计算矩阵乘积态边际分布张量

以下代码示例说明了如何定义张量网络状态，计算其矩阵乘积态 (MPS) 分解，然后计算 MPS 分解状态的边际分布张量。完整的代码可以在 NVIDIA/cuQuantum 存储库中找到（此处）。

头文件和错误处理

#include <cstdlib>
#include <cstdio>
#include <cassert>
#include <complex>
#include <vector>
#include <bitset>
#include <iostream>

#include <cuda_runtime.h>
#include <cutensornet.h>


#define HANDLE_CUDA_ERROR(x) \
{ const auto err = x; \
  if( err != cudaSuccess ) \
  { printf("CUDA error %s in line %d\n", cudaGetErrorString(err), __LINE__); fflush(stdout); std::abort(); } \
};

#define HANDLE_CUTN_ERROR(x) \
{ const auto err = x; \
  if( err != CUTENSORNET_STATUS_SUCCESS ) \
  { printf("cuTensorNet error %s in line %d\n", cutensornetGetErrorString(err), __LINE__); fflush(stdout); std::abort(); } \
};


int main()
{
  static_assert(sizeof(size_t) == sizeof(int64_t), "Please build this sample on a 64-bit architecture!");

  constexpr std::size_t fp64size = sizeof(double);

定义张量网络状态和所需的边际分布张量

让我们定义一个对应于 16 量子比特量子线路的张量网络状态，并请求量子比特 0 和 1 的边际分布张量。

  // Quantum state configuration
  constexpr int32_t numQubits = 16;
  const std::vector<int64_t> qubitDims(numQubits,2); // qubit dimensions
  constexpr int32_t numMarginalModes = 2; // rank of the marginal (reduced density matrix)
  const std::vector<int32_t> marginalModes({0,1}); // open qubits (must be in acsending order)
  std::cout << "Quantum circuit: " << numQubits << " qubits\n";

初始化 cuTensorNet 库句柄

  // Initialize the cuTensorNet library
  HANDLE_CUDA_ERROR(cudaSetDevice(0));
  cutensornetHandle_t cutnHandle;
  HANDLE_CUTN_ERROR(cutensornetCreate(&cutnHandle));
  std::cout << "Initialized cuTensorNet library on GPU 0\n";

在 GPU 上定义量子门

  // Define necessary quantum gate tensors in Host memory
  const double invsq2 = 1.0 / std::sqrt(2.0);
  //  Hadamard gate
  const std::vector<std::complex<double>> h_gateH {{invsq2, 0.0},  {invsq2, 0.0},
                                                   {invsq2, 0.0}, {-invsq2, 0.0}};
  //  CX gate
  const std::vector<std::complex<double>> h_gateCX {{1.0, 0.0}, {0.0, 0.0}, {0.0, 0.0}, {0.0, 0.0},
                                                    {0.0, 0.0}, {1.0, 0.0}, {0.0, 0.0}, {0.0, 0.0},
                                                    {0.0, 0.0}, {0.0, 0.0}, {0.0, 0.0}, {1.0, 0.0},
                                                    {0.0, 0.0}, {0.0, 0.0}, {1.0, 0.0}, {0.0, 0.0}};

  // Copy quantum gates to Device memory
  void *d_gateH{nullptr}, *d_gateCX{nullptr};
  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateH, 4 * (2 * fp64size)));
  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateCX, 16 * (2 * fp64size)));
  std::cout << "Allocated quantum gate memory on GPU\n";
  HANDLE_CUDA_ERROR(cudaMemcpy(d_gateH, h_gateH.data(), 4 * (2 * fp64size), cudaMemcpyHostToDevice));
  HANDLE_CUDA_ERROR(cudaMemcpy(d_gateCX, h_gateCX.data(), 16 * (2 * fp64size), cudaMemcpyHostToDevice));
  std::cout << "Copied quantum gates to GPU memory\n";

分配 MPS 张量

在这里，我们设置 MPS 张量的形状，并为其存储分配 GPU 内存。

  // Determine the MPS representation and allocate buffers for the MPS tensors
  const int64_t maxExtent = 2; // GHZ state can be exactly represented with max bond dimension of 2
  std::vector<std::vector<int64_t>> extents;
  std::vector<int64_t*> extentsPtr(numQubits); 
  std::vector<void*> d_mpsTensors(numQubits, nullptr);
  for (int32_t i = 0; i < numQubits; i++) {
    if (i == 0) { // left boundary MPS tensor
      extents.push_back({2, maxExtent});
      HANDLE_CUDA_ERROR(cudaMalloc(&d_mpsTensors[i], 2 * maxExtent * 2 * fp64size));
    } 
    else if (i == numQubits-1) { // right boundary MPS tensor
      extents.push_back({maxExtent, 2});
      HANDLE_CUDA_ERROR(cudaMalloc(&d_mpsTensors[i], 2 * maxExtent * 2 * fp64size));
    }
    else { // middle MPS tensors
      extents.push_back({maxExtent, 2, maxExtent});
      HANDLE_CUDA_ERROR(cudaMalloc(&d_mpsTensors[i], 2 * maxExtent * maxExtent * 2 * fp64size));
    }
    extentsPtr[i] = extents[i].data();
  }

在 GPU 上分配边际分布张量

在这里，我们分配边际分布张量，即量子比特 0 和 1 的约化密度矩阵，在 GPU 上。

  // Allocate the specified quantum circuit reduced density matrix (marginal) in Device memory
  void *d_rdm{nullptr};
  std::size_t rdmDim = 1;
  for(const auto & mode: marginalModes) rdmDim *= qubitDims[mode];
  const std::size_t rdmSize = rdmDim * rdmDim;
  HANDLE_CUDA_ERROR(cudaMalloc(&d_rdm, rdmSize * (2 * fp64size)));

在 GPU 上分配暂存缓冲区

  // Query the free memory on Device
  std::size_t freeSize{0}, totalSize{0};
  HANDLE_CUDA_ERROR(cudaMemGetInfo(&freeSize, &totalSize));
  const std::size_t scratchSize = (freeSize - (freeSize % 4096)) / 2; // use half of available memory with alignment
  void *d_scratch{nullptr};
  HANDLE_CUDA_ERROR(cudaMalloc(&d_scratch, scratchSize));
  std::cout << "Allocated " << scratchSize << " bytes of scratch memory on GPU\n";

创建纯张量网络状态

现在，让我们为一个 16 量子比特量子线路创建一个纯张量网络状态。

  // Create the initial quantum state
  cutensornetState_t quantumState;
  HANDLE_CUTN_ERROR(cutensornetCreateState(cutnHandle, CUTENSORNET_STATE_PURITY_PURE, numQubits, qubitDims.data(),
                    CUDA_C_64F, &quantumState));
  std::cout << "Created the initial quantum state\n";

应用量子门

让我们通过应用相应的量子门来构建 GHZ 量子线路。

  // Construct the final quantum circuit state (apply quantum gates) for the GHZ circuit
  int64_t id;
  HANDLE_CUTN_ERROR(cutensornetStateApplyTensorOperator(cutnHandle, quantumState, 1, std::vector<int32_t>{{0}}.data(),
                    d_gateH, nullptr, 1, 0, 1, &id));
  for(int32_t i = 1; i < numQubits; ++i) {
    HANDLE_CUTN_ERROR(cutensornetStateApplyTensorOperator(cutnHandle, quantumState, 2, std::vector<int32_t>{{i-1,i}}.data(),
                      d_gateCX, nullptr, 1, 0, 1, &id));
  }
  std::cout << "Applied quantum gates\n";

请求最终量子线路状态的 MPS 分解

在这里，我们表达了使用 MPS 分解来分解最终量子线路状态的意图。提供的 MPS 张量的形状指的是 MPS 重整化过程中的最大尺寸限制。最终 MPS 张量的实际计算形状可能更小。此处尚未进行任何计算。

  // Specify the final target MPS representation (use default fortran strides)
  HANDLE_CUTN_ERROR(cutensornetStateFinalizeMPS(cutnHandle, quantumState, 
                    CUTENSORNET_BOUNDARY_CONDITION_OPEN, extentsPtr.data(), /*strides=*/nullptr));
  std::cout << "Requested the final MPS factorization of the quantum circuit state\n";

配置 MPS 分解过程

在表达了执行最终量子线路状态的 MPS 分解的意图后，我们还可以通过重置不同的选项（例如，SVD 算法）来配置 MPS 分解过程。

  // Optional, set up the SVD method for MPS truncation.
  cutensornetTensorSVDAlgo_t algo = CUTENSORNET_TENSOR_SVD_ALGO_GESVDJ; 
  HANDLE_CUTN_ERROR(cutensornetStateConfigure(cutnHandle, quantumState, 
                    CUTENSORNET_STATE_CONFIG_MPS_SVD_ALGO, &algo, sizeof(algo)));
  std::cout << "Configured the MPS factorization computation\n";

准备 MPS 分解的计算

让我们创建一个工作区描述符并准备 MPS 分解的计算。

  // Prepare the MPS computation and attach workspace
  cutensornetWorkspaceDescriptor_t workDesc;
  HANDLE_CUTN_ERROR(cutensornetCreateWorkspaceDescriptor(cutnHandle, &workDesc));
  std::cout << "Created the workspace descriptor\n";
  HANDLE_CUTN_ERROR(cutensornetStatePrepare(cutnHandle, quantumState, scratchSize, workDesc, 0x0));
  std::cout << "Prepared the computation of the quantum circuit state\n";
  double flops {0.0};
  HANDLE_CUTN_ERROR(cutensornetStateGetInfo(cutnHandle, quantumState,
                    CUTENSORNET_STATE_INFO_FLOPS, &flops, sizeof(flops)));
  if(flops > 0.0) {
    std::cout << "Total flop count = " << (flops/1e9) << " GFlop\n";
  }else if(flops < 0.0) {
    std::cout << "ERROR: Negative Flop count!\n";
    std::abort();
  }

  int64_t worksize {0};
  HANDLE_CUTN_ERROR(cutensornetWorkspaceGetMemorySize(cutnHandle,
                                                      workDesc,
                                                      CUTENSORNET_WORKSIZE_PREF_RECOMMENDED,
                                                      CUTENSORNET_MEMSPACE_DEVICE,
                                                      CUTENSORNET_WORKSPACE_SCRATCH,
                                                      &worksize));
  std::cout << "Scratch GPU workspace size (bytes) for MPS computation = " << worksize << std::endl;
  if(worksize <= scratchSize) {
    HANDLE_CUTN_ERROR(cutensornetWorkspaceSetMemory(cutnHandle, workDesc, CUTENSORNET_MEMSPACE_DEVICE,
                      CUTENSORNET_WORKSPACE_SCRATCH, d_scratch, worksize));
  }else{
    std::cout << "ERROR: Insufficient workspace size on Device!\n";
    std::abort();
  }
  std::cout << "Set the workspace buffer for the MPS factorization computation\n";

计算 MPS 分解

一旦 MPS 分解过程被配置和准备好，让我们计算最终量子线路状态的 MPS 分解。

  // Execute MPS computation
  HANDLE_CUTN_ERROR(cutensornetStateCompute(cutnHandle, quantumState, 
                    workDesc, extentsPtr.data(), /*strides=*/nullptr, d_mpsTensors.data(), 0));
  std::cout << "Computed the MPS factorization\n";

创建边际分布对象

一旦量子线路构建完成，让我们创建边际分布对象，该对象将计算量子比特 0 和 1 的边际分布张量（约化密度矩阵）。

  // Specify the desired reduced density matrix (marginal)
  cutensornetStateMarginal_t marginal;
  HANDLE_CUTN_ERROR(cutensornetCreateMarginal(cutnHandle, quantumState, numMarginalModes, marginalModes.data(),
                    0, nullptr, std::vector<int64_t>{{1,2,4,8}}.data(), &marginal)); // using explicit strides
  std::cout << "Created the specified quantum circuit reduced densitry matrix (marginal)\n";

配置边际分布对象

现在我们可以通过设置张量网络收缩路径查找器要使用的超样本数量来配置边际分布对象。

  // Configure the computation of the specified quantum circuit reduced density matrix (marginal)
  const int32_t numHyperSamples = 8; // desired number of hyper samples used in the tensor network contraction path finder
  HANDLE_CUTN_ERROR(cutensornetMarginalConfigure(cutnHandle, marginal,
                    CUTENSORNET_MARGINAL_CONFIG_NUM_HYPER_SAMPLES, &numHyperSamples, sizeof(numHyperSamples)));
  std::cout << "Configured the specified quantum circuit reduced density matrix (marginal) computation\n";

准备边际分布张量的计算

让我们准备边际分布张量的计算。

  // Prepare the specified quantum circuit reduced densitry matrix (marginal)
  HANDLE_CUTN_ERROR(cutensornetMarginalPrepare(cutnHandle, marginal, scratchSize, workDesc, 0x0));
  std::cout << "Prepared the specified quantum circuit reduced density matrix (marginal)\n";
  flops = 0.0;
  HANDLE_CUTN_ERROR(cutensornetMarginalGetInfo(cutnHandle, marginal,
                    CUTENSORNET_MARGINAL_INFO_FLOPS, &flops, sizeof(flops)));
  std::cout << "Total flop count = " << (flops/1e9) << " GFlop\n";
  if(flops <= 0.0) {
    std::cout << "ERROR: Invalid Flop count!\n";
    std::abort();
  }

设置工作区

现在我们可以设置所需的工作区缓冲区。

  // Attach the workspace buffer
  HANDLE_CUTN_ERROR(cutensornetWorkspaceGetMemorySize(cutnHandle,
                                                      workDesc,
                                                      CUTENSORNET_WORKSIZE_PREF_RECOMMENDED,
                                                      CUTENSORNET_MEMSPACE_DEVICE,
                                                      CUTENSORNET_WORKSPACE_SCRATCH,
                                                      &worksize));
  std::cout << "Required scratch GPU workspace size (bytes) for marginal computation = " << worksize << std::endl;
  if(worksize <= scratchSize) {
    HANDLE_CUTN_ERROR(cutensornetWorkspaceSetMemory(cutnHandle, workDesc, CUTENSORNET_MEMSPACE_DEVICE,
                      CUTENSORNET_WORKSPACE_SCRATCH, d_scratch, worksize));
  }else{
    std::cout << "ERROR: Insufficient workspace size on Device!\n";
    std::abort();
  }
  std::cout << "Set the workspace buffer\n";
  

计算边际分布张量

一旦一切都设置好，我们计算请求的量子比特 0 和 1 的边际分布张量（约化密度矩阵），将其复制回主机内存，并打印出来。

  // Compute the specified quantum circuit reduced densitry matrix (marginal)
  HANDLE_CUTN_ERROR(cutensornetMarginalCompute(cutnHandle, marginal, nullptr, workDesc, d_rdm, 0));
  std::cout << "Computed the specified quantum circuit reduced density matrix (marginal)\n";
  std::vector<std::complex<double>> h_rdm(rdmSize);
  HANDLE_CUDA_ERROR(cudaMemcpy(h_rdm.data(), d_rdm, rdmSize * (2 * fp64size), cudaMemcpyDeviceToHost));
  std::cout << "Reduced density matrix for " << numMarginalModes << " qubits:\n";
  for(std::size_t i = 0; i < rdmDim; ++i) {
    for(std::size_t j = 0; j < rdmDim; ++j) {
      std::cout << " " << h_rdm[i + j * rdmDim];
    }
    std::cout << std::endl;
  }

释放资源

  // Destroy the workspace descriptor
  HANDLE_CUTN_ERROR(cutensornetDestroyWorkspaceDescriptor(workDesc));
  std::cout << "Destroyed the workspace descriptor\n";

  // Destroy the quantum circuit reduced density matrix
  HANDLE_CUTN_ERROR(cutensornetDestroyMarginal(marginal));
  std::cout << "Destroyed the quantum circuit state reduced density matrix (marginal)\n";

  // Destroy the quantum circuit state
  HANDLE_CUTN_ERROR(cutensornetDestroyState(quantumState));
  std::cout << "Destroyed the quantum circuit state\n";

  for (int32_t i = 0; i < numQubits; i++) {
    HANDLE_CUDA_ERROR(cudaFree(d_mpsTensors[i]));
  }
  HANDLE_CUDA_ERROR(cudaFree(d_scratch));
  HANDLE_CUDA_ERROR(cudaFree(d_rdm));
  HANDLE_CUDA_ERROR(cudaFree(d_gateCX));
  HANDLE_CUDA_ERROR(cudaFree(d_gateH));
  std::cout << "Freed memory on GPU\n";

  // Finalize the cuTensorNet library
  HANDLE_CUTN_ERROR(cutensornetDestroy(cutnHandle));
  std::cout << "Finalized the cuTensorNet library\n";

  return 0;
}