计算矩阵乘积态期望值

以下代码示例说明了如何定义张量网络状态，然后计算其矩阵乘积态 (MPS) 分解，然后计算 MPS 分解状态上张量网络算符的期望值。 完整的代码可以在 NVIDIA/cuQuantum 仓库中找到（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

让我们定义一个对应于 16 量子比特量子电路的张量网络状态。

  // Quantum state configuration
  constexpr int32_t numQubits = 16; // number of qubits
  const std::vector<int64_t> qubitDims(numQubits,2); // qubit dimensions
  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}};
  //  Pauli X gate
  const std::vector<std::complex<double>> h_gateX {{0.0, 0.0}, {1.0, 0.0},
                                                   {1.0, 0.0}, {0.0, 0.0}};
  //  Pauli Y gate
  const std::vector<std::complex<double>> h_gateY {{0.0, 0.0}, {0.0, -1.0},
                                                   {0.0, 1.0}, {0.0, 0.0}};
  //  Pauli Z gate
  const std::vector<std::complex<double>> h_gateZ {{1.0, 0.0}, {0.0, 0.0},
                                                   {0.0, 0.0}, {-1.0, 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_gateX{nullptr}, *d_gateY{nullptr}, *d_gateZ{nullptr}, *d_gateCX{nullptr};
  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateH, 4 * (2 * fp64size)));
  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateX, 4 * (2 * fp64size)));
  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateY, 4 * (2 * fp64size)));
  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateZ, 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_gateX, h_gateX.data(), 4 * (2 * fp64size), cudaMemcpyHostToDevice));
  HANDLE_CUDA_ERROR(cudaMemcpy(d_gateY, h_gateY.data(), 4 * (2 * fp64size), cudaMemcpyHostToDevice));
  HANDLE_CUDA_ERROR(cudaMemcpy(d_gateZ, h_gateZ.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

这里我们设置 MPS 张量的形状，并为它们的存储分配 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 16 量子比特量子电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造 GHZ 量子电路。

  // 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";

请求最终量子电路状态的 MPS 分解

这里我们表达了我们使用 MPS 分解来分解最终量子电路状态的意图。 提供的 MPS 张量的形状指的是 MPS 重整化过程中它们的最大尺寸限制。 最终 MPS 张量的实际计算形状可能更小。 这里尚未进行任何计算。

  // Specify the final target MPS representation (use default fortran strides)
  HANDLE_CUTN_ERROR(cutensornetStateFinalizeMPS(cutnHandle, quantumState, 
                    CUTENSORNET_BOUNDARY_CONDITION_OPEN, extentsPtr.data(), /*strides=*/nullptr ));

配置 MPS 分解过程

在表达了我们执行最终量子电路状态的 MPS 分解的意图之后，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 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 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 MPS 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电路状态的 MPS 分解。

  // Execute MPS computation
  HANDLE_CUTN_ERROR(cutensornetStateCompute(cutnHandle, quantumState, 
                    workDesc, extentsPtr.data(), /*strides=*/nullptr, d_mpsTensors.data(), 0));

构造张量网络算符

现在让我们为 16 量子比特状态创建一个空的张量网络算符，然后向其附加三个组件，其中每个组件都是 Pauli 矩阵的直积，并按某个复系数缩放（如 Jordan-Wigner 表示法中所示）。

  // Create an empty tensor network operator
  cutensornetNetworkOperator_t hamiltonian;
  HANDLE_CUTN_ERROR(cutensornetCreateNetworkOperator(cutnHandle, numQubits, qubitDims.data(), CUDA_C_64F, &hamiltonian));
  // Append component (0.5 * Z1 * Z2) to the tensor network operator
  {
    const int32_t numModes[] = {1, 1}; // Z1 acts on 1 mode, Z2 acts on 1 mode
    const int32_t modesZ1[] = {1}; // state modes Z1 acts on
    const int32_t modesZ2[] = {2}; // state modes Z2 acts on
    const int32_t * stateModes[] = {modesZ1, modesZ2}; // state modes (Z1 * Z2) acts on
    const void * gateData[] = {d_gateZ, d_gateZ}; // GPU pointers to gate data
    HANDLE_CUTN_ERROR(cutensornetNetworkOperatorAppendProduct(cutnHandle, hamiltonian, cuDoubleComplex{0.5,0.0},
                      2, numModes, stateModes, NULL, gateData, &id));
  }
  // Append component (0.25 * Y3) to the tensor network operator
  {
    const int32_t numModes[] = {1}; // Y3 acts on 1 mode
    const int32_t modesY3[] = {3}; // state modes Y3 acts on
    const int32_t * stateModes[] = {modesY3}; // state modes (Y3) acts on
    const void * gateData[] = {d_gateY}; // GPU pointers to gate data
    HANDLE_CUTN_ERROR(cutensornetNetworkOperatorAppendProduct(cutnHandle, hamiltonian, cuDoubleComplex{0.25,0.0},
                      1, numModes, stateModes, NULL, gateData, &id));
  }
  // Append component (0.13 * Y0 X2 Z3) to the tensor network operator
  {
    const int32_t numModes[] = {1, 1, 1}; // Y0 acts on 1 mode, X2 acts on 1 mode, Z3 acts on 1 mode
    const int32_t modesY0[] = {0}; // state modes Y0 acts on
    const int32_t modesX2[] = {2}; // state modes X2 acts on
    const int32_t modesZ3[] = {3}; // state modes Z3 acts on
    const int32_t * stateModes[] = {modesY0, modesX2, modesZ3}; // state modes (Y0 * X2 * Z3) acts on
    const void * gateData[] = {d_gateY, d_gateX, d_gateZ}; // GPU pointers to gate data
    HANDLE_CUTN_ERROR(cutensornetNetworkOperatorAppendProduct(cutnHandle, hamiltonian, cuDoubleComplex{0.13,0.0},
                      3, numModes, stateModes, NULL, gateData, &id));
  }
  std::cout << "Constructed a tensor network operator: (0.5 * Z1 * Z2) + (0.25 * Y3) + (0.13 * Y0 * X2 * Z3)" << std::endl;

创建期望值对象

一旦量子电路和张量网络算符被构造，并且最终量子电路状态已使用 MPS 表示分解，让我们创建期望值对象，它将计算指定张量网络算符在指定量子电路的最终 MPS 分解状态上的期望值。

  // Specify the quantum circuit expectation value
  cutensornetStateExpectation_t expectation;
  HANDLE_CUTN_ERROR(cutensornetCreateExpectation(cutnHandle, quantumState, hamiltonian, &expectation));
  std::cout << "Created the specified quantum circuit expectation value\n";

配置期望值计算

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期望值对象。

  // Configure the computation of the specified quantum circuit expectation value
  const int32_t numHyperSamples = 8; // desired number of hyper samples used in the tensor network contraction path finder
  HANDLE_CUTN_ERROR(cutensornetExpectationConfigure(cutnHandle, expectation,
                    CUTENSORNET_EXPECTATION_CONFIG_NUM_HYPER_SAMPLES, &numHyperSamples, sizeof(numHyperSamples)));

准备期望值计算

让我们准备所需期望值的计算。

  // Prepare the specified quantum circuit expectation value for computation
  HANDLE_CUTN_ERROR(cutensornetExpectationPrepare(cutnHandle, expectation, scratchSize, workDesc, 0x0));
  std::cout << "Prepared the specified quantum circuit expectation value\n";
  flops = 0.0;
  HANDLE_CUTN_ERROR(cutensornetExpectationGetInfo(cutnHandle, expectation,
                    CUTENSORNET_EXPECTATION_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) = " << 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";

计算请求的期望值

一旦一切都设置好，我们计算请求的期望值并打印它。 请注意，返回的期望值未归一化。 张量网络状态的 2-范数作为单独的参数返回。

  // Compute the specified quantum circuit expectation value
  std::complex<double> expectVal{0.0,0.0}, stateNorm2{0.0,0.0};
  HANDLE_CUTN_ERROR(cutensornetExpectationCompute(cutnHandle, expectation, workDesc,
                    static_cast<void*>(&expectVal), static_cast<void*>(&stateNorm2), 0x0));
  std::cout << "Computed the specified quantum circuit expectation value\n";
  expectVal /= stateNorm2;
  std::cout << "Expectation value = (" << expectVal.real() << ", " << expectVal.imag() << ")\n";
  std::cout << "Squared 2-norm of the state = (" << stateNorm2.real() << ", " << stateNorm2.imag() << ")\n";

释放资源

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

  // Destroy the quantum circuit expectation value
  HANDLE_CUTN_ERROR(cutensornetDestroyExpectation(expectation));
  std::cout << "Destroyed the quantum circuit state expectation value\n";

  // Destroy the tensor network operator
  HANDLE_CUTN_ERROR(cutensornetDestroyNetworkOperator(hamiltonian));
  std::cout << "Destroyed the tensor network operator\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_gateCX));
  HANDLE_CUDA_ERROR(cudaFree(d_gateZ));
  HANDLE_CUDA_ERROR(cudaFree(d_gateY));
  HANDLE_CUDA_ERROR(cudaFree(d_gateX));
  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;
}