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Classical shadows (CS) offer a resource-efficient means to estimate quantum observables, circumventing the need for exhaustive state tomography. Here, we clarify and explore the connection between CS techniques and least squares (LS) and regularized least squares (RLS) methods commonly used in machine learning and data analysis. By formal identification of LS and RLS ``shadows'' completely analogous to those in CS---namely, point estimators calculated from the empirical frequencies of single measurements---we show that both RLS and CS can be viewed as regularizers for the underdetermined regime, replacing the pseudoinverse with invertible alternatives. Through numerical simulations, we evaluate RLS and CS from three distinct angles: the tradeoff in bias and variance, mismatch between the expected and actual measurement distributions, and the interplay between the number of measurements and number of shots per measurement.Compared to CS, RLS attains lower variance at the expense of bias, is robust to distribution mismatch, and is more sensitive to the number of shots for a fixed number of state copies---differences that can be understood from the distinct approaches taken to regularization. Conceptually, our integration of LS, RLS, and CS under a unifying ``shadow'' umbrella aids in advancing the overall picture of CS techniques, while practically our results highlight the tradeoffs intrinsic to these measurement approaches, illuminating the circumstances under which either RLS or CS would be preferred, such as unverified randomness for the former or unbiased estimation for the latter. 

Discrete frequency modes, or bins, present a blend of opportunities and challenges for photonic quantum information processing. Frequency-bin-encoded photons are readily generated by integrated quantum light sources, naturally high-dimensional, stable in optical fiber, and massively parallelizable in a single spatial mode. Yet quantum operations on frequency-bin states require coherent and controllable multifrequency interference, making them significantly more challenging to manipulate than more traditional spatial degrees of freedom. In this mini-review, we describe recent developments that have transformed these challenges and propelled frequency bins forward. Focusing on sources, manipulation schemes, and detection approaches, we introduce the basics of frequency-bin encoding, summarize the state of the art, and speculate on the field’s next phases. Given the combined progress in integrated photonics, high-fidelity quantum gates, and proof-of-principle demonstrations, frequency-bin quantum information is poised to emerge from the lab and leave its mark on practical quantum information processing—particularly in networking where frequency bins offer unique tools for multiplexing, interconnects, and high-dimensional communications. 

We report on the manipulation of the time-resolved biphoton correlation function using a sub-GHz resolution silicon nitride microresonator-based spectral shaper capa-ble of programmable amplitude and phase modulation. 

We showcase a fully on-chip CMOS-fabricated silicon photonic integrated circuit employing a bidirectionally pumped microring and polarization splitter-rotators tailored for the generation of broadband (>9 THz), high-fidelity (90–98%) polarization-entangled photons. Spanning the optical C+L-band and producing over 116 frequency-bin pairs on a 38.4-GHz-spaced grid, this source is ideal for flex-grid wavelength-multiplexed entanglement distribution in multiuser networks. 

We demonstrate a silicon photonic integrated circuit fabricated through the CMOS manufacturing process, which features a bidirectionally pumped microring to achieve over 116 high-fidelity polarization entangled channels covering the entire optical C+L-band for flex-grid entanglement distribution.