Unitarization Through Approximate Basis

In this paper, we introduce the problem of unitarization. Unitarization is the problem of taking several input quantum circuits that produce orthogonal states from the all 0 state, and create an output circuit implementing a unitary with its first columns as those states. We allow the resulting circuits, and even the input circuits, to use ancilla qubits initialized to 0. But ancilla qubits must always be returned to 0 for ANY input, and we are only guaranteed the input circuits returns ancilla qubits to 0 on the all 0 input. The unitarization problem seems hard if the output states are neither orthogonal to or in the span of the computational basis vectors that need to map to them. In this work, we approximately solve this problem in the case where input circuits are given as black box oracles by probably finding an approximate basis for our states. This method may be more interesting than the application. Specifically, we find an approximate basis in polynomial time for the following parameters. Take any natural $n$, $k = O\left(\frac{\ln(n)}{\ln(\ln(n))}\right)$, and $\epsilon = 2^{-O(\sqrt{\ln(n)})}$. Take any $k$ input unit vectors, $(\lvert \psi_i \rangle)_{i\in [k]}$ on polynomial in $n$ qubits prepared by quantum oracles, $(V_i)_{i \in [k]}$, (that we can control call and control invert). Then there is a quantum circuit with polynomial size in $n$ and access to the oracles $(V_i)_{i \in [k]}$ that with at least $1 - \epsilon$ probability, computes at most $k$ circuits with size polynomial in $n$ and oracle access to $(V_i)_{i \in [k]}$ that $\epsilon$ approximately computes an $\epsilon$ approximate orthonormal basis for $(\lvert \psi_i \rangle)_{i\in [k]}$.