Abstract. We present a new scheme for quantum homomorphic encryption which is compact and allows for efficient evaluation of arbitrary polynomial-sized quantum circuits. Building on the framework of Broadbent and Jeffery [BJ15] and recent results in the area of instantaneous non-local quantum computation [Spe15], we show how to construct quantum gadgets that allow perfect correction of the errors which occur during the homomorphic evaluation of T gates on encrypted quantum data. Our scheme can be based on any classical (leveled) fully homomorphic encryption (FHE) scheme and requires no computational assumptions besides those already used by the classical scheme. The size of our quantum gadget depends on the space complexity of the classical decryption function -which aligns well with the current efforts to minimize the complexity of the decryption function. Our scheme (or slight variants of it) offers a number of additional advantages such as ideal compactness, the ability to supply gadgets "on demand", circuit privacy for the evaluator against passive adversaries, and a three-round scheme for blind delegated quantum computation which puts only very limited demands on the quantum abilities of the client.

In device-independent quantum key distribution (DIQKD), an adversary prepares a device consisting of two components, distributed to Alice and Bob, who use the device to generate a secure key. The security of existing DIQKD schemes holds under the assumption that the two components of the device cannot communicate with one another during the protocol execution. This is called the no-communication assumption in DIQKD. Here, we show how to replace this assumption, which can be hard to enforce in practice, by a standard computational assumption from post-quantum cryptography: we give a protocol that produces secure keys even when the components of an adversarial device can exchange arbitrary quantum communication, assuming the device is computationally bounded. Importantly, the computational assumption only needs to hold during the protocol execution—the keys generated at the end of the protocol are information-theoretically secure as in standard DIQKD protocols.