Doubling Efficiency of Hamiltonian Simulation via Generalized Quantum Signal Processing

The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error ε is O[td∥H[over ^]∥_{max}+log(1/ε)/loglog(1/ε)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i)transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii)transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii)projecting this ancilla with near unity success probability.

Quantum signal processing (QSP) is a powerful toolbox for the design of quantum algorithms and can lead to asymptotically optimal computational costs. Its realization on noisy quantum computers without fault tolerance, however, is challenging because it requires a deep quantum circuit in general. We propose a strategy to run an entire QSP protocol on noisy quantum hardware by carefully reducing overhead costs at each step. To illustrate the approach, we consider the application of Hamiltonian simulation for which QSP implements a polynomial approximation of the time evolution operator. We test the protocol by running the algorithm on the Quantinuum H1-1 trapped-ion quantum computer powered by Honeywell. In particular, we compute the time dependence of bipartite entanglement entropies for Ising spin chains and find good agreements with exact numerical simulations. To make the best use of the device, we determine optimal experimental parameters by using a simplified error model for the hardware and numerically studying the trade-off between Hamiltonian simulation time, polynomial degree, and total accuracy. Our results are the first step in the experimental realization of QSP-based quantum algorithms.

In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is “large” and another part is “small”). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas.

We propose a quantum algorithm for calculating the structural properties of complex networks and graphs. The corresponding protocol——is designed to perform large-scale community and botnet detection, where a specific subgraph of a larger graph is identified based on its properties. We construct a workflow relying on ground state preparation of the network modularity matrix or graph Laplacian. The corresponding maximum modularity vector is encoded into a log 2 ( N ) -qubit register that contains community information. We develop a strategy for “signing” this vector via quantum signal processing, such that it closely resembles a hypergraph state, and project it onto a suitable linear combination of such states to detect botnets. As part of the workflow, and of potential independent interest, we present a readout technique that allows filtering out the incorrect solutions deterministically. This can reduce the scaling for the number of samples from exponential to polynomial. The approach serves as a building block for graph analysis with quantum speedup and enables the cybersecurity of large-scale networks.

Solving Computational Fluid Dynamics (CFD) problems requires the inversion of a linear system of equations, which can be done using a quantum algorithm for matrix inversion.
 However, the number of shots required to measure the output of the system can be prohibitive and remove any advantage obtained by quantum computing.
 In this work we propose a scheme for measuring the output of a Quantum Singular Value Transform (QSVT) matrix inversion algorithm specifically for the CFD use case.
 We use a Quantum Signal Processing (QSP) based amplitude estimation algorithm and show how it can be combined with the QSVT matrix inversion algorithm.
 We perform a detailed resource estimation of the amount of computational resources required for a single iteration of amplitude estimation, and compare the costs of amplitude estimation with the cost of not doing amplitude estimation and measuring the whole wavefunction.
 We also propose a measurement scheme to reduce the number of amplitudes measured in the CFD example by focussing on large amplitudes only. We simulate the whole CFD loop, finding that thus measuring only a small number of the total amplitudes in the output vector still results in an acceptable level of overall error.

Generalized uncertainty relations based upon Fourier transforms of both discrete and continuous functions are briefly reviewed. We extend these results in order to establish discrete versions of the angular momentum uncertainty relations, based upon SU(2) transformations. Possible applications in quantum signal processing and computations are briefly discussed.

Quantum signal processing (QSP) is a framework which was proven to unify and simplify a large number of known quantum algorithms, as well as discovering new ones. QSP allows one to transform a signal embedded in a given unitary using polynomials. Characterizing which polynomials can be achieved with QSP protocols is an important part of the power of this technique, and while such a characterization is well-understood in the case of univariate signals, it is unclear which multivariate polynomials can be constructed when the signal is a vector, rather than a scalar. This work uses a slightly different formalism than what is found in the literature, and uses it to find simpler necessary conditions for decomposability, as well as a sufficient condition – the first, to the best of our knowledge, proven for a (generally inhomogeneous) multivariate polynomial in the context of quantum signal processing.

Quantum computing simulation platform can simulate the computation results of the quantum computer based on traditional computers, which is an effective way to promote the development of quantum computing software, algorithms and hardware at the current immature stage of the real quantum computer. Since quantum computers have exponential calculation acceleration compared with traditional computers, the main problems in implementing quantum computing simulation on traditional computers are low computational efficiency and long time-consuming. A quantum circuit which is a sequence of quantum gates acting on a collection of qubits is the general quantum computing model. So by the means of quantum circuit optimization, the calculation speed can be significantly increased while keeping the calculation result unchanged. The existing empirical rules of quantum circuit optimization methods have limitations and there is no common and automatic quantum circuit optimization method. In this paper, a general and automatic quantum circuit optimization method based on the genetic algorithm is proposed, by which the equivalent optimal quantum circuit is obtained through a finite number of searching in a large searching space. This method is not limited by the hardware of the quantum computing simulation and the composition of the quantum circuit. The experimental results show that for the QFT algorithm of 29 qubits, the running time can be shortened by 41.4% and for the variational circuit of 6 qubits, the running time can be shortened by 18.8% compared with the state-of-the-art quantum circuit optimization method. So this method can improve the quantum computing simulation capability and operating efficiency and provide a rapid development way for quantum algorithms and applications.

A Busy March Meeting Is Brewing in Seattle

Two fundamental problems exist in the use of quantum computation to process an image or signal. The first one is how to represent giant data, such as image data, using quantum state without losing information. The second one is how to load a colossal volume of data into the quantum registers of a quantum CPU from classical electronic memory. Researches on these two questions are rarely reported. Here an entangled state is used to represent an image (or vector) for which two entangled registers are used to store a vector component and its classical address. Using the representation, n 1 + n 2 + 8 qubits are used to store the whole information of the gray image that has a $2^{n_1 } \times 2^{n_2 } $ size at a superposition of states, a feat is not possible with a classic computer. The way of designing a unitary operation to load data, such as a vector (or image), into the quantum registers of a quantum CPU from electronic memory is defined herein as a quantum loading scheme (QLS). In this paper, the QLS with time complexity O(log2 N) is presented where N denotes the number of vector components, a solution that would break through the efficiency bottleneck of loading data. QLS would enable a quantum CPU to be compatible with electronic memory and make possible quantum image compression and quantum signal processing that has classical input and output.

Grover's Adaptative Search (GAS) algorithm combined with the High Order Binary Optimization (HOBO) oracle proposal is currently the Fault Tolerant Quantum Computer (FTQC)'s main algorithm to solve binary optimization problems. In this paper, we propose a new way to obtain the Grover's oracle needed which is costless in terms of gates and ancilla qubits. This oracle proposal also has zero additional costs linked to decimal numbers of the problem weights and a very small overcost due to high order problems with respect to the other proposals. Nevertheless, it cannot be easily adapted to constrained problems. This new oracle is based on a completely different approach, which uses Quantum Signal Processing (QSP) techniques and amplitude encoding. This method can also be described in the Quantum Singular Value Transformation (QSVT) framework as a variant of eigenstate search algorithms. Our oracle combines the benefits of both the QSVT amplitude encoding of the problem and GAS strategies to seek eigenvalues.

In this paper, the fractional Riemann wave equation with M-truncated derivative (FRWE-MTD) is considered. The Jacobi elliptic function method and the modified extended tanh function method are applied to acquire new elliptic, rational, hyperbolic, and trigonometric functions solutions. Moreover, we expand some earlier studies. The obtained solutions are important in explaining some exciting physical phenomena, since the Riemann wave equation is used in various fields, including quantum mechanics, optics, signal processing, and general relativity. Also, this equation is used to describe the propagation of waves in various dispersive systems, where wave motion is affected by the medium through which it travels. Several 3D and 2D graphs are shown to demonstrate how the M-truncated derivative affects the exact solutions of the FRWE-MTD.

This study examines the intricate interplay between Hamiltonian dynamics and spectral analysis within the context of algebraic structures and symmetry groups. By integrating perspectives from symplectic geometry, Lie theory, and nonlinear dynamics, we introduce a framework that aims to unify classical conservation laws with advanced spectral decomposition techniques. Our approach synthesizes geometric mechanics with adaptive numerical strategies, revealing novel relationships between phase-space invariants and multi-scale Fourier analysis. Applications in quantum mechanics, signal processing, and machine learning illustrate the potential of Hamiltonian-preserving algorithms to maintain spectral integrity while ensuring long-term stability. These findings assist in detecting chaotic transitions and demonstrate compatibility with established FFT methodologies, thereby offering new insights into the synergy between geometric structures and spectral methods.

Lithium niobate (LiNbO 3 ) on insulator (LNOI) is a promising material platform for integrated photonics due to single crystal LiNbO 3 film’s wide transparent window, high refractive index, and high second-order nonlinearity. Based on LNOI, the fast-developing ridge-waveguide fabrication techniques enabled various structures, devices, systems, and applications. We review the basic structures including waveguides, cavities, periodically poled LiNbO 3 , and couplers, along with their fabrication methods and optical properties. Treating those basic structures as building blocks, we review several integrated devices including electro-optic modulators, nonlinear optical devices, and optical frequency combs with each device’s operating mechanism, design principle and methodology, and performance metrics. Starting from these integrated devices, we review how integrated LNOI devices boost the performance of LiNbO 3 ’s traditional applications in optical communications and data center, integrated microwave photonics, and quantum optics. Beyond those traditional applications, we also review integrated LNOI devices’ novel applications in metrology including ranging system and frequency comb spectroscopy. Finally, we envision integrated LNOI photonics’ potential in revolutionizing nonlinear and quantum optics, optical computing and signal processing, and devices in ultraviolet, visible, and mid-infrared regimes. Beyond this outlook, we discuss the challenges in integrated LNOI photonics and the potential solutions.

Covariance shaping least squares estimator is a new kind of linear estimator based on covariance shaping in quantum signal processing (QSP) framework, it is a biased estimator directed at improving the performance of traditional leastsquares (LS) estimator. In this paper, we discuss the covariance shaping least squares (CSLS) estimator, and apply it in multi-user detection that leads to a new detection algorithm, and then we employ it in the detection of MIMO systems. The numerical results in synchronous DS-CDMA communications system show that the performance of covariance shaping multi-user detection which depends only on the users' signatures is better than that of traditional MF and decorrelation detection, and is an approach to the MMSE detection method. The results in MIMO systems also show that the performance of system is better than that of detection algorithm with Zero-forcing and MMSE solutions.