We present a typed quantum lambda calculus, PUNQ (Polytime UNitarity with Quantum control), featuring full quantum control with higher-order programs that can be superposed via quantum conditionals. PUNQ as a programming language ensures unitarity and polynomial-time normalization. Unitarity is achieved by using a special modality for superpositions while requiring orthogonality among superposed terms. Polynomial-time normalization is achieved using a linear-logic-based type discipline employing Barber and Plotkin duality along with a specific modality to account for potential duplications. This type discipline also guarantees that derived values have polynomial size. PUNQ seamlessly combines the two modalities: quantum circuit programs uphold unitarity, and all programs are evaluated in polynomial time, ensuring their feasibility.