# Computing Matrix Product State (MPS) Amplitudes¶

The following code example illustrates how to define a tensor network state, factorize it as a Matrix Product State (MPS), and then compute a slice of amplitudes of the factorized MPS 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 slice of state amplitudes¶

Let’s define a tensor network state corresponding to a 6-qubit quantum circuit and request a slice of state amplitudes where qubits 0 and 1 are fixed at value 1.

```40  // Quantum state configuration
41  constexpr int32_t numQubits = 6; // number of qubits
42  const std::vector<int64_t> qubitDims(numQubits,2); // qubit dimensions
43  const std::vector<int32_t> fixedModes({0,1}); // fixed modes in the output amplitude tensor (must be in acsending order)
44  const std::vector<int64_t> fixedValues({1,1}); // values of the fixed modes in the output amplitude tensor
45  const int32_t numFixedModes = fixedModes.size(); // number of fixed modes in the output amplitude tensor
46  std::cout << "Quantum circuit: " << numQubits << " qubits\n";
```

# Initialize the cuTensorNet library handle¶

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

# Define quantum gates on GPU¶

```58  // Define necessary quantum gate tensors in Host memory
59  const double invsq2 = 1.0 / std::sqrt(2.0);
61  const std::vector<std::complex<double>> h_gateH {{invsq2, 0.0},  {invsq2, 0.0},
62                                                   {invsq2, 0.0}, {-invsq2, 0.0}};
63  //  CX gate
64  const std::vector<std::complex<double>> h_gateCX {{1.0, 0.0}, {0.0, 0.0}, {0.0, 0.0}, {0.0, 0.0},
65                                                    {0.0, 0.0}, {1.0, 0.0}, {0.0, 0.0}, {0.0, 0.0},
66                                                    {0.0, 0.0}, {0.0, 0.0}, {0.0, 0.0}, {1.0, 0.0},
67                                                    {0.0, 0.0}, {0.0, 0.0}, {1.0, 0.0}, {0.0, 0.0}};
68
69  // Copy quantum gates to Device memory
70  void *d_gateH{nullptr}, *d_gateCX{nullptr};
71  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateH, 4 * (2 * fp64size)));
72  HANDLE_CUDA_ERROR(cudaMalloc(&d_gateCX, 16 * (2 * fp64size)));
73  std::cout << "Allocated quantum gate memory on GPU\n";
74  HANDLE_CUDA_ERROR(cudaMemcpy(d_gateH, h_gateH.data(), 4 * (2 * fp64size), cudaMemcpyHostToDevice));
75  HANDLE_CUDA_ERROR(cudaMemcpy(d_gateCX, h_gateCX.data(), 16 * (2 * fp64size), cudaMemcpyHostToDevice));
76  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.

```80  // Determine the MPS representation and allocate buffers for the MPS tensors
81  const int64_t maxExtent = 2; // GHZ state can be exactly represented with max bond dimension of 2
82  std::vector<std::vector<int64_t>> extents;
83  std::vector<int64_t*> extentsPtr(numQubits);
84  std::vector<void*> d_mpsTensors(numQubits, nullptr);
85  for (int32_t i = 0; i < numQubits; i++) {
86    if (i == 0) { // left boundary MPS tensor
87      extents.push_back({2, maxExtent});
88      HANDLE_CUDA_ERROR(cudaMalloc(&d_mpsTensors[i], 2 * maxExtent * 2 * fp64size));
89    }
90    else if (i == numQubits-1) { // right boundary MPS tensor
91      extents.push_back({maxExtent, 2});
92      HANDLE_CUDA_ERROR(cudaMalloc(&d_mpsTensors[i], 2 * maxExtent * 2 * fp64size));
93    }
94    else { // middle MPS tensors
95      extents.push_back({maxExtent, 2, maxExtent});
96      HANDLE_CUDA_ERROR(cudaMalloc(&d_mpsTensors[i], 2 * maxExtent * maxExtent * 2 * fp64size));
97    }
98    extentsPtr[i] = extents[i].data();
99  }
```

# Allocate the amplitudes slice tensor on GPU¶

Here we allocate GPU memory for the requested amplitudes slice tensor.

```103  // Allocate Device memory for the specified slice of the quantum circuit amplitudes tensor
104  void *d_amp{nullptr};
105  std::size_t ampSize = 1;
106  for(const auto & qubitDim: qubitDims) ampSize *= qubitDim; // all state modes (full size)
107  for(const auto & fixedMode: fixedModes) ampSize /= qubitDims[fixedMode]; // fixed state modes reduce the slice size
108  HANDLE_CUDA_ERROR(cudaMalloc(&d_amp, ampSize * (2 * fp64size)));
109  std::cout << "Allocated memory for the specified slice of the quantum circuit amplitude tensor of size "
110            << ampSize << " elements\n";
```

# Allocate the scratch buffer on GPU¶

```114  // Query the free memory on Device
115  std::size_t freeSize{0}, totalSize{0};
116  HANDLE_CUDA_ERROR(cudaMemGetInfo(&freeSize, &totalSize));
117  const std::size_t scratchSize = (freeSize - (freeSize % 4096)) / 2; // use half of available memory with alignment
118  void *d_scratch{nullptr};
119  HANDLE_CUDA_ERROR(cudaMalloc(&d_scratch, scratchSize));
120  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 6-qubit quantum circuit.

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

# Apply quantum gates¶

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

```132  // Construct the final quantum circuit state (apply quantum gates) for the GHZ circuit
133  int64_t id;
134  HANDLE_CUTN_ERROR(cutensornetStateApplyTensor(cutnHandle, quantumState, 1, std::vector<int32_t>{{0}}.data(),
135                    d_gateH, nullptr, 1, 0, 1, &id));
136  for(int32_t i = 1; i < numQubits; ++i) {
137    HANDLE_CUTN_ERROR(cutensornetStateApplyTensor(cutnHandle, quantumState, 2, std::vector<int32_t>{{i-1,i}}.data(),
138                      d_gateCX, nullptr, 1, 0, 1, &id));
139  }
140  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.

```144  // Specify the final target MPS representation (use default fortran strides)
145  HANDLE_CUTN_ERROR(cutensornetStateFinalizeMPS(cutnHandle, quantumState,
146                    CUTENSORNET_BOUNDARY_CONDITION_OPEN, extentsPtr.data(), /*strides=*/nullptr));
147  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.

```151  // Optional, set up the SVD method for MPS truncation.
152  cutensornetTensorSVDAlgo_t algo = CUTENSORNET_TENSOR_SVD_ALGO_GESVDJ;
153  HANDLE_CUTN_ERROR(cutensornetStateConfigure(cutnHandle, quantumState,
154                    CUTENSORNET_STATE_MPS_SVD_CONFIG_ALGO, &algo, sizeof(algo)));
155  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.

```159  // Prepare the MPS computation and attach workspace
160  cutensornetWorkspaceDescriptor_t workDesc;
161  HANDLE_CUTN_ERROR(cutensornetCreateWorkspaceDescriptor(cutnHandle, &workDesc));
162  std::cout << "Created the workspace descriptor\n";
163  HANDLE_CUTN_ERROR(cutensornetStatePrepare(cutnHandle, quantumState, scratchSize, workDesc, 0x0));
164  int64_t worksize {0};
165  HANDLE_CUTN_ERROR(cutensornetWorkspaceGetMemorySize(cutnHandle,
166                                                      workDesc,
167                                                      CUTENSORNET_WORKSIZE_PREF_RECOMMENDED,
168                                                      CUTENSORNET_MEMSPACE_DEVICE,
169                                                      CUTENSORNET_WORKSPACE_SCRATCH,
170                                                      &worksize));
171  std::cout << "Scratch GPU workspace size (bytes) for MPS computation = " << worksize << std::endl;
172  if(worksize <= scratchSize) {
173    HANDLE_CUTN_ERROR(cutensornetWorkspaceSetMemory(cutnHandle, workDesc, CUTENSORNET_MEMSPACE_DEVICE,
174                      CUTENSORNET_WORKSPACE_SCRATCH, d_scratch, worksize));
175  }else{
176    std::cout << "ERROR: Insufficient workspace size on Device!\n";
177    std::abort();
178  }
179  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.

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

# Create the state amplitudes accessor¶

Once the factorized MPS representation of the final quantum circuit state has been computed, let’s create the amplitudes accessor object that will compute the requested slice of state amplitudes.

```190  // Specify the quantum circuit amplitudes accessor
191  cutensornetStateAccessor_t accessor;
192  HANDLE_CUTN_ERROR(cutensornetCreateAccessor(cutnHandle, quantumState, numFixedModes, fixedModes.data(),
193                    nullptr, &accessor)); // using default strides
194  std::cout << "Created the specified quantum circuit amplitudes accessor\n";
```

# Configure the state amplitudes accessor¶

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

```198  // Configure the computation of the slice of the specified quantum circuit amplitudes tensor
199  const int32_t numHyperSamples = 8; // desired number of hyper samples used in the tensor network contraction path finder
200  HANDLE_CUTN_ERROR(cutensornetAccessorConfigure(cutnHandle, accessor,
201                    CUTENSORNET_ACCESSOR_OPT_NUM_HYPER_SAMPLES, &numHyperSamples, sizeof(numHyperSamples)));
```

# Prepare the computation of the amplitudes slice tensor¶

Let’s prepare the computation of the amplitudes slice tensor.

```205  // Prepare the computation of the specified slice of the quantum circuit amplitudes tensor
206  HANDLE_CUTN_ERROR(cutensornetAccessorPrepare(cutnHandle, accessor, scratchSize, workDesc, 0x0));
207  std::cout << "Prepared the computation of the specified slice of the quantum circuit amplitudes tensor\n";
```

# Set up the workspace¶

Now we can set up the required workspace buffer.

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

# Compute the specified slice of state amplitudes¶

Once everything has been set up, we compute the requested slice of state amplitudes, copy it back to Host memory, and print it.

```230  // Compute the specified slice of the quantum circuit amplitudes tensor
231  std::complex<double> stateNorm{0.0,0.0};
232  HANDLE_CUTN_ERROR(cutensornetAccessorCompute(cutnHandle, accessor, fixedValues.data(),
233                    workDesc, d_amp, static_cast<void*>(&stateNorm), 0x0));
234  std::cout << "Computed the specified slice of the quantum circuit amplitudes tensor\n";
235  std::vector<std::complex<double>> h_amp(ampSize);
236  HANDLE_CUDA_ERROR(cudaMemcpy(h_amp.data(), d_amp, ampSize * (2 * fp64size), cudaMemcpyDeviceToHost));
237  std::cout << "Amplitudes slice for " << (numQubits - numFixedModes) << " qubits:\n";
238  for(std::size_t i = 0; i < ampSize; ++i) {
239    std::cout << " " << h_amp[i] << std::endl;
240  }
241  std::cout << "State 2-norm = (" << stateNorm.real() << ", " << stateNorm.imag() << ")\n";
```

# Free resources¶

```245  // Destroy the workspace descriptor
246  HANDLE_CUTN_ERROR(cutensornetDestroyWorkspaceDescriptor(workDesc));
247  std::cout << "Destroyed the workspace descriptor\n";
248
249  // Destroy the quantum circuit amplitudes accessor
250  HANDLE_CUTN_ERROR(cutensornetDestroyAccessor(accessor));
251  std::cout << "Destroyed the quantum circuit amplitudes accessor\n";
252
253  // Destroy the quantum circuit state
254  HANDLE_CUTN_ERROR(cutensornetDestroyState(quantumState));
255  std::cout << "Destroyed the quantum circuit state\n";
256
257  for (int32_t i = 0; i < numQubits; i++) {
258    HANDLE_CUDA_ERROR(cudaFree(d_mpsTensors[i]));
259  }
260  HANDLE_CUDA_ERROR(cudaFree(d_scratch));
261  HANDLE_CUDA_ERROR(cudaFree(d_amp));
262  HANDLE_CUDA_ERROR(cudaFree(d_gateCX));
263  HANDLE_CUDA_ERROR(cudaFree(d_gateH));
264  std::cout << "Freed memory on GPU\n";
265
266  // Finalize the cuTensorNet library
267  HANDLE_CUTN_ERROR(cutensornetDestroy(cutnHandle));
268  std::cout << "Finalized the cuTensorNet library\n";
269
270  return 0;
271}
```