Despite progress towards achieving low error rates with superconducting qubits, error-prone two-qubit gates remain a bottleneck for realizing large-scale quantum computers. Therefore, a systematic framework to design high-fidelity gates becomes imperative. One type of two-qubit gate in superconducting qubits is the controlled-phase (CPHASE) gate, which utilizes a conditional interaction between higher energy levels of the qubits controlled by a baseband flux pulse on one of the qubits or a tunable coupler. In this work, we study an adiabatic implementation of CPHASE gates and formulate the design of the control trajectory for the gate as a pulse-design problem. We show in simulation that the Chebyshev-based trajectory can, in certain cases, enable gates with gate infidelity lower by an average of 23.3% when compared to the widely used Slepian-based trajectory.

Chebyshev approximation problems are fundamental in approximation theory and are integral to various engineering applications. In this study, we suggest that weighted Chebyshev approximation (WCA) can be effectively applied to designing quantum gates. Specifically, we examine a class of two-qubit gates in superconducting qubits known as controlled-hase (CPHASE) gates, which leverage interactions between higher levels controlled by baseband flux pulses. We frame the design of CPHASE gates as a pulse design problem and propose Chebyshev pulses, generated through WCA, as an alternative solution. We utilize an illustrative example to demonstrate our process of designing a Chebyshev pulse with WCA, then contrast it with a benchmark Slepian pulse for comparison. We find that Chebyshev pulses can facilitate faster CPHASE gates while ensuring leakage errors remain below the desired threshold.

A qubit pure state can be specified by a point on the Bloch sphere, and similarly certain quantum measurements can be specified by M points on a sphere which we refer to as an Etro sphere. A key objective of this paper is to compare the performance for discriminating between two qubit pure states using POVM designs based on the distribution of M points on the Etro sphere. We specifically address the case where the alignment of the M points relative to the coordinates of the sphere is unknown. Of particular interest is the insensitivity of the POVM designs as measured by the difference between the maximum and minimum probabilities of error over all alignments. We consider distributions of the M points corresponding to Platonic solids as well as optimal distributions with respect to several criteria including, for example, maximum nearest neighbor distance and minimum Riesz s-energy. We provide evidence through simulation of various performance tradeoffs such as the tradeoff between stability and best case performance.

Our field has had a rich history, and clearly it has incredible potential going forward. There is always an opportunity for discovering or rediscovering mathe-matical principles that have not yet been fully exploited in the context of signal processing. And physics will continue to provide us with new ways of imple-menting signal processing systems. While digital platforms have played an increasingly important part in signal processing system implementation, the role of analog platforms also continues to grow, as does a mix of both. And quite likely, as the technology advances,it will become increasingly difficult to define precisely which parts of a sys-tem are considered analog and which are digital.

The articles in this special section focus on ways to deliver both innovative educational content and the core signal processing curriculum to current electrical engineering and computer science degrees in order to provide high-quality and hands on multidisciplinary skills, experience, and inspiration for students at all levels.

This monograph addresses operating characteristics for binary hypothesis testing in both classical and quantum settings and overcomplete quantum measurements for quantum binary state discrimination. We specifically explore decision and measurement operating characteristics defined as the tradeoff between probability of detection and probability of false alarm as parameters of the pre-decision operator and the binary decision rule are varied. In the classical case we consider in detail the Neyman-Pearson optimality of the operating characteristics when they are generated using threshold tests on a scalar score variable rather than threshold tests on the likelihood ratio. In the quantum setting, informationally overcomplete POVMs are explored to provide robust quantum binary state discrimination. We focus on equal trace rank one POVMs which can be specified by arrangements of points on a sphere that we refer to as an Etro sphere. Catherine A. Medlock and Alan V. Oppenheim (2021), “Operating Characteristics for Classical and Quantum Binary Hypothesis Testing”, Foundations and Trends® in Signal Processing: Vol. 15, No. 1, pp 1–120. DOI: 10.1561/2000000106. Full text available at: http://dx.doi.org/10.1561/2000000106

It is well-known in classical frame theory that overcomplete representations of a given vector space provide robustness to additive noise on the frame coefficients of an unknown vector. We describe how the same robustness can be shown to exist in the context of quantum state estimation. A key element of the discussion is the application of classical frame theory to operator-valued vector spaces, or operator spaces, which arise naturally in quantum mechanics. Specifically, in the problem we describe the frame vectors are represented by the elements of an informationally complete or overcomplete (IC or IOC) POVM, the frame coefficients are represented by the outcome probabilities of a quantum measurement made on an unknown state, and the error on the frame coefficients arises from finite sample size estimations of the probabilities. We show that with this formulation of the problem, there is a tradeoff in estimation performance between the number of copies of the unknown system and the number of POVM elements. Lastly, we present evidence through simulation that the same tradeoff is present in the context of quantum binary state detection -- the probability of error can be reduced either by increasing the number of copies of the unknown system or by increasing the number of POVM elements.

The theoretical basis for conventional acquisition of bandlimited signals typically relies on uniform time sampling and assumes infinite-precision amplitude values. In this paper, we explore signal representation and recovery based on uniform amplitude sampling with assumed infinite precision timing information. The approach is based on the delta-ramp encoder which consists of applying a one-level level-crossing detector to the result of adding an appropriate sawtooth-like waveform to the input signal. The output samples are the time instants of these level crossings, thus representing a time-encoded version of the input signal. For theoretical purposes, this system can be equivalently analyzed by reversibly transforming through ramp addition a nonmonotonic input signal into a monotonic one, which is then uniformly sampled in amplitude. The monotonic function is then represented by the times at which the signal crosses a predefined and equally-spaced set of amplitude values. We refer to this technique as amplitude sampling. The time sequence generated can be interpreted alternatively as nonuniform time sampling of the original source signal. We derive duality and frequency-domain properties for the functions involved in the transformation. Iterative algorithms are proposed and implemented for recovery of the original source signal. As indicated in the simulations, the proposed iterative amplitude-sampling algorithm achieves a faster convergence rate than frame-based reconstruction for nonuniform sampling. The performance can also be improved by appropriate choice of the parameters while maintaining the same sampling density.

The Receiver Operating Characteristic (ROC) is a well-established representation of the tradeoff between detection and false alarm probabilities in binary hypothesis testing. In many practical contexts ROC's are generated by thresholding a measured score variable - applying score variable threshold tests (SVT's). In many cases the resulting curve is different from the likelihood ratio test (LRT) ROC and is therefore not Neyman-Pearson optimal. While it is well-understood that concavity is a necessary condition for an ROC to be Neyman-Pearson optimal, this paper establishes that it is also a sufficient condition in the case where the ROC was generated using SVT's. It further defines a constructive procedure by which the LRT ROC can be generated from a non-concave SVT ROC, without requiring explicit knowledge of the conditional PDF's of the score variable. If the conditional PDF's are known, the procedure implicitly provides a way of redesigning the test so that it is equivalent to an LRT.

Analog-to-digital (A/D) converters are the common interface between analog signals and the domain of digital discrete-time signal processing. In essence, this domain simultaneously incorporates quantization both in amplitude and time, i. e. amplitude quantization and uniform time sampling. Thus, we view A/D conversion as a sampling process in both the time and amplitude domains based on the observation that the underlying continuous-time signals representing digital sequences can be sampled in a lattice-i.e. at points restricted to lie on a uniform grid both in time and amplitude. We refer to them as lattice functions. This is in contrast with the traditional approach based on the classical sampling theorem and quantization error analysis. The latter has been mainly addressed with the help of probabilistic models, or deterministic ones either confined to very particular scenarios or considering worst-case assumptions. In this paper, we provide a deterministic theoretical analysis and framework for the functions involved in digital discrete-time processing. We show that lattice functions possess a rich analytic structure in the context of integral-valued entire functions of exponential type. We derive set and spectral properties of this class of functions. This allows us to prove in a deterministic way and for general bandlimited functions a fundamental lower bound on the maximum frequency component introduced by quantization that is independent of the resolution of the quantizer.

Receiver operating characteristics (ROCs) are a well-established representation of the tradeoff between detection and false alarm probabilities in classical binary hypothesis testing. We use classical ROCs as motivation for two types of operating characteristics for binary hypothesis testing in quantum systems - decision operating characteristics (QDOCs) and measurement operating characteristics (QMOCs). Both are described in the context of a framework we propose that encompasses the typical formulations of binary hypothesis testing in both the classical and quantum scenarios. We interpret Helstrom's well-known result [1] regarding discrimination between two quantum density operators with minimum probability of error in this framework. We also present a generalization of previous results [2], [3] regarding the correspondence between classical Parseval frames and quantum measurements. The derivation naturally leads to a constructive procedure for generating many different measurements besides Helstrom's optimal measurement, some standard and others non-standard, that achieve minimum probability of error.