Abstract: Quantum computing methods and systems are described. A computing device receives an encrypted state from another device. The encrypted state is stored on a quantum register, and a sequence of operations is applied to the encrypted state in the quantum register. The sequence of operations includes an operation parameterized by a control message from the other device. Applying the sequence of operations manipulates the state of the quantum register and an auxiliary quantum system. The auxiliary quantum system can be, for example, a qubit selected from four specified quantum states. Applying the sequence of operations produces encryption-key-update information. The computing device may send an encrypted output state and the encryption-key-update message to the other device.