Siddhartha Jain

We present a new primitive for quantum algorithms that implements a discrete Hermite transform efficiently, in time that depends logarithmically in both the dimension and the inverse of the allowable error. This transform, which maps basis states to states whose amplitudes are proportional to the Hermite functions, can be interpreted as the Gaussian analogue of the Fourier transform. Our algorithm is based on a method to exponentially fast forward the evolution of the quantum harmonic oscillator, which significantly improves over prior art. We apply this Hermite transform to give examples of provable quantum query advantage in property testing and learning. In particular, we show how to efficiently test the property of being close to a low-degree in the Hermite basis when inputs are sampled from the Gaussian distribution, and how to solve a Gaussian analogue of the Goldreich-Levin learning task efficiently. We also comment on other potential uses of this transform to simulating time dynamics of quantum systems in the continuum.

We present efficient quantum circuits that implement high-dimensional unitary irreducible representations (irreps) of SU(n), where n>=2 is constant. For dimension N and error eps, the number of quantum gates in our circuits is polynomial in log(N) and log(1/eps). Our construction relies on the Jordan-Schwinger representation, which allows us to realize irreps of SU(n) in the Hilbert space of n quantum harmonic oscillators. Together with a recent efficient quantum Hermite transform, which allows us to map the computational basis states to the eigenstates of the quantum harmonic oscillator, this allows us to implement these irreps efficiently. Our quantum circuits can be used to construct explicit Ramanujan quantum expanders, a longstanding open problem. They can also be used to fast-forward the evolution of certain quantum systems.

The no-cloning theorem in quantum mechanics has been used as a basis for quantum money constructions, which guarantee unconditionally unforgeable currency. Existing schemes, however, either (i) require long-term quantum memory and quantum communication between the user and the bank in order to verify the validity of a bill or (ii) fail to protect user privacy due to the uniqueness of each bill issued by the bank, which can allow its usage to be tracked. We introduce a construction of single-use quantum money that gives users the ability to detect whether the issuing authority is tracking them, employing an auditing procedure for which we prove unconditional security. The use of our scheme does not require long-term quantum memory or quantum communication from the users themselves since their validation is a purely classical operation, making the protocol relatively practical to deploy. We discuss potential applications beyond money, including anonymous one-time pads and voting.