Project: 069_FlexPrefill_A_Context-Aware_Sparse_Attention_Mecha | Function: square_root_js_divergence (iclr)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Mathematical implementation of Jensen-Shannon divergence calculation that forms the foundation for Query-Aware pattern determination, measuring distributional difference between query and uniform distributions

**KEY TERMS:** {
  "P": "first probability distribution (query)",
  "Q": "second probability distribution (uniform)",
  "M": "(P+Q)/2 average distribution",
  "kl_divergence": "KL(P||Q) Kullback-Leibler divergence",
  "js_divergence": "Jensen-Shannon divergence metric"
}

**CONTEXT CODE:**     mat_strided = mat_padded.as_strided(
        (b, h, m, n), (h * n * (n + m - 1), n * (n + m - 1), 1, n + m)
    )
    sum_diags = torch.sum(mat_strided, -1)
    return sum_diags


def transform_veritcal_slash_idx(v_idx, s_idx, num_blocks):
    batch_size, num_heads, _ = v_idx.shape
    range_blocks = torch.arange(num_blocks, device=s_idx.device)[None, None, :, None]
    # vertical
    v_idx = (
        torch.arange(0, num_blocks, device=v_idx.device)[None, None, :, None]
        * num_blocks
        + v_idx[:, :, None, :]
    ).view(batch_size, num_heads, -1)
    v_idx[v_idx // num_blocks < v_idx % num_blocks] = 0
    # slash
    s_idx = (
        range_blocks * num_blocks + range_blocks + s_idx[:, :, None, :] * num_blocks
    ).view(batch_size, num_heads, -1)
    s_idx[s_idx >= num_blocks * num_blocks] = 0
    # union
    vs_idx = torch.cat((s_idx, v_idx), dim=-1)
    block_idx = [
        [torch.unique(vs_idx[b, h]) for h in range(num_heads)]
        for b in range(batch_size)
    ]
    return block_idx


causal_mask = None


def get_block_vertical_slash_from_qk(
    qk: torch.Tensor,
    block_size: int,
):
    batch_size, num_heads, last_q_len, seq_len = qk.shape
    # slash shape: [batch_size, num_heads, seq_len] -> [batch_size, num_heads, num_blocks]
    slash = sum_all_diagonal_matrix(qk)
    slash = torch_bhn_sumpool(slash, block_size)
    slash = slash / last_q_len
    # vertical shape: [batch_size, num_heads, seq_len] -> [batch_size, num_heads, num_blocks]
    vertical = qk.sum(-2)
    vertical = torch_bhn_sumpool(vertical, block_size)
    vertical = vertical / last_q_len
    return vertical, slash


def square_root_js_divergence(p: torch.Tensor, q: torch.Tensor):
    raise NotImplementedError("Function implementation has been masked for testing")

@triton.jit
def prefill_kernel(
    q_ptr,  # Q: b x n x h x d
    k_ptr,  # K: b x n x h x d
    v_ptr,  # V: b x n x h x d
    o_ptr,
    # shape
    BATCH_SIZE,
    NUM_HEADS,
    NUM_KV_HEADS,
    NUM_SHARE_Q_HEADS,
    Q_LEN,
    K_LEN,
    HEAD_DIM: tl.constexpr,
    # softmax_scale
    softmax_scale,
    # causal
    causal,
    # gqa
    gqa_interleave,
    # stride
    stride_qb,
    stride_qn,
    stride_qh,
    stride_qd,
    stride_kb,
    stride_kn,
    stride_kh,
    stride_kd,
    stride_vb,
    stride_vn,
    stride_vh,
    stride_vd,
    stride_ob,
    stride_on,
    stride_oh,
    stride_od,
    # META parameters
    BLOCK_SIZE_Q: tl.constexpr,  # q block size
    BLOCK_SIZE_K: tl.constexpr,  # k block size
):
    raise NotImplementedError("Function implementation has been masked for testing")



from transformers.utils import is_flash_attn_2_available

if is_flash_attn_2_available():
    from flash_attn import flash_attn_func
else:
    flash_attn_func = triton_flash_attention


@torch.no_grad()
def flex_prefill_attention(
    q: torch.Tensor,
    k: torch.Tensor,
    v: torch.Tensor,
    gamma: float,
    tau: float = 0,
    min_budget: int = None,
    max_budget: int = None,
    gqa_interleave: bool = False,
    softmax_scale: Optional[float] = None,
    block_size: int = 128,
    return_computational_ratio: bool = False,
) -> Union[torch.Tensor, Tuple[torch.Tensor, float]]:
    """Flex Prefill sparse attention function. If query length is 1, will use flash decoding attention.

    Args:
        q (torch.Tensor): query tensor, shape [batch_size, q_len, num_q_heads, head_dim]
        k (torch.Tensor): key tensor, shape [batch_size, kv_len, num_kv_heads, head_dim]
        v (torch.Tensor): value tensor, shape [batch_size, kv_len, num_kv_heads, head_dim]
        gamma (float): attention coverage ratio, (0, 1).
        tau (float, optional): query aware head threshold, [0, 1]. Defaults to 0.
        min_budget (int, optional): minimum number of calculated tokens. Defaults to None.
        max_budget (int, optional): maximum number of calculated tokens. Defaults to None.
        gqa_interleave (bool, optional): GQA pattern. Defaults to False.
        softmax_scale (Optional[float], optional): softmax scale. Defaults to None, which means sqrt(head_dim).
        block_size (int, optional): block size. Defaults to 128.
        return_computational_ratio (bool, optional): whether to return computation ratio. Defaults to False.

    Returns:
        Union[torch.Tensor, Tuple[torch.Tensor, float]]: if return_computational_ratio is True, return attention output, else return attention output and computation ratio.
    """
    raise NotImplementedError("Function implementation has been masked for testing")


if __name__ == "__main__":
    torch.manual_seed(0)
    B, N, H, D = 1, 64000, 32, 128
    gamma = 0.9
    tau = 0.1

    q = torch.randn(B, N, H, D, device="cuda", dtype=torch.bfloat16) / 0.5
    k = torch.randn(B, N, H // 4, D, device="cuda", dtype=torch.bfloat16) / 0.5
    v = torch.randn(B, N, H // 4, D, device="cuda", dtype=torch.bfloat16)


Complete the function implementation replacing "raise NotImplementedError".