Random unitaries serve as indispensable toy models for complex processes in quantum many-body physics and form the backbone of numerous components of quantum technologies. This idea raises a fundamental question: In what time-scale (i.e. circuit depth) can a quantum circuit behave like a random unitary? I will present recent work in which we show that local quantum circuits can realize random unitaries in exponentially shorter time-scales than previously thought. We prove that random quantum circuits on any geometry, including a 1D line, can form so-called approximate unitary designs over n qubits in log n depth. 

 

Similarly, we show that 1D circuits can form pseudorandom unitaries, a recent object of interest in quantum cryptography, in poly(log n) circuit depth. Both depths can be further exponentially reduced using long-range connectivity. These shallow quantum circuits have low complexity and create only short-range entanglement, yet are indistinguishable from unitaries with exponential complexity. Applications of our results include:

Time pending, I will also discuss recent extensions of our results to symmetric quantum systems and time-reversal experiments.