  The creation complexity of a quantum state is the minimum number of
elementary gates required to create it from a basic initial state. The creation
complexity of quantum states is closely related to the complexity of quantum
circuits, which is crucial in developing efficient quantum algorithms that can
outperform classical algorithms. A major question unanswered so far is what
quantum states can be created with a number of elementary gates that scales
polynomially with the number of qubits. In this work we first show for an
entirely general quantum state it is exponentially hard (requires a number of
steps that scales exponentially with the number of qubits) to determine if the
creation complexity is polynomial. We then show it is possible for a large
class of quantum states with polynomial creation complexity to have common
coefficient features such that given any candidate quantum state we can design
an efficient coefficient sampling procedure to determine if it belongs to the
class or not with arbitrarily high success probability. Consequently partial
knowledge of a quantum state's creation complexity is obtained, which can be
useful for designing quantum circuits and algorithms involving such a state.

Quantum Physics