chunks, grid_x):
                U_tmem = U_tmem_base + V_dim * stage_id
                W_tmem = U_tmem | (16 << 16)
                Ab_tmem = Ab_tmem_base + (BT // 2) * stage_id
                Abg_tmem = Ab_tmem | (16 << 16)

                ##### KKT MMA: KKT = K @ K.T #####
                kaddr = sK[None, None, stage_id].iterator.toint()
                kdesc_base = sdesc_template | (kaddr >> 4)

                # wait for TMA data to arrive
                # kkt tmem is guaranteed to be free as this is issued
                # after the previous kkt's consumer (inv warps)
                cute.arch.mbarrier_wait(tma_mbar + stage_id, parity)
                _tcgen05.fence_after_thread_sync()

                with cute.arch.elect_one():
                    for i in cutlass.range_constexpr(K_dim // 64):
                        for j in cutlass.range_constexpr(64 // 16):
                            kdesc = kdesc_base | ((i * BT * 128 + j * 32) >> 4)
                            _tcgen05.mma_f16(
                                kkt_tmem,
                                kdesc,
                                kdesc,
                                kkt_idesc,
                                (i > 0) or (j > 0),
                            )
                    _tcgen05.commit(mma_kkt_mbar + stage_id)

                ##### U/W MMA: U = Ab @ V, W = Abg @ K #####
                vaddr = sV[None, None, stage_id].iterator.toint()
                vdesc = sdesc_template | (vaddr >> 4)
                kdesc = sdesc_template | (kaddr >> 4)

                # wait for epilogue to release tmem buffer
                cute.arch.mbarrier_wait(epi_mbar + stage_id, parity ^ 1)
                cute.arch.mbarrier_wait(inv_mbar + stage_id, parity)
                _tcgen05.fence_after_thread_sync()

                with cute.arch.elect_one():
                    for i in cutlass.range_constexpr(BT // 16):
                        _tcgen05.mma_ts_f16(
                            W_tmem, Abg_tmem + i * 8, kdesc, w_idesc, i > 0
                        )
                        kdesc += (16 * 128) >> 4
                    _tcgen05.commit(mma_w_mbar + stage_id)

                    for i in cutlass.range_constexpr(BT // 16):
                        _tcgen05.mma_ts_f16(
                            U_tmem, Ab_tmem + i * 8, vdesc, u_idesc, i > 0
                        )
                        vdesc += (16 * 128) >> 4
                    _tcgen05.commit(mma_u_mbar + stage_id)

                stage_id = (stage_id + 1) % num_stages
                if stage_id == 0:
                    parity ^= 1

            cute.arch.mbarrier_wait(epi_mbar + stage_id, parity ^ 1)
            _tcgen05.dealloc()

        elif warp_id >= 4:
            # inv warps
            tid_ = tid % 128
            warp_id_ = warp_id % 4

            stage_id = 0
            parity = 0

            # view into (16,16) sub-tiles, then ldmatrix layout
            sA_ldsm = cute.logical_divide(sA, (16, cute.make_layout((8, 2))))
            sAi_ldsm = cute.logical_divide(sAi, (16, cute.make_layout((8, 2))))
            sA_ldsm = sA_ldsm[(lane_id % 16, None), ((None, lane_id // 16), None)]
            sAi_ldsm = sAi_ldsm[(lane_id % 16, None), ((None, lane_id // 16), None)]

            # init Ai smem buffer with zeros (only the first 48 rows)
            for i in cutlass.range_constexpr((BT // 4 * 3) * BT // 128):
                idx = i * 128 + tid_
                sAi[idx // BT, idx % BT] = BFloat16(0.0)

            # indices for ldmatrix layout later
            row_indices = cute.make_rmem_tensor((1, 2, 1), Int32)
            row_indices[0, 0, 0] = warp_id_ * 16 + (lane_id // 4)
            row_indices[0, 1, 0] = warp_id_ * 16 + (lane_id // 4) + 8
            row_indices = row_indices.load()

            col_indices = cute.make_rmem_tensor((2, 1, 2), Int32)
            col_indices[0, 0, 0] = (lane_id % 4) * 2 + 0
            col_indices[1, 0, 0] = (lane_id % 4) * 2 + 1
            col_indices[0, 0, 1] = (lane_id % 4) * 2 + 8
            col_indices[1, 0, 1] = (lane_id % 4) * 2 + 9
            col_indices = col_indices.load()

            for global_chunk_id in range(bid, num_global_chunks, grid_x):
                seq_id = chunk_indices[global_chunk_id, 0]
                chunk_id = chunk_indices[global_chunk_id, 1]
                bos = cu_seqlens[seq_id]
                eos = cu_seqlens[seq_id + 1]
                off_t = bos + chunk_id * BT

                t = off_t + tid_

                ##### Phase 1: load g and beta #####
                if tid_ < BT:
                    in_bounds = t < eos
                    beta_val = beta[t, head_id] if in_bounds else Float32(0.0)
                    g_val = g[t, head_id] if in_bounds else Float32(0.0)

                    s_beta[tid_] = beta_val

                    # compute cumsum(g)
                    # parallel scan within a warp
                    for i in cutlass.range_constexpr(5):
                        offset = cutlass.const_expr(1 << i)
                        lower = cute.arch.shuffle_sync_up(
                            g_val, offset, mask_and_clamp=0
                        )
                        if lane_id >= offset:
                            g_val += lower

                    # store warp sum
                    if lane_id == 31:
                        s_g_cu[warp_id_] = g_val
                    cute.arch.barrier(barrier_id=3, number_of_threads=BT)

                    # add warp sum from lower warps
                    for i in cutlass.range_constexpr(1, BT // 32):
                        if warp_id_ >= i:
                            g_val += s_g_cu[i - 1]
                    cute.arch.barrier(barrier_id=3, number_of_threads=BT)

                    # store g_cu to gmem for H and O kernels
                    if in_bounds:
                        g_cu[t, head_id] = g_val

                    # store g and g_cu to smem for later
                    s_g_cu[tid_] = g_val
                    s_g_cu_exp[tid_] = cute.math.exp(g_val) if in_bounds else 0.0

                ##### Phase 2: A = strictLower(beta * kkt * Gamma) #####
                if warp_id_ == 0:
                    cute.arch.mbarrier_wait(mma_kkt_mbar + stage_id, parity)
                cute.arch.barrier(barrier_id=1, number_of_threads=128)
                _tcgen05.fence_after_thread_sync()

                # tmem 16x256b layout / ldmatrix layout
                # mode0 is 8 rows together
                # mode1 is top and bottom 8 rows
                # mode2 is groups of 16 rows
                row_coord = (lane_id // 4, None, warp_id_)
                s_beta_view = cute.make_tensor(s_beta, (8, 2, 4))
                beta_row = s_beta_view[row_coord].load().reshape((1, 2, 1))

                s_g_cu_view = cute.make_tensor(s_g_cu, (8, 2, 4))
                g_cu_row = s_g_cu_view[row_coord].load().reshape((1, 2, 1))

                # mode0 is 2 consecutive elems
                # mode1 is top and bottom 8 rows
                # mode2 is next 8 columns
                # mode3 is repeating that 16x16 tile pattern
                kkt = _tcgen05.ld(kkt_tmem, 0, "16x256b", BT // 8)
                kkt = kkt.reshape((2, 2, 2, BT // 16))

                for i in cutlass.range_constexpr(BT // 16):
                    # mode0 is 2 elems next to each other
                    # mode1 is 4 pairs of elems on 1 row
                    # mode2 is top and bottom 8 rows
                    # mode3 is next 16 columns
                    col_coord = (None, lane_id % 4, None, i)
                    s_g_cu_view = cute.make_tensor(s_g_cu, (2, 4, 2, BT // 16))
                    g_cu_col = s_g_cu_view[col_coord].load().reshape((2, 1, 2))

                    Gamma = cute.math.exp(g_cu_row - g_cu_col, fastmath=True)
                    A = kkt[None, None, None, i] * beta_row * Gamma

                    # strict lower mask
                    # NOTE: for OOB t position, s_beta is filled with zeros.
                    # hence, we don't need to apply bounds check for columns.
                    A_masked = cute.where(row_indices > col_indices + i * 16, A, 0.0)

                    # pack to BF16
                    # CuteDSL doesn't generate cvt.bf16x2.f32 here for some reasons
                    packed = cute.make_rmem_tensor(4, Uint32)
                    packed[0] = cvt.fp32x2_to_bf16x2(
                        A_masked[0, 0, 0], A_masked[1, 0, 0]
                    )
                    packed[1] = cvt.fp32x2_to_bf16x2(
                        A_masked[0, 1, 0], A_masked[1, 1, 0]
                    )
                    packed[2] = cvt.fp32x2_to_bf16x2(
                        A_masked[0, 0, 1], A_masked[1, 0, 1]
                    )
                    packed[3] = cvt.fp32x2_to_bf16x2(
                        A_masked[0, 1, 1], A_masked[1, 1, 1]
                    )

                    # store to smem
                    cute.copy(
                        stsm_atom,
                        cute.recast_tensor(packed, BFloat16),
                        sA_ldsm[warp_id_, None, i],
                    )

                cute.arch.barrier(barrier_id=1, number_of_threads=128)

                ##### Phase 3: matrix inverse #####
                # we use Newton-Schulz iterations to compute the inverse
                # of the four 16x16 diagonal blocks.
                #   Ai_new = 2 Ai - Ai @ M @ Ai
                #   where M = I + A
                #
                # we do this with 2 MMAs:
                # 1. -AiM = Ai @ (-M)
                # 2. Ai_new = 2 Ai + (-AiM) @ Ai
                zeros_f32 = cute.make_rmem_tensor(4, Float32)
                zeros_f32.fill(0.0)

                def set_diagonal(A: cute.Tensor, lane_id: Int32):
                    "Set the diagonal to 1s"
                    if lane_id % 9 == 0:
                        A[0] = (A[0] & Uint32(0xFFFF0000)) | Uint32(0x00003F80)
                        A[3] = (A[3] & Uint32(0xFFFF0000)) | Uint32(0x00003F80)
                    elif lane_id % 9 == 4:
                        A[0] = (A[0] & Uint32(0x0000FFFF)) | Uint32(0x3F800000)
                        A[3] = (A[3] & Uint32(0x0000FFFF)) | Uint32(0x3F800000)

                Ai_bf16 = cute.make_rmem_tensor(8, BFloat16)
                mma_B_bf16 = cute.make_rmem_tensor(8, BFloat16)
                M_bf16 = cute.make_rmem_tensor(8, BFloat16)
                acc = cute.make_rmem_tensor((4, 2), Float32)

                # share the same storage
                Ai = cute.recast_tensor(Ai_bf16, Uint32)
                mma_B = cute.logical_divide(cute.recast_tensor(mma_B_bf16, Uint32), 2)
                M = cute.logical_divide(cute.recast_tensor(M_bf16, Uint32), 2)

                # initial guess: Ai = I-A
                cute.copy(ldsm_atom, sA_ldsm[warp_id_, None, warp_id_], Ai_bf16)
                for i in cutlass.range_constexpr(4):
                    Ai[i] ^= Uint32(0x80008000)  # negate A
                set_diagonal(Ai, lane_id)

                # (4, 2)
                Ai_f32 = cute.logical_divide(cvt.bf16x2_to_fp32x2(Ai), 4)

                # M is holding -(I+A), stay constant throughout the iterations
                cute.copy(ldsm_trans_atom, sA_ldsm[warp_id_, None, warp_id_], M_bf16)
                set_diagonal(M, lane_id)
                for i in cutlass.range_constexpr(4):
                    M[i] ^= Uint32(0x80008000)

                # 3 rounds of Newton-Schulz
                for _ in cutlass.range_constexpr(3):
                    # First MMA: -AiM = Ai @ (-M)
                    cute.copy(stsm_atom, Ai_bf16, sA_ldsm[warp_id_, None, warp_id_])
                    cute.arch.sync_warp()
                    acc[None, 0] = mma_bf16(Ai, M[None, 0], zeros_f32)
                    acc[None, 1] = mma_bf16(Ai, M[None, 1], zeros_f32)
                    Ai_bf16.store(acc.load().to(BFloat16))

                    # Second MMA: Ai_new = 2Ai + (-AiM) @ Ai
                    for j in cutlass.range_constexpr(8):
                        Ai_f32[j] *= 2.0
                    cute.copy(
                        ldsm_trans_atom,
                        sA_ldsm[warp_id_, None, warp_id_],
                        mma_B_bf16,
                    )
                    Ai_f32[None, 0] = mma_bf16(Ai, mma_B[None, 0], Ai_f32[None, 0])
                    Ai_f32[None, 1] = mma_bf16(Ai, mma_B[None, 1], Ai_f32[None, 1])
                    Ai_bf16.store(Ai_f32.load().to(BFloat16))

                cute.copy(stsm_atom, Ai_bf16, sAi_ldsm[warp_id_, None, warp_id_])
                cute.arch.barrier(barrier_id=1, number_of_threads=128)

                # off-diagonal by 1
                # given
                # [ Ai00               ]
                # [  A10 Ai11          ]
                # [  A20  A21 Ai22     ]
                # [  A30  A31  A32 Ai33]
                # warp1: Ai10 = -Ai11 @ A10 @ Ai00
                # warp2: Ai21 = -Ai22 @ A21 @ Ai11
                # warp3: Ai32 = -Ai33 @ A32 @ Ai22
                if warp_id_ > 0:
                    neg_Ai = cute.make_rmem_tensor(4, Uint32)
                    for i in cutlass.range_constexpr(4):
                        neg_Ai[i] = Ai[i] ^ Uint32(0x80008000)

                    cute.copy(
                        ldsm_trans_atom,
                        sA_ldsm[warp_id_, None, warp_id_ - 1],
                        mma_B_bf16,
                    )
                    acc[None, 0] = mma_bf16(neg_Ai, mma_B[None, 0], zeros_f32)
                    acc[None, 1] = mma_bf16(neg_Ai, mma_B[None, 1], zeros_f32)
                    Ai_bf16.store(acc.load().to(BFloat16))

                    cute.copy(
                        ldsm_trans_atom,
                        sAi_ldsm[warp_id_ - 1, None, warp_id_ - 1],
                        mma_B_bf16,
                    )
                    acc[None, 0] = mma_bf16(Ai, mma_B[None, 0], zeros_f32)
                    acc[None, 1] = mma_bf16(Ai, mma_B[None, 1], zeros_f32)
                    Ai_bf16.store(acc.load().to(BFloat16))
                    cute.copy(
                        stsm_atom,
                        Ai_bf16,
                        sAi_ldsm[warp_id_, None, warp_id_ - 1],
                    )
                cute.arch.barrier(barrier_id=1, number_of_threads=128)

                # off-diagonal by 2
                # warp0: Ai20 = -Ai22 @ (A20 @ Ai00 + A21 @ Ai10)
                # warp1: Ai31 = -Ai33 @ (A31 @ Ai11 + A32 @ Ai21)
                if warp_id_ < 2:
                    cute.copy(
                        ldsm_atom,
                        sA_ldsm[warp_id_ + 2, None, warp_id_],
                        Ai_bf16,
                    )
                    cute.copy(
                        ldsm_trans_atom,
                        sAi_ldsm[warp_id_, None, warp_id_],
                        mma_B_bf16,
                    )
                    acc[None, 0] = mma_bf16(Ai, mma_B[None, 0], zeros_f32)
                    acc[None, 1] = mma_bf16(Ai, mma_B[None, 1], zeros_f32)

                    cute.copy(
                        ldsm_atom,
                        sA_ldsm[warp_id_ + 2, None, warp_id_ + 1],
                        Ai_bf16,
                    )
                    cute.copy(
                        ldsm_trans_atom,
                        sAi_ldsm[warp_id_ + 1, None, warp_id_],
                        mma_B_bf16,
                    )
                    acc[None, 0] = mma_bf16(Ai, mma_B[None, 0], acc[None, 0])
                    acc[None, 1] = mma_bf16(Ai, mma_B[None, 1], acc[None, 1])

                    tmp = cute.make_rmem_tensor(8, BFloat16)
                    tmp.store(acc.load().to(BFloat16))
                    cute.copy(stsm_atom, tmp, sAi_ldsm[warp_id_ + 2, None, warp_id_])
                    cute.arch.sync_warp()

                    cute.copy(
                        ldsm_atom, sAi_ldsm[warp_id_ + 2, None, warp_id_ + 2], Ai_bf16
                    )
                    for i in cutlass.range_constexpr(4):
                        Ai[i] ^= Uint32(0x80008000)
                    cute.copy(
                        ldsm_trans_atom,
                        sAi_ldsm[warp_id_ + 2, None, warp_id_],
                        mma_B_bf16,
                    )
                    acc[None, 0] = mma_bf16(Ai, mma_B[None, 0], zeros_f32)
                    acc[None, 1] = mma_bf16(Ai, mma_B[None, 1], zeros_f32)
                    tmp.store(acc.load().to(BFloat16))
                    cute.copy(stsm_atom, tmp, sAi_ldsm[warp_id_ + 2, None, warp_id_])
                cute.arch.barrier(barrier_id=1, number_of_threads=128)

                # off-diagonal by 3
                # warp0: Ai30 = -Ai33 @ (A30 @ Ai00 + A31 @ Ai10 + A32 @ Ai20)
                if warp_id_ == 0:
                    cute.copy(ldsm_atom, sA_ldsm[3, None, 0], Ai_bf16)
                    cute.copy(ldsm_trans_atom, sAi_ldsm[0, None, 0], mma_B_bf16)
                    acc[None, 0] = mma_bf16(Ai, mma_B[None, 0], zeros_f32)
                    acc[None, 1] = mma_bf16(Ai, mma_B[None, 1], zeros_f32)

                    for i in cutlass.range_constexpr(1, 3):
                        cute.copy(ldsm_atom, sA_ldsm[3, None, i], Ai_bf16)
                        cute.copy(ldsm_trans_atom, sAi_ldsm[i, None, 0], mma_B_bf16)
                        acc[None, 0] = mma_bf16(Ai, mma_B[None, 0], acc[None, 0])
                        acc[None, 1] = mma_bf16(Ai, mma_B[None, 1], acc[None, 1])

                    tmp = cute.make_rmem_tensor(8, BFloat16)
                    tmp.store(acc.load().to(BFloat16))
                    cute.copy(stsm_atom, tmp, sAi_ldsm[3, None, 0])
                    cute.arch.sync_warp()

                    cute.copy(ldsm_atom, sAi_ldsm[3, None, 3], Ai_bf16)
                    for i in cutlass.range_constexpr(4):
                        Ai[i] ^= Uint32(0x80008000)
                    cute.copy(ldsm_trans_atom, sAi_ldsm[3, None, 0], mma_B_bf16)
                    acc[None, 0] = mma_bf16(Ai, mma_B[None, 0], zeros_f32)
                    acc[None, 1] = mma_bf16(Ai, mma_B[None, 1], zeros_f32)
                    tmp.store(acc.load().to(BFloat16))
                    cute.copy(stsm_atom, tmp, sAi_ldsm[3, None, 0])

                ##### Phase 4: compute Ab, Abg #####
                if warp_id_ == 3:
                    cute.arch.mbarrier_wait(mma_u_mbar + stage_id, parity ^ 1)
                cute.arch.barrier(barrier_id=1, number_of_threads=128)

                for i in cutlass.range_constexpr(BT // 16):
                    cute.copy(ldsm_atom, sAi_ldsm[warp_id_, None, i], Ai_bf16)

                    col_coord = (None, lane_id % 4, None, i)
                    s_beta_view = cute.make_tensor(s_beta, (2, 4, 2, BT // 16))
                    beta_col = s_beta_view[col_coord].load().reshape((2, 1, 2))

                    s_g_cu_view = cute.make_tensor(s_g_cu_exp, (2, 4, 2, BT // 16))
                    g_cu_col = s_g_cu_view[col_coord].load().reshape((2, 1, 2))

                    Ai_f32 = cvt.bf16x2_to_fp32x2(Ai).load().reshape((2, 2, 2))

                    Ab_f32 = Ai_f32 * beta_col
                    Ab = Ab_f32.to(BFloat16)
                    Ab_tmem = Ab_tmem_base + (BT // 2) * stage_id + i * 8
                    _tcgen05.st(warp_id_ * 32, Ab_tmem, "16x128b", 2, Ab)

                    Abg_f32 = Ab_f32 * g_cu_col
                    Abg = Abg_f32.to(BFloat16)
                    _tcgen05.st(warp_id_ * 32 + 16, Ab_tmem, "16x128b", 2, Abg)

                _tcgen05.wait_st()
                _tcgen05.fence_before_thread_sync()
                cute.arch.mbarrier_arrive(inv_mbar + stage_id)

                stage_id = (stage_id + 1) % num_stages
                if stage_id == 0:
                    parity ^= 1

        elif warp_id < 4:
            # epi warps
            stage_id = 0
            parity = 0

            # ((BT, num_global_chunks), V_dim)
            gU_tiles = cute.logical_divide(tmaU[None, head_id, None], (BT, None))
            gW_tiles = cute.logical_divide(tmaW[None, head_id, None], (BT, None))

            # sW shape: [BT, (64, K_dim/64)]
            # sW_view shape: [(8, 2), (4, K_dim/64)]
            s_row = warp_id * 16 + lane_id % 16  # select the rows of [16,16] tile
            sW_view = cute.zipped_divide(
                sW[s_row, None],
                tiler=cute.make_layout((8, 2)),
            )
            sU_view = cute.zipped_divide(
                sU[s_row, None],
                tiler=cute.make_layout((8, 2)),
            )

            # select the 8 columns within [16,16] tile
            sW_view = sW_view[(None, lane_id // 16), None]
            sU_view = sU_view[(None, lane_id // 16), None]

            for global_chunk_id in range(bid, num_global_chunks, grid_x):
                # wait for W MMA + previous TMA store to finish
                U_tmem = U_tmem_base + V_dim * stage_id
                if warp_id == 0:
                    cute.arch.mbarrier_wait(mma_w_mbar + stage_id, parity)
                elif warp_id == 1:
                    with cute.arch.elect_one():
                        cute.arch.cp_async_bulk_wait_group(0, read=True)
                cute.arch.barrier(barrier_id=2, number_of_threads=128)
                _tcgen05.fence_after_thread_sync()

                w_f32 = _tcgen05.ld(warp_id * 32 + 16, U_tmem, "16x256b", K_dim // 8)
                _tcgen05.wait_ld()
                w_bf16 = cute.make_rmem_tensor((8, K_dim // 16), BFloat16)
                w_bf16.store(w_f32.to(BFloat16))
                cute.copy(stsm_atom, w_bf16, sW_view)

                # wait for U MMA + issue W TMA store
                cute.arch.barrier(barrier_id=2, number_of_threads=128)
                fence_before_tma_store()
                if warp_id == 0:
                    cute.arch.mbarrier_wait(mma_u_mbar + stage_id, parity)
                elif warp_id == 1:
                    # don't need to commit
                    simple_tma_copy(
                        W_tma_atom, sW, gW_tiles[(None, global_chunk_id), None]
                    )
                cute.arch.barrier(barrier_id=2, number_of_threads=128)
                _tcgen05.fence_after_thread_sync()

                u_f32 = _tcgen05.ld(warp_id * 32, U_tmem, "16x256b", V_dim // 8)
                _tcgen05.wait_ld()
                _tcgen05.fence_before_thread_sync()
                cute.arch.mbarrier_arrive(epi_mbar + stage_id)
                u_bf16 = cute.make_rmem_tensor((8, V_dim // 16), BFloat16)
                u_bf16.store(u_f32.to(BFloat16))
                cute.copy(stsm_atom, u_bf16, sU_view)

                cute.arch.barrier(barrier_id=2, number_of_threads=128)
                fence_before_tma_store()
                if warp_id == 1:
                    simple_tma_copy(
                        U_tma_atom, sU, gU_tiles[(None, global_chunk_id), None]
                    )
                    with cute.arch.elect_one():
                        cute.arch.cp_async_bulk_commit_group()

                stage_id = (stage_id + 1) % num_stages
                if stage_id == 0:
                    parity ^= 1

    @cache
    @staticmethod
    def compile(H: int, Hv: int, K_dim: int, V_dim: int, num_stages: int = 2):
        total_t = cute.sym_int()
        pad_t = cute.sym_int()
        total_chunks_n = cute.sym_int()
        num_sequences = cute.sym_int()

        K = make_fake_tensor(BFloat16, (total_t, H, K_dim), divisibility=16)
        V = make_fake_tensor(BFloat16, (total_t, Hv, V_dim), divisibility=16)
        U = make_fake_tensor(BFloat16, (pad_t, Hv, V_dim), divisibility=16)
        W = make_fake_tensor(BFloat16, (pad_t, Hv, K_dim), divisibility=16)
        g = make_fake_tensor(Float32, (total_t, Hv), divisibility=4)
        beta = make_fake_tensor(Float32, (total_t, Hv), divisibility=4)
        g_cu = make_fake_tensor(Float32, (total_t, Hv), divisibility=4)
        cu_seqlens = make_fake_tensor(Int32, (num_sequences,), divisibility=1)
        chunk_indices = make_fake_tensor(Int32, (total_chunks_n, 2), divisibility=2)
        total_chunks = make_fake_tensor(Int32, (1,), divisibility=1)

        kernel = Sm100ChunkUWKernel(H, Hv, K_dim, V_dim, num_stages)
        stream = cute.runtime.make_fake_stream(use_tvm_ffi_env_stream=True)
        return cute.compile(
            kernel,
            K,
            V,
            U,
            W,
            g,
            beta,
            g_cu,
            cu_seqlens,
            chunk_indices,
            total_chunks,
            Int32(148),
            stream,
            options="--enable-tvm-ffi",
        )