No abstract

The classical shadow estimation protocol is a noise-resilient and sample-efficient quantum algorithm for learning the properties of quantum systems. Its performance depends on the choice of a unitary ensemble, which must be chosen by a user in advance. What is the weakest assumption that can be made on the chosen unitary ensemble that would still yield meaningful and interesting results? To address this question, we consider the class of Pauli-invariant unitary ensembles, i.e. unitary ensembles that are invariant under multiplication by a Pauli operator. This class includes many previously studied ensembles like the local and global Clifford ensembles as well as locally scrambled unitary ensembles. For this class of ensembles, we provide an explicit formula for the reconstruction map corresponding to the shadow channel and give explicit sample complexity bounds. In addition, we provide two applications of our results. Our first application is to locally scrambled unitary ensembles, where we give explicit formulas for the reconstruction map and sample complexity bounds that circumvent the need to solve an exponential-sized linear system. Our second application is to the classical shadow tomography of quantum channels with Pauli-invariant unitary ensembles. Our results pave the way for more efficient or robust protocols for predicting important properties of quantum states, such as their fidelity, entanglement entropy, and quantum Fisher information.

Quantum circuit complexity-a measure of the minimum number of gates needed to implement a given unitary transformation-is a fundamental concept in quantum computation, with widespread applications ranging from determining the running time of quantum algorithms to understanding the physics of black holes. In this work, we study the complexity of quantum circuits using the notions of sensitivity, average sensitivity (also called influence), magic, and coherence. We characterize the set of unitaries with vanishing sensitivity and show that it coincides with the family of matchgates. Since matchgates are tractable quantum circuits, we have proved that sensitivity is necessary for a quantum speedup. As magic is another measure to quantify quantum advantage, it is interesting to understand the relation between magic and sensitivity. We do this by introducing a quantum version of the Fourier entropy-influence relation. Our results are pivotal for understanding the role of sensitivity, magic, and coherence in quantum computation.

We quantify the role of scrambling in quantum machine learning. We characterize a quantum neural network’s (QNNs) error in terms of the network’s scrambling properties via the out-of-time-ordered correlator (OTOC). A network can be trained by minimizing a loss function. We show that the loss function can be bounded by the OTOC. We prove that the gradient of the loss function can be bounded by the gradient of the OTOC. This demonstrates that the OTOC landscape regulates the trainability of a QNN. We show numerically that this landscape is flat for maximally scrambling QNNs, which can pose a challenge to training. Our results pave the way for the exploration of quantum chaos in quantum neural networks.

Spatial and temporal small-angle X-ray scattering (SAXS) scans show that 8.6 and 11.8 nm iron oxide nanoparticles (NPs) in heptane drop-cast on top of a heptane layer atop a diethylene glycol (DEG) layer are trapped at the DEG/heptane interface to generally form a single ordered, hexagonal-close packed monolayer (ML), and this occurs long before the heptane evaporates. Many NPs remain dispersed in the heptane after this NP assembly. Assembly occurs faster than expected from considering only the diffusion of NPs from the drop-cast site to this liquid/liquid interface. The formation of the ordered NP ML occurs within 100 s of drop-casting, as followed by using the (10) ordered NP SAXS peak, and on the same time scale there is a concomitant decrease in the SAXS form factor from disordered NPs that is apparently from disordered NPs at the meniscus. Usually, most of the ordered NPs are close packed, but there is evidence that some are ordered although not close packed. After the heptane evaporates, a close-packed ordered NP ML remains at the DEG/vapor interface, though with smaller NP−NP separation, as expected due to less van der Waals shielding caused by the upper medium in the interface. X-ray beam transmission at different vertical heights characterizes the heptane and DEG bulk and interfacial regions, while monitoring the time dependence of SAXS at and near the DEG/heptane interface gives a clear picture of the evolution of NP assembly at this liquid/liquid interface. These SAXS observations of self-limited NP ML formation at the DEG/heptane interface are consistent with those using the less direct method of real-time optical reflection monitoring of that interface.