code: def backward(ctx, grad_output):
      if ctx.is_empty:
          bias_grad = (
              None if ctx.bias is None else torch.zeros_like(ctx.bias)
          )
          return (
              torch.zeros_like(ctx.A),
              torch.zeros_like(ctx.B),
              None,
              bias_grad,
              None,
          )
      req_gradA, req_gradB, _, req_gradBias, _ = ctx.needs_input_grad
      CAt, subA, A = ctx.tensors
      SCAt, idx = ctx.tensor_states
      formatB = ctx.formatB
      state = ctx.state
      grad_A = grad_B = grad_bias = None
  
      if req_gradBias:
          # compute grad_bias first before changing grad_output dtype
          grad_bias = grad_output.sum(0, dtype=ctx.dtype_bias)
  
      # Cast grad_output to fp16
      if len(grad_output.shape) == 3:
          grad_output = grad_output.reshape(
              -1, grad_output.shape[-1]
          ).contiguous()
  
      Cgrad, Cgradt, SCgrad, SCgradt, coo_tensor = F.double_quant(
          grad_output.to(torch.float16)
      )
      if req_gradB:
          CxAt, SAt = F.transform(CAt, formatB, transpose=True)
          C32grad, Sgrad = F.transform(Cgradt, "col32", transpose=True)
          gradB32, SgradB32 = F.igemmlt(C32grad, CxAt, Sgrad, SAt)
          grad_B = F.mm_dequant(gradB32, SgradB32, SCgradt, SCAt)
          if state.threshold > 0.0 and subA is not None:
              grad_B[:, idx] += torch.matmul(grad_output.t(), subA)
  
      if req_gradA:
          if state.CBt is not None:
              C32grad, Sgrad = F.transform(Cgrad, "col32")
              if state.CxBt is None:
                  state.CxBt, state.SBt = F.transform(
                      state.CBt, to_order=formatB, transpose=True
                  )
              gradA32, SgradA32 = F.igemmlt(
                  C32grad, state.CxBt, Sgrad, state.SBt
              )
              grad_A = (
                  F.mm_dequant(gradA32, SgradA32, SCgrad, state.SCBt)
                  .view(ctx.grad_shape)
                  .to(ctx.dtype_A)
              )
  
          elif state.CB is not None:
              CB = state.CB.to(ctx.dtype_A, copy=True).mul_(
                  state.SCB.unsqueeze(1).mul(1.0 / 127.0)
              )
              grad_A = (
                  torch.matmul(grad_output, CB)
                  .view(ctx.grad_shape)
                  .to(ctx.dtype_A)
              )
          elif state.CxB is not None:
              CB = (
                  undo_layout(state.CxB, state.tile_indices)
                  .to(ctx.dtype_A)
                  .mul_(state.SCB.unsqueeze(1).mul(1.0 / 127.0))
              )
              grad_A = (
                  torch.matmul(grad_output, CB)
                  .view(ctx.grad_shape)
                  .to(ctx.dtype_A)
              )
          else:
              raise Exception(
                  "State must contain either CBt or CB or CxB matrix for backward"
              )
  
      return grad_A, grad_B, None, grad_bias, None
  
imports: import operator
  import warnings
  from dataclasses import dataclass
  from functools import reduce
  from typing import Tuple, Optional, List
  from warnings import warn
  import torch
  import bitsandbytes.functional
label_desc: Use image node_id 2 for calling the MatMul8bitLt obj's underlying member method code with example usage: obj.backward(ctx, grad_output) and returns: grad_A,  grad_B,  None,  grad_bias,  None, torch,  torch,  None,  bias_grad,  None