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Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…

Current self-supervised learning (SSL) methods (e.g., SimCLR, DINO, VICReg,MOCOv3) target primarily on representations at instance level and do not generalize well to dense prediction tasks, such as object detection and segmentation.Towards…

Quantitative separation logic (QSL) is an extension of separation logic (SL) for the verification of probabilistic pointer programs. In QSL, formulae evaluate to real numbers instead of truth values, e.g., the probability of memory-safe…

Quantum signal processing (QSP) has emerged as a unifying subroutine in quantum algorithms. In QSP, we are given a function $f$ and a unitary black-box $U$, and the goal is to construct a quantum circuit for implementing $f(U)$ to a given…

We describe two implementations of the optimal error correction algorithm known as the maximum likelihood decoder (MLD) for the 2D surface code with a noiseless syndrome extraction. First, we show how to implement MLD exactly in time…

We present a general strategy for mapping fermionic systems to quantum hardware with square qubit connectivity which yields low-depth quantum circuits, counted in the number of native two-qubit fSIM gates. We achieve this by leveraging…

Quantum error detection (QED) offers a promising pathway to fault tolerance in near-term quantum devices by balancing error suppression with minimal resource overhead. However, its practical utility hinges on optimizing design…

Quantum algorithms reformulate computational problems as quantum evolutions in a large Hilbert space. Most quantum algorithms assume that the time-evolution is perfectly unitary and that the full Hilbert space is available. However, in…

Superconducting quantum circuits are promising candidate for building scalable quantum computers. Here, we use a four-qubit superconducting quantum processor to solve a two-dimensional system of linear equations based on a quantum algorithm…

Efficient state preparation is a challenging and important problem in quantum computing. In this work, we present a recursive state preparation algorithm that combines logarithmic-depth Dicke state circuits with Hamming weight encoders for…

Universal compilation is a training process that compiles a trainable unitary into a target unitary and it serves vast potential applications from quantum dynamic simulations to optimal circuits with deep-compressing, device benchmarking,…

The rigorous solution to the grating diffraction problem is a cornerstone step in many scientific fields and industrial applications ranging from the study of the fundamental properties of metasurfaces to the simulation of photolithography…

The most scalable proposed methods of simulating lattice fermions on noisy quantum computers employ encodings that eliminate nonlocal operators using a constant factor more qubits and a nontrivial stabilizer group. In this work, we…

Universal quantum computers require fault-tolerant logical qudits, as qudits naturally align with the simulation of multi-level physical systems. Here, we present a general framework and working examples for encoding fault-tolerant logical…

Understanding the effects of noise on quantum computations is fundamental to the development of quantum hardware and quantum algorithms. Simulation tools are essential for quantitatively modelling these effects, yet unless artificial…

Simulation of materials is one of the most promising applications of quantum computers. On near-term hardware the crucial constraint on these simulations is circuit depth. Many quantum simulation algorithms rely on a layer of unitary…

Quantum Layout Synthesis (QLS) is a critical compilation stage that adapts quantum circuits to hardware constraints with an objective of minimizing the SWAP overhead. While heuristic tools demonstrate good efficiency, they often produce…

We present quantum algorithms for the simulation of quantum systems in one spatial dimension, which result in quantum speedups that range from superpolynomial to polynomial. We first describe a method to simulate the evolution of the…

We present a protocol to encode and decode arbitrary quantum states in the parity architecture with constant circuit depth using measurements, local nearest-neighbor and single-qubit operations only. While this procedure typically requires…

The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process---it is the algorithm which computes…