Quantum Linear Algebra Research Articles

Solving linear systems on quantum hardware with hybrid HHL++

The limited capabilities of current quantum hardware significantly constrain the scale of experimental demonstrations of most quantum algorithmic primitives. This makes it challenging to perform benchmarking of the current hardware using useful quantum algorithms, i.e., application-oriented benchmarking. In particular, the Harrow-Hassidim-Lloyd (HHL) algorithm is a critical quantum linear algebra primitive, but the majority of the components of HHL are far out of the reach of noisy intermediate-scale quantum devices, which has led to the proposal of hybrid classical-quantum variants. The goal of this work is to further bridge the gap between proposed near-term friendly implementations of HHL and the kinds of quantum circuits that can be executed on noisy hardware. Our proposal adds to the existing literature of hybrid quantum algorithms for linear algebra that are more compatible with the current scale of quantum devices. Specifically, we propose two modifications to the Hybrid HHL algorithm proposed by Lee et al., leading to our algorithm Hybrid HHL++\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {HHL}^{++}$$\\end{document}: (1) propose a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL, and (2) introduce a heuristic for compressing the HHL circuit. We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date.

Ideals in Bipolar Quantum Linear Algebra

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists c∈(0,1] such that I=(−x,y)∈R2|cx≤y≤xcandx,y≥0. This shows that its shape has to be symmetric with the graph y=−x.

Improved Resource‐Tunable Near‐Term Quantum Algorithms for Transition Probabilities, with Applications in Physics and Variational Quantum Linear Algebra

AbstractTransition amplitudes and transition probabilities are relevant to many areas of physics simulation, including the calculation of response properties and correlation functions. These quantities can also be related to solving linear systems of equations. This study presents three related algorithms for calculating transition probabilities. First, a previously published short‐depth algorithm is extended, allowing for the two input states to be non‐orthogonal. Building on this first procedure, a higher‐depth algorithm based on Trotterization and Richardson extrapolation is derived that requires fewer circuit evaluations. Third, a tunable algorithm that allows for trading off circuit depth and measurement complexity is introduced, yielding an algorithm that can be tailored to specific hardware characteristics. Finally, proof‐of‐principle numerics are presented for models in physics and chemistry and for a subroutine in variational quantum linear solving (VQLS). The primary benefits of these approaches are that a) arbitrary non‐orthogonal states may now be used with small increases in quantum resources, b) they (like another recently proposed method) entirely avoid subroutines such as the Hadamard test that may require three‐qubit gates to be decomposed, and c) in some cases fewer quantum circuit evaluations are required as compared to the previous state‐of‐the‐art in NISQ algorithms for transition probabilities.

Design Space Exploration for Efficient Quantum Most-Significant Digit-First Arithmetic

Quantum computing has been considered as an emerging approach in addressing problems which are not easily solvable using classical computers. In parallel to the physical implementation of quantum processors, quantum algorithms have been actively developed for real-life applications to show quantum advantages, many of which benefit from quantum arithmetic algorithms and their efficient implementations. As one of the most important operations, quantum addition has been adopted in Shor's algorithm and quantum linear algebra algorithms. Although various least-significant digit-first quantum adders have been introduced in previous work, interest in investigating the efficient implementation of most-significant digit-first addition is growing. In this work, we propose a novel design method for most-significant digit-first addition with several quantum circuit optimisations to reduce the number of quantum bits ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> qubits), quantum gates, and circuit depth. An open-source library of different arithmetic operators based on our proposed method is presented, where all circuits are implemented on IBM Qiskit SDK. Extensive experiments demonstrate that our proposed design, together with the optimisation techniques, reduces T-depth by up-to 4.0×, T-count by 3.5×, and qubit consumption by 1.2×.

Sampling-based Sublinear Low-rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ 2 -norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

Quantum algorithms for SVD-based data representation and analysis

This paper narrows the gap between previous literature on quantum linear algebra and practical data analysis on a quantum computer, formalizing quantum procedures that speed-up the solution of eigenproblems for data representations in machine learning. The power and practical use of these subroutines is shown through new quantum algorithms, sublinear in the input matrix’s size, for principal component analysis, correspondence analysis, and latent semantic analysis. We provide a theoretical analysis of the run-time and prove tight bounds on the randomized algorithms’ error. We run experiments on multiple datasets, simulating PCA’s dimensionality reduction for image classification with the novel routines. The results show that the run-time parameters that do not depend on the input’s size are reasonable and that the error on the computed model is small, allowing for competitive classification performances.

The Metallic Ratio of Pulsating Fibonacci Sequences

The golden ratio and the Fibonacci sequence (Fn) are well known, as is the fact that the ratio Fn+1Fn converges to the golden ratio for sufficiently large n. In this paper, we investigate the metallic ratio—a generalized version of the golden ratio—of pulsating Fibonacci sequences in three forms. Two of these forms are considered in the sense of pulsating recurrence relations, and their diagrams can be represented by symmetry, which is one of their distinguishing characteristics. The third form is the Fibonacci sequence in bipolar quantum linear algebra (BQLA), which also pulsates.

Fast algorithm for quantum polar decomposition and applications

The polar decomposition of a matrix is a key element in the quantum linear algebra toolbox. We show that the problem of quantum polar decomposition, recently studied in Lloyd et al. [arXiv:2006.00841], has a simple and concise implementation via the quantum singular value transform (QSVT). We focus on the applications to pretty-good measurements, a close-to-optimal measurement to distinguish quantum states, and the quantum Procrustes problem, the task of learning an optimal unitary mapping between given ``input'' and ``output'' quantum states. By transforming the state-preparation unitaries into a block-encoding, a prerequisite for QSVT, we develop algorithms for these problems whose gate complexity exhibits a polynomial advantage in the size and condition number of the input compared to alternative approaches for the same problem settings [Lloyd et al., arXiv:2006.00841; Gily\'en et al., arXiv:2006.16924]. For these applications of the polar decomposition, we also obtain an exponential speedup in precision compared to Lloyd et al. [arXiv:2006.00841], as the block-encodings remove the need for the costly density matrix exponentiation step. We contribute a rigorous analysis of the approach of Lloyd et al. [arXiv:2006.00841].

Quantum Principal Component Analysis Only Achieves an Exponential Speedup Because of Its State Preparation Assumptions.

A central roadblock to analyzing quantum algorithms on quantum states is the lack of a comparable input model for classical algorithms. Inspired by recent work of the author [E. Tang, STOC 2019.], we introduce such a model, where we assume we can efficiently perform ℓ^{2}-norm samples of input data, a natural analog to quantum algorithms that assume efficient state preparation of classical data. Though this model produces less practical algorithms than the (stronger) standard model of classical computation, it captures versions of many of the features and nuances of quantum linear algebra algorithms. With this model, we describe classical analogs to Lloyd, Mohseni, and Rebentrost's quantum algorithms for principal component analysis [S. Lloyd, M. Mohseni, and P. Rebentrost, Nat. Phys. 10, 631 (2014).NPAHAX1745-247310.1038/nphys3029] and nearest-centroid clustering [S. Lloyd, M. Mohseni, and P. Rebentrost, Quantum algorithms for supervised and unsupervised machine learning]. Since they are only polynomially slower, these algorithms suggest that the exponential speedups of their quantum counterparts are simply an artifact of state preparation assumptions.

Quantum locally linear embedding for nonlinear dimensionality reduction

Reducing the dimension of nonlinear data is crucial in data processing and visualization. The locally linear embedding algorithm (LLE) is specifically a representative nonlinear dimensionality reduction method with maintaining well the original manifold structure. In this paper, we present two implementations of the quantum locally linear embedding (QLLE) algorithm to perform the nonlinear dimensionality reduction on quantum devices. One implementation, the linear-algebra-based QLLE algorithm, utilizes quantum linear algebra subroutines to reduce the dimension of the given data. The other implementation, the variational quantum locally linear embedding (VQLLE) algorithm, utilizes a variational hybrid quantum-classical procedure to acquire the low-dimensional data. The classical LLE algorithm requires polynomial time complexity of N, where N is the global number of the original high-dimensional data. Compared with the classical LLE, the linear-algebra-based QLLE achieves quadratic speedup in the number and dimension of the given data. The VQLLE can be implemented on the near-term quantum devices in two different designs. In addition, the numerical experiments are presented to demonstrate that the two implementations in our work can achieve the procedure of locally linear embedding.

YinYang bipolar quantum geometry and bipolar quantum superposition Part II – Toward an equilibrium-based analytical paradigm of quantum mechanics and quantum biology

In Part I of this paper, YinYang bipolar quantum agent (BQA), bipolar quantum geometry (BQG) and 2-dimensional generic bipolar quantum superposition are introduced with a geometrical and logical exposition of Dirac 3-polarizer experiment. While the exposition qualifies BQG as a geometry of light, it is shown in this paper that the logical exposition can be extended to an analytical paradigm of quantum mechanics and quantum biology. It is shown that BQG as the geometry of light is also the geometry of Nature with a logical unification of matter and antimatter atoms into a bipolar quantum cellular automaton (BQCA) through multidimensional YinYang bipolar quantum superposition using bipolar quantum linear algebra (BQLA). With the BQCA interpretation of quantum mechanics, it is shown that matter and antimatter self-organization and spacetime emergence is logically possible within BQG. A scalable BQCA model for biological repressionactivation and/or degeneration-regeneration is introduced. Bipolar cellular division and bipolar fractality are proposed. Background independent normal and abnormal bipolar fractal branching is proposed. A discussion on quantum gravity and mathematical abstraction is presented. A few challenges and predictions are posted. It is contended that this work leads to an analytical paradigm of quantum mechanics and quantum biology that may contribute to equilibrium-based analysis of quantum decoherence and collapse as associated with quantum measurement.

Causality Is Logically Definable—Toward an Equilibrium-Based Computing Paradigm of Quantum Agent and Quantum Intelligence (QAQI) (Survey and Research)

A survey on agents, causality and intelligence is presented and an equilibrium-based computing paradigm of quantum agents and quantum intelligence (QAQI) is proposed. In the survey, Aristotle’s causality principle and its historical extensions by David Hume, Bertrand Russell, Lotfi Zadeh, Donald Rubin, Judea Pearl, Niels Bohr, Albert Einstein, David Bohm, and the causal set initiative are reviewed; bipolar dynamic logic (BDL) is introduced as a causal logic for bipolar inductive and deductive reasoning; bipolar quantum linear algebra (BQLA) is introduced as a causal algebra for quantum agent interaction and formation. Despite the widely held view that causality is undefinable with regularity, it is shown that equilibrium-based bipolar causality is logically definable using BDL and BQLA for causal inference in physical, social, biological, mental, and philosophical terms. This finding leads to the paradigm of QAQI where agents are modeled as quantum ensembles; intelligence is revealed as quantum intelligence. It is shown that the ensembles formation, mutation and interaction of agents can be described as direct or indirect results of quantum causality. Some fundamental laws of causation are presented for quantum agent entanglement and quantum intelligence. Applicability is illustrated; major challenges are identified in equilibrium based causal inference and quantum data mining.

Bipolar Quantum Logic Gates and Quantum Cellular Combinatorics—A Logical Extension to Quantum Entanglement

Based on bipolar dynamic logic (BDL) and bipolar quantum linear algebra (BQLA) this work introduces bipolar quantum logic gates and quantum cellular combinatorics with a logical interpretation to quantum entanglement. It is shown that: 1) BDL leads to logically definable causality and generic particle-antiparticle bipolar quantum entanglement; 2) BQLA makes composite atom-atom bipolar quantum entanglement reachable. Certain logical equivalence is identified between the new interpretation and established ones. A logical reversibility theorem is presented for ubiquitous quantum computing. Physical reversibility is briefly discussed. It is shown that a bipolar matrix can be either a modular generalization of a quantum logic gate matrix or a cellular connectivity matrix. Based on this observation, a scalable graph theory of quantum cellular combinatorics is proposed. It is contended that this work constitutes an equilibrium-based logical extension to Bohr’s particle-wave complementarity principle, Bohm’s wave function and Bell’s theorem. In the meantime, it is suggested that the result may also serve as a resolution, rather than a falsification, to the EPR paradox and, therefore, a equilibrium-based logical unification of local realism and quantum non-locality.

YinYang Bipolar Atom—An Eastern Road toward Quantum Gravity

Based on bipolar dynamic logic and bipolar quantum linear algebra, a causal theory of YinYang bipolar atom is intro-duced in a completely background independent geometry that transcends spacetime. The causal theory leads to an equilibrium-based super symmetrical quantum cosmology of negative-positive energies. It is contended that the new theory has opened an Eastern road toward quantum gravity with bipolar logical unifications of particle and wave, matter and antimatter, relativity and quantum entanglement. Information recovery after a black hole is discussed. It is shown that not only can the new theory be applied in physical worlds but also in logical, mental, social and biological worlds. Falsifiability of the theory is discussed.

BRST and anti-BRST operators for quantum linear algebra U q( gl( N)).

For a quantum Lie algebra U q ( gl( N)) we construct BRST, anti-BRST and Laplace operators. The (anti)commutator with the BRST operator defines the differential on the de Rham complex over the quantum group GL q ( N). The Hodge decomposition theorem for this complex is formulated.