Toward Mixed Analog-Digital Quantum Signal Processing: Quantum AD/DA Conversion and the Fourier Transform

Signal processing stands as a pillar of classical computation and modern information technology, applicable to both analog and digital signals. Recently, advancements in quantum information science have suggested that quantum signal processing (QSP) can enable more powerful signal processing capabilities. However, the developments in QSP have primarily leveraged \emph{digital} quantum resources, such as discrete-variable (DV) systems like qubits, rather than \emph{analog} quantum resources, such as continuous-variable (CV) systems like quantum oscillators. Consequently, there remains a gap in understanding how signal processing can be performed on hybrid CV-DV quantum computers. Here we address this gap by developing a new paradigm of mixed analog-digital QSP. We demonstrate the utility of this paradigm by showcasing how it naturally enables analog-digital conversion of quantum signals -- specifically, the transfer of states between DV and CV quantum systems. We then show that such quantum analog-digital conversion enables new implementations of quantum algorithms on CV-DV hardware. This is exemplified by realizing the quantum Fourier transform of a state encoded on qubits via the free-evolution of a quantum oscillator, albeit with a runtime exponential in the number of qubits due to information theoretic arguments. Collectively, this work marks a significant step forward in hybrid CV-DV quantum computation, providing a foundation for scalable analog-digital signal processing on quantum processors.

Biological error correction codes generate fault-tolerant neural networks

It has been an open question in deep learning if fault-tolerant computation is possible: can arbitrarily reliable computation be achieved using only unreliable neurons? In the mammalian cortex, analog error correction codes known as grid codes have been observed to protect states against neural spiking noise, but their role in information processing is unclear. Here, we use these biological codes to show that a universal fault-tolerant neural network can be achieved if the faultiness of each neuron lies below a sharp threshold, which we find coincides in order of magnitude with noise observed in biological neurons. The discovery of a sharp phase transition from faulty to fault-tolerant neural computation opens a path towards understanding noisy analog systems in artificial intelligence and neuroscience.