# Computing Matrix Product State Marginal Distribution Tensor¶

The following code example illustrates how to define a tensor network state, compute its Matrix Product State (MPS) factorization, and then compute a marginal distribution tensor for the MPS-factorized state. The full code can be found in the NVIDIA/cuQuantum repository (here).

``` 7#include <cstdlib>
8#include <cstdio>
9#include <cassert>
10#include <complex>
11#include <vector>
12#include <bitset>
13#include <iostream>
14
15#include <cuda_runtime.h>
16#include <cutensornet.h>
17
18
19#define HANDLE_CUDA_ERROR(x) \
20{ const auto err = x; \
21  if( err != cudaSuccess ) \
22  { printf("CUDA error %s in line %d\n", cudaGetErrorString(err), __LINE__); fflush(stdout); std::abort(); } \
23};
24
25#define HANDLE_CUTN_ERROR(x) \
26{ const auto err = x; \
27  if( err != CUTENSORNET_STATUS_SUCCESS ) \
28  { printf("cuTensorNet error %s in line %d\n", cutensornetGetErrorString(err), __LINE__); fflush(stdout); std::abort(); } \
29};
30
31
32int main(int argc, char **argv)
33{
34  static_assert(sizeof(size_t) == sizeof(int64_t), "Please build this sample on a 64-bit architecture!");
35
36  constexpr std::size_t fp64size = sizeof(double);
```

# Define the tensor network state and the desired marginal distribution tensor¶

Let’s define a tensor network state corresponding to a 16-qubit quantum circuit and request the marginal distribution tensor for qubits 0 and 1.

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

# Initialize the cuTensorNet library handle¶

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

# Define quantum gates on GPU¶

```57  // Define necessary quantum gate tensors in Host memory
58  const double invsq2 = 1.0 / std::sqrt(2.0);
60  const std::vector<std::complex<double>> h_gateH {{invsq2, 0.0},  {invsq2, 0.0},
61                                                   {invsq2, 0.0}, {-invsq2, 0.0}};
62  //  CX gate
63  const std::vector<std::complex<double>> h_gateCX {{1.0, 0.0}, {0.0, 0.0}, {0.0, 0.0}, {0.0, 0.0},
64                                                    {0.0, 0.0}, {1.0, 0.0}, {0.0, 0.0}, {0.0, 0.0},
65                                                    {0.0, 0.0}, {0.0, 0.0}, {0.0, 0.0}, {1.0, 0.0},
66                                                    {0.0, 0.0}, {0.0, 0.0}, {1.0, 0.0}, {0.0, 0.0}};
67
68  // Copy quantum gates to Device memory
69  void *d_gateH{nullptr}, *d_gateCX{nullptr};
70  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateH, 4 * (2 * fp64size)));
71  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateCX, 16 * (2 * fp64size)));
72  std::cout << "Allocated quantum gate memory on GPU\n";
73  HANDLE_CUDA_ERROR(cudaMemcpy(d_gateH, h_gateH.data(), 4 * (2 * fp64size), cudaMemcpyHostToDevice));
74  HANDLE_CUDA_ERROR(cudaMemcpy(d_gateCX, h_gateCX.data(), 16 * (2 * fp64size), cudaMemcpyHostToDevice));
75  std::cout << "Copied quantum gates to GPU memory\n";
```

# Allocate MPS tensors¶

Here we set the shapes of MPS tensors and allocate GPU memory for their storage.

```79  // Determine the MPS representation and allocate buffers for the MPS tensors
80  const int64_t maxExtent = 2; // GHZ state can be exactly represented with max bond dimension of 2
81  std::vector<std::vector<int64_t>> extents;
82  std::vector<int64_t*> extentsPtr(numQubits);
83  std::vector<void*> d_mpsTensors(numQubits, nullptr);
84  for (int32_t i = 0; i < numQubits; i++) {
85    if (i == 0) { // left boundary MPS tensor
86      extents.push_back({2, maxExtent});
87      HANDLE_CUDA_ERROR(cudaMalloc(&d_mpsTensors[i], 2 * maxExtent * 2 * fp64size));
88    }
89    else if (i == numQubits-1) { // right boundary MPS tensor
90      extents.push_back({maxExtent, 2});
91      HANDLE_CUDA_ERROR(cudaMalloc(&d_mpsTensors[i], 2 * maxExtent * 2 * fp64size));
92    }
93    else { // middle MPS tensors
94      extents.push_back({maxExtent, 2, maxExtent});
95      HANDLE_CUDA_ERROR(cudaMalloc(&d_mpsTensors[i], 2 * maxExtent * maxExtent * 2 * fp64size));
96    }
97    extentsPtr[i] = extents[i].data();
98  }
```

# Allocate the marginal distribution tensor on GPU¶

Here we allocate the marginal distribution tensor, that is, the reduced density matrix for qubits 0 and 1, on GPU.

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

# Allocate the scratch buffer on GPU¶

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

# Create a pure tensor network state¶

Now let’s create a pure tensor network state for a 16-qubit quantum circuit.

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

# Apply quantum gates¶

Let’s construct the GHZ quantum circuit by applying the corresponding quantum gates.

```129  // Construct the final quantum circuit state (apply quantum gates) for the GHZ circuit
130  int64_t id;
131  HANDLE_CUTN_ERROR(cutensornetStateApplyTensor(cutnHandle, quantumState, 1, std::vector<int32_t>{{0}}.data(),
132                    d_gateH, nullptr, 1, 0, 1, &id));
133  for(int32_t i = 1; i < numQubits; ++i) {
134    HANDLE_CUTN_ERROR(cutensornetStateApplyTensor(cutnHandle, quantumState, 2, std::vector<int32_t>{{i-1,i}}.data(),
135                      d_gateCX, nullptr, 1, 0, 1, &id));
136  }
137  std::cout << "Applied quantum gates\n";
```

# Request MPS factorization for the final quantum circuit state¶

Here we express our intent to factorize the final quantum circuit state using MPS factorization. The provided shapes of the MPS tensors refer to their maximal size limit during the MPS renormalization procedure. The actually computed shapes of the final MPS tensors may be smaller. No computation is done here yet.

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

# Configure MPS factorization procedure¶

After expressing our intent to perform MPS factorization of the final quantum circuit state, we can also configure the MPS factorization procedure by resetting different options, for example, the SVD algorithm.

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

# Prepare the computation of MPS factorization¶

Let’s create a workspace descriptor and prepare the computation of MPS factorization.

```156  // Prepare the MPS computation and attach workspace
157  cutensornetWorkspaceDescriptor_t workDesc;
158  HANDLE_CUTN_ERROR(cutensornetCreateWorkspaceDescriptor(cutnHandle, &workDesc));
159  std::cout << "Created the workspace descriptor\n";
160  HANDLE_CUTN_ERROR(cutensornetStatePrepare(cutnHandle, quantumState, scratchSize, workDesc, 0x0));
161  int64_t worksize {0};
162  HANDLE_CUTN_ERROR(cutensornetWorkspaceGetMemorySize(cutnHandle,
163                                                      workDesc,
164                                                      CUTENSORNET_WORKSIZE_PREF_RECOMMENDED,
165                                                      CUTENSORNET_MEMSPACE_DEVICE,
166                                                      CUTENSORNET_WORKSPACE_SCRATCH,
167                                                      &worksize));
168  std::cout << "Scratch GPU workspace size (bytes) for MPS computation = " << worksize << std::endl;
169  if(worksize <= scratchSize) {
170    HANDLE_CUTN_ERROR(cutensornetWorkspaceSetMemory(cutnHandle, workDesc, CUTENSORNET_MEMSPACE_DEVICE,
171                      CUTENSORNET_WORKSPACE_SCRATCH, d_scratch, worksize));
172  }else{
173    std::cout << "ERROR: Insufficient workspace size on Device!\n";
174    std::abort();
175  }
176  std::cout << "Set the workspace buffer for the MPS factorization computation\n";
```

# Compute MPS factorization¶

Once the MPS factorization procedure has been configured and prepared, let’s compute the MPS factorization of the final quantum circuit state.

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

# Create the marginal distribution object¶

Once the quantum circuit has been constructed, let’s create the marginal distribution object that will compute the marginal distribution tensor (reduced density matrix) for qubits 0 and 1.

```187  // Specify the desired reduced density matrix (marginal)
188  cutensornetStateMarginal_t marginal;
189  HANDLE_CUTN_ERROR(cutensornetCreateMarginal(cutnHandle, quantumState, numMarginalModes, marginalModes.data(),
190                    0, nullptr, std::vector<int64_t>{{1,2,4,8}}.data(), &marginal)); // using explicit strides
191  std::cout << "Created the specified quantum circuit reduced densitry matrix (marginal)\n";
```

# Configure the marginal distribution object¶

Now we can configure the marginal distribution object by setting the number of hyper-samples to be used by the tensor network contraction path finder.

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

# Prepare the computation of the marginal distribution tensor¶

Let’s prepare the computation of the marginal distribution tensor.

```203  // Prepare the specified quantum circuit reduced densitry matrix (marginal)
204  HANDLE_CUTN_ERROR(cutensornetMarginalPrepare(cutnHandle, marginal, scratchSize, workDesc, 0x0));
205  std::cout << "Prepared the specified quantum circuit reduced density matrix (marginal)\n";
```

# Set up the workspace¶

Now we can set up the required workspace buffer.

```209  // Attach the workspace buffer
210  HANDLE_CUTN_ERROR(cutensornetWorkspaceGetMemorySize(cutnHandle,
211                                                      workDesc,
212                                                      CUTENSORNET_WORKSIZE_PREF_RECOMMENDED,
213                                                      CUTENSORNET_MEMSPACE_DEVICE,
214                                                      CUTENSORNET_WORKSPACE_SCRATCH,
215                                                      &worksize));
216  std::cout << "Required scratch GPU workspace size (bytes) for marginal computation = " << worksize << std::endl;
217  if(worksize <= scratchSize) {
218    HANDLE_CUTN_ERROR(cutensornetWorkspaceSetMemory(cutnHandle, workDesc, CUTENSORNET_MEMSPACE_DEVICE,
219                      CUTENSORNET_WORKSPACE_SCRATCH, d_scratch, worksize));
220  }else{
221    std::cout << "ERROR: Insufficient workspace size on Device!\n";
222    std::abort();
223  }
224  std::cout << "Set the workspace buffer\n";
225
```

# Compute the marginal distribution tensor¶

Once everything has been set up, we compute the requested marginal distribution tensor (reduced density matrix) for qubits 0 and 1, copy it back to Host memory, and print it.

```228  // Compute the specified quantum circuit reduced densitry matrix (marginal)
229  HANDLE_CUTN_ERROR(cutensornetMarginalCompute(cutnHandle, marginal, nullptr, workDesc, d_rdm, 0));
230  std::cout << "Computed the specified quantum circuit reduced density matrix (marginal)\n";
231  std::vector<std::complex<double>> h_rdm(rdmSize);
232  HANDLE_CUDA_ERROR(cudaMemcpy(h_rdm.data(), d_rdm, rdmSize * (2 * fp64size), cudaMemcpyDeviceToHost));
233  std::cout << "Reduced density matrix for " << numMarginalModes << " qubits:\n";
234  for(std::size_t i = 0; i < rdmDim; ++i) {
235    for(std::size_t j = 0; j < rdmDim; ++j) {
236      std::cout << " " << h_rdm[i + j * rdmDim];
237    }
238    std::cout << std::endl;
239  }
```

# Free resources¶

```243  // Destroy the workspace descriptor
244  HANDLE_CUTN_ERROR(cutensornetDestroyWorkspaceDescriptor(workDesc));
245  std::cout << "Destroyed the workspace descriptor\n";
246
247  // Destroy the quantum circuit reduced density matrix
248  HANDLE_CUTN_ERROR(cutensornetDestroyMarginal(marginal));
249  std::cout << "Destroyed the quantum circuit state reduced density matrix (marginal)\n";
250
251  // Destroy the quantum circuit state
252  HANDLE_CUTN_ERROR(cutensornetDestroyState(quantumState));
253  std::cout << "Destroyed the quantum circuit state\n";
254
255  for (int32_t i = 0; i < numQubits; i++) {
256    HANDLE_CUDA_ERROR(cudaFree(d_mpsTensors[i]));
257  }
258  HANDLE_CUDA_ERROR(cudaFree(d_scratch));
259  HANDLE_CUDA_ERROR(cudaFree(d_rdm));
260  HANDLE_CUDA_ERROR(cudaFree(d_gateCX));
261  HANDLE_CUDA_ERROR(cudaFree(d_gateH));
262  std::cout << "Freed memory on GPU\n";
263
264  // Finalize the cuTensorNet library
265  HANDLE_CUTN_ERROR(cutensornetDestroy(cutnHandle));
266  std::cout << "Finalized the cuTensorNet library\n";
267
268  return 0;
269}
```