Quantum metamaterials are optical media comprised of controllable artificial quantum scatterers, which maintain quantum coherence for times exceeding the characteristic travel time of an electromagnetic wave through the system. Being an extended quantum object, a quantum metamaterial is predicted to have unusual properties, such as quantum birefringence – the superposition of states with different refractive indices, n₁ and n₂. Here we present analytical and numerical results on quantum birefringence in a realistic test system – an array of superconducting qubits. In particular, the case of ambidextrous quantum metamaterials (with n₁n₂<0) is investigated.