[RFC] Unitary Synthesis

Describe the feature

Problem

Given a user provided arbitrary quantum unitary, synthesize it into a sequence of quantum gates.

Expectations

• User provides an arbitrary unitary matrix as a custom quantum operation.
• The custom operation can be used as a regular CUDA-Q supported quantum operation.
  • Q: Broadcast (same operation on multiple qubits): Out of scope
• The allowed set of quantum gates for synthesis depends on the backend target.
  • Q: Allow user to specify set of allowed gates: Out of scope
• CUDA-Q throws error if a unitary cannot be synthesized (reasonably).
  • 'reasonably' to account for time limit (timeout), gate count limit (upper threshold), and how close the synthesized "circuit" is to the input unitary (tolerance)
• Parameterized custom operations will be covered in a follow-up RFC.

User API

• Python

import cudaq 

cudaq.register_operation("custom_h", 1. / np.sqrt(2.) *  np.array([[1, 1], [1, -1]])) 
cudaq.register_operation("custom_x", np.array([[0, 1], [1, 0]])) 

@cudaq.kernel 
def bell(): 
  qubits = cudaq.qvector(2) 
  custom_h(qubits[0]) 
  custom_x.ctrl(qubits[0], qubits[1]) 

counts = cudaq.sample(bell) 
counts.dump()

• C++

// Macro to specify the custom unitary operation
cudaq_register_operation(custom_h, 1, 0,
                         (std::vector<std::vector<std::complex<double>>>{
                             {M_SQRT1_2, M_SQRT1_2}, {M_SQRT1_2, -M_SQRT1_2}}));
cudaq_register_operation(
    custom_x, 1, 0, (std::vector<std::vector<std::complex<double>>>{{0, 1}, {1, 0}}));

void custom_operation() __qpu__ {
  cudaq::qvector qubits(2);
  custom_h(qubits[0]);
  custom_x.ctrl(qubits[0], qubits[1]);
}

int main() {
  auto result = cudaq::sample(custom_operation);
  std::cout << result.most_probable() << '\n';
  return 0;
}

• The user must provide valid unitary matrix (CUDA-Q will not check / enforce this requirement)
• Ordering: The user provided matrix must be in row-major format
• Endianness: The user provided matrix is interpreted as Big-endian (often followed by Physics textbooks).

Constraints

• Size of unitary matrix: limit to 8 qubits, (2^8 = 256), 256 x 256
• The custom operation must be defined outside of a quantum kernel. (for e.g. call to register_operation cannot be inside a function decorated with @cudaq.kernel)
• The tolerance for the synthesized circuit and the gate count limit will be default values determined by CUDA-Q
• The custom operation definition is restricted to qubit (cudaq::qudit<2>).

Workflow

• In simulation, no synthesis will happen.
• Compiler will automatically synthesize the matrix when targeting hardware.
• Explicit synthesis mechanism (API or command-line argument) - Out of scope for the first iteration
• NVQC target behaves same as when running locally

Work items / TO-DOs

• Support in simulation for Python -
  • Kernel mode
  • Builder mode
  • State vector simulators
  • Tensornet simulators
• Support in simulation for C++
  • Library mode
  • MLIR mode
• Add generic synthesis for emulation
• Error handling: Gracefully handle user errors, feature constraints and runtime errors
• Support synthesis per hardware backend
• Comprehensive documentation and useful example(s): Covered in issue #2002