Random unitaries are a central object of study in quantum information, with applications to quantum computation, quantum many-body physics, and quantum cryptography. Recent work has constructed unitary designs and pseudorandom unitaries (PRUs) using Θ(log log n)-depth unitary circuits with two-qubit gates. In this work, we show that unitary designs and PRUs can be efficiently constructed in several well-studied models of constant-time quantum computation (i.e., the time complexity on the quantum computer is independent of the system size). These models are constant-depth circuits augmented with certain nonlocal operations, such as (a) many-qubit TOFFOLI gates, (b) many-qubit FANOUT gates, or (c) mid-circuit measurements with classical feedforward control. Recent advances in quantum computing hardware suggest experimental feasibility of these models in the near future. Our results demonstrate that unitary designs and PRUs can be constructed in much weaker circuit models than previously thought. Furthermore, our construction of PRUs in constant-depth with many-qubit TOFFOLI gates shows that, under cryptographic assumptions, there is no polynomial-time learning algorithm for the circuit class QAC⁰. Finally, our results suggest a new approach towards proving that PARITY is not computable in QAC⁰, a long-standing question in quantum complexity theory.

In this work, we establish the strongest known lower bounds against QAC0, while allowing its full power of polynomially many ancillae and gates. Our two main results show that: (1) Depth 3 QAC0 circuits cannot compute PARITY regardless of size, and require at least Ω(exp(√n)) many gates to compute MAJORITY. (2) Depth 2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage, regardless of size.

The circuit class QAC0 was introduced by Moore (1999) as a model for constant depth quantum circuits where the gate set includes many-qubit Toffoli gates. Proving lower bounds against such circuits is a longstanding challenge in quantum circuit complexity; in particular, showing that polynomial-size QAC0 cannot compute the parity function has remained an open question for over 20 years. In this work, we identify a notion of the Pauli spectrum of QAC0 circuits, which can be viewed as the quantum analogue of the Fourier spectrum of classical AC0 circuits. We conjecture that the Pauli spectrum of QAC0 circuits satisfies low-degree concentration, in analogy to the famous Linial, Mansour, Nisan (LMN) theorem on the low-degree Fourier concentration of AC0 circuits. If true, this conjecture immediately implies that polynomial-size QAC0 circuits cannot compute parity. We prove this conjecture for the class of depth-d, polynomial-size QAC0 circuits with at most nO(1/d) auxiliary qubits. We obtain new circuit lower bounds and learning results as applications: this class of circuits cannot correctly compute the n-bit parity function on more than (1/2 + 2−Ω(n1/d))-fraction of inputs, and the n-bit majority function on more than (1/2 + O(n−1/4))-fraction of inputs. Additionally we show that this class of QAC0 circuits with limited auxiliary qubits can be learned with quasipolynomial sample complexity, giving the first learning result for QAC0 circuits. More broadly, our results add evidence that “Pauli-analytic” techniques can be a powerful tool in studying quantum circuits.

We consider the design of vision-based control algorithms for unmanned aerial vehicles (UAVs), so as to enable a UAV to autonomously follow a person. A new vision-based control architecture is proposed with the goals of 1) robustly following the user and 2) implementing following behaviors programmed by manipulation of visual patterns. This is achieved within a detection/tracking paradigm, where the target is a programmable badge worn by the user. This badge contains a visual pattern with two components. The first is fixed and used to locate the user. The second is variable and implements a code used to program the UAV behavior. A biologically inspired tracking/recognition architecture, combining bottom-up and top-down saliency mechanisms, a novel image similarity measure, and an affine validation procedure, is proposed to detect the badge in the scene. The badge location is used by a control algorithm to adjust the UAV flight parameters so as to maintain the user in the center of the field of view. The detected badge is further analyzed to extract the visual code that commands the UAV behavior This is used to control the height and distance of the UAV relative to the user.