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At least 253 records · Page 14

Quantum optimization algorithms: Energetic implications

Since the dawn of quantum computing (QC), theoretical developments like Shor's algorithm proved the conceptual superiority of QC over traditional computing. However, such quantum supremacy claims are difficult to achieve in practice because of the technical challenges of realizing noiseless qubits. In the near future, QC applications will need to rely on noisy quantum devices that offload part of their work to classical devices. One way to achieve this is by using parameterized quantum circuits in optimization or even in machine learning tasks. The energy requirements of quantum algorithms have not yet been studied extensively. Here in this article, we explore several optimization algorithms using both theoretical insights and numerical experiments to understand their impact on energy consumption. Specifically, we highlight why and how algorithms like quantum natural gradient descent, simultaneous perturbation stochastic approximations or circuit learning methods, are at least 2x to 4x more energy efficient than their classical counterparts; why feedback-based quantum optimization is energy-inefficient; and how techniques like Rosalin can improve the energy efficiency of other algorithms by a factor of ≥2 0 x. Finally, we use the NchooseK high-level programming model to run optimization problems on both gate-based quantum computers and quantum annealers. Empirical data indicate that these optimization problems run faster, have better success rates, and consume less energy on quantum annealers than on their gate-based counterparts.

97 MATHEMATICS AND COMPUTING↗

Optimized Quantum Program Execution Ordering to Mitigate Errors in Simulations of Quantum Systems

Simulating the time evolution of a physical system at quantum mechanical levels of detail - known as Hamiltonian Simulation (HS) - is an important and interesting problem across physics and chemistry. For this task, algorithms that run on quantum computers are known to be exponentially faster than classical algorithms; in fact, this application motivated Feynman to propose the construction of quantum computers. Nonetheless, there are challenges in reaching this performance potential. Prior work has focused on compiling circuits (quantum programs) for HS with the goal of maximizing either accuracy or gate cancellation. Our work proposes a compilation strategy that simultaneously advances both goals. At a high level, we use classical optimizations such as graph coloring and travelling salesperson to order the execution of quantum programs. Specifically, we group together mutually commuting terms in the Hamiltonian (a matrix characterizing the quantum mechanical system) to improve the accuracy of the simulation. We then rearrange the terms within each group to maximize gate cancellation in the final quantum circuit. Furthermore, these optimizations work together to improve HS performance and result in an average 40% reduction in circuit depth. This work advances the frontier of HS which in turn can advance physical and chemical modeling in both basic and applied sciences.

97 MATHEMATICS AND COMPUTING↗

Quantum Fourier transform revisited

Summary The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank‐1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix‐2 QFT decomposition to a radix‐ d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.

Camps, Daan↗

Q4Q: Quantum Computation for Quantum Prediction of Materials and Molecular Properties

We aim at exploring the potential of quantum computation to solve practical problems that are currently targeted by “traditional” high performance computing (HPC). Along the way, we will develop theoretical frameworks and algorithmic strategies. We propose computational research activities that will advance the application of quantum computers to selected materials/molecular properties: (1) Band structure of solids; (2) Molecular electronic/vibronic properties; (3) Free energies and phase transitions.

36 MATERIALS SCIENCE↗

Sampling frequency thresholds for the quantum advantage of the quantum approximate optimization algorithm

We compare the performance of the Quantum Approximate Optimization Algorithm (QAOA) with state-of-the-art classical solvers Gurobi and MQLib to solve the MaxCut problem on 3-regular graphs. We identify the minimum noiseless sampling frequency and depth p required for a quantum device to outperform classical algorithms. There is potential for quantum advantage on hundreds of qubits and moderate depth with a sampling frequency of 10 kHz. We observe, however, that classical heuristic solvers are capable of producing high-quality approximate solutions in linear time complexity. In order to match this quality for large graph sizes N, a quantum device must support depth p > 11. Additionally, multi-shot QAOA is not efficient on large graphs, indicating that QAOA p ≤ 11 does not scale with N. These results limit achieving quantum advantage for QAOA MaxCut on 3-regular graphs. Other problems, such as different graphs, weighted MaxCut, and 3-SAT, may be better suited for achieving quantum advantage on near-term quantum devices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Resilience–runtime tradeoff relations for quantum algorithms

Abstract A leading approach to algorithm design aims to minimize the number of operations in an algorithm’s compilation. One intuitively expects that reducing the number of operations may decrease the chance of errors. This paradigm is particularly prevalent in quantum computing, where gates are hard to implement and noise rapidly decreases a quantum computer’s potential to outperform classical computers. Here, we find that minimizing the number of operations in a quantum algorithm can be counterproductive, leading to a noise sensitivity that induces errors when running the algorithm in non-ideal conditions. To show this, we develop a framework to characterize the resilience of an algorithm to perturbative noises (including coherent errors, dephasing, and depolarizing noise). Some compilations of an algorithm can be resilient against certain noise sources while being unstable against other noises. We condense these results into a tradeoff relation between an algorithm’s number of operations and its noise resilience. We also show how this framework can be leveraged to identify compilations of an algorithm that are better suited to withstand certain noises.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Variational Quantum Fidelity Estimation

Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F ( ρ , σ ) based on the ``truncated fidelity'' F ( ρ m , σ ) , which is evaluated for a state ρ m obtained by projecting ρ onto its m -largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with m . To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize ρ , (2) compute matrix elements of σ in the eigenbasis of ρ , and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where σ is arbitrary and ρ is low rank, which we call low-rank fidelity estimation, and we prove that no classical algorithm can efficiently solve this problem under reasonable assumptions. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.

97 MATHEMATICS AND COMPUTING↗

Optimized Quantum Compilation for Near-Term Algorithms with OpenPulse

Quantum computers are traditionally operated by programmers at the granularity of a gate-based instruction set. However, the actual device-level control of a quantum computer is performed via analog pulses. Here, we introduce a compiler that exploits direct control at this microarchitectural level to achieve significant improvements for quantum programs. Unlike quantum optimal control, our approach is bootstrapped from existing gate calibrations and the resulting pulses are simple. Our techniques are applicable to any quantum computer and realizable on current devices. We validate our techniques with millions of experimental shots on IBM quantum computers, controlled via the OpenPulse control interface. For representative benchmarks, our pulse control techniques achieve both 1.6x lower error rates and 2x faster execution time, relative to standard gate-based compilation. These improvements are critical in the near-term era of quantum computing, which is bottlenecked by error rates and qubit lifetimes.

Gokhale, Pranav↗

Dimension Reduction and Redundancy Removal through Successive Schmidt Decompositions

Quantum computers are believed to have the ability to process huge data sizes, which can be seen in machine learning applications. In these applications, the data, in general, are classical. Therefore, to process them on a quantum computer, there is a need for efficient methods that can be used to map classical data on quantum states in a concise manner. On the other hand, to verify the results of quantum computers and study quantum algorithms, we need to be able to approximate quantum operations into forms that are easier to simulate on classical computers with some errors. Motivated by these needs, in this paper, we study the approximation of matrices and vectors by using their tensor products obtained through successive Schmidt decompositions. We show that data with distributions such as uniform, Poisson, exponential, or similar to these distributions can be approximated by using only a few terms, which can be easily mapped onto quantum circuits. The examples include random data with different distributions, the Gram matrices of iris flower, handwritten digits, 20newsgroup, and labeled faces in the wild. Similarly, some quantum operations, such as quantum Fourier transform and variational quantum circuits with a small depth, may also be approximated with a few terms that are easier to simulate on classical computers. Furthermore, we show how the method can be used to simplify quantum Hamiltonians: In particular, we show the application to randomly generated transverse field Ising model Hamiltonians. The reduced Hamiltonians can be mapped into quantum circuits easily and, therefore, can be simulated more efficiently.

97 MATHEMATICS AND COMPUTING↗

Characterizing and mitigating coherent errors in a trapped ion quantum processor using hidden inverses

Quantum computing testbeds exhibit high-fidelity quantum control over small collections of qubits, enabling performance of precise, repeatable operations followed by measurements. Currently, these noisy intermediate-scale devices can support a sufficient number of sequential operations prior to decoherence such that near term algorithms can be performed with proximate accuracy (like chemical accuracy for quantum chemistry problems). While the results of these algorithms are imperfect, these imperfections can help bootstrap quantum computer testbed development. Demonstrations of these algorithms over the past few years, coupled with the idea that imperfect algorithm performance can be caused by several dominant noise sources in the quantum processor, which can be measured and calibrated during algorithm execution or in post-processing, has led to the use of noise mitigation to improve typical computational results. Conversely, benchmark algorithms coupled with noise mitigation can help diagnose the nature of the noise, whether systematic or purely random. Here, we outline the use of coherent noise mitigation techniques as a characterization tool in trapped-ion testbeds. We perform model-fitting of the noisy data to determine the noise source based on realistic physics focused noise models and demonstrate that systematic noise amplification coupled with error mitigation schemes provides useful data for noise model deduction. Further, in order to connect lower level noise model details with application specific performance of near term algorithms, we experimentally construct the loss landscape of a variational algorithm under various injected noise sources coupled with error mitigation techniques. This type of connection enables application-aware hardware codesign, in which the most important noise sources in specific applications, like quantum chemistry, become foci of improvement in subsequent hardware generations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Quantum algorithms for geologic fracture networks

Abstract Solving large systems of equations is a challenge for modeling natural phenomena, such as simulating subsurface flow. To avoid systems that are intractable on current computers, it is often necessary to neglect information at small scales, an approach known as coarse-graining. For many practical applications, such as flow in porous, homogenous materials, coarse-graining offers a sufficiently-accurate approximation of the solution. Unfortunately, fractured systems cannot be accurately coarse-grained, as critical network topology exists at the smallest scales, including topology that can push the network across a percolation threshold. Therefore, new techniques are necessary to accurately model important fracture systems. Quantum algorithms for solving linear systems offer a theoretically-exponential improvement over their classical counterparts, and in this work we introduce two quantum algorithms for fractured flow. The first algorithm, designed for future quantum computers which operate without error, has enormous potential, but we demonstrate that current hardware is too noisy for adequate performance. The second algorithm, designed to be noise resilient, already performs well for problems of small to medium size (order 10–1000 nodes), which we demonstrate experimentally and explain theoretically. We expect further improvements by leveraging quantum error mitigation and preconditioning.

58 GEOSCIENCES↗

Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems

We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal $\mathcal{\tilde{O}}$(dκlog(1/ϵ)) query complexity for a d -sparse matrix, where κ is the condition number, and ϵ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Computable and Operationally Meaningful Multipartite Entanglement Measures

Multipartite entanglement is an essential resource for quantum communication, quantum computing, quantum sensing, and quantum networks. The utility of a quantum state |ψ$\rangle$ for these applications is often directly related to the degree or type of entanglement present in |ψ$\rangle$. Therefore, efficiently quantifying and characterizing multipartite entanglement is of paramount importance. Here, in this work, we introduce a family of multipartite entanglement measures, called concentratable entanglements. Several well-known entanglement measures are recovered as special cases of our family of measures, and hence we provide a general framework for quantifying multipartite entanglement. We prove that the entire family does not increase, on average, under local operations and classical communications. We also provide an operational meaning for these measures in terms of probabilistic concentration of entanglement into Bell pairs. Finally, we show that these quantities can be efficiently estimated on a quantum computer by implementing a parallelized SWAP test, opening up a research direction for measuring multipartite entanglement on quantum devices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

A NASA Perspective on Quantum AI, Error Correction, and Beyond

Quantum computing is one of the most enticing computational paradigms with the potential to revolutionize diverse areas of future-generation computational systems. While quantum computing hardware has advanced rapidly, from tiny laboratory experiments to quantum chips that can outperform even the largest supercomputers on specialized computational tasks, these noisy-intermediate scale quantum (NISQ) processors are still too small and non-robust to be directly useful for any real-world applications. We discuss the prospects for quantum computing and AI, highlighting advances in algorithms, both near- and longer-term. Quantum error correction is critical to the realization of any such vision. The talk with touch on some recent exciting advanced in quantum error correction, particularly in dynamical codes. The talk will conclude with an example of how the combination of quantum computing and artificial intelligence can help probe fundamental aspect of quantum physics.

quantum optimization algorithms and sampling↗

Scalable qubit representations of neutrino mixing matrices

Oscillating neutrino beams exhibit quantum coherence over distances of thousands of kilometers. Their unambiguously quantum nature suggests an appealing test system for direct quantum simulation. Such techniques may enable presently analytically intractable calculations involving multi-neutrino entanglements, such as collective neutrino oscillations in supernovae, but only once oscillation phenomenology is properly re-expressed in the language of quantum circuits. Here we resolve outstanding conceptual issues regarding encoding of arbitrarily mixed neutrino flavor states in the Hilbert space of an n-qubit quantum computer. We introduce algorithms to encode mixing and oscillation of any number of flavor-mixed neutrinos, both with and without CP-violation, with an efficient number of prescriptive input parameters in terms of sub-rotations of the PMNS matrix in standard form. As a result, examples encoded for an IBM-Q quantum computer are shown to converge to analytic predictions both with and without CP-violation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

A universal variational quantum eigensolver for non-Hermitian systems

Abstract Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional $$O(log_{2}{N})$$ O ( l o g 2 N ) quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications.

97 MATHEMATICS AND COMPUTING↗

A Polynomial-Time Classical Algorithm for Noisy Quantum Circuits

We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g., the computational basis). Our approach is based upon the intuition that noise exponentially damps nonlocal correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by keeping track of only the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasipolynomial time, so long as the distribution anticoncentrates. A number of implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: For constant noise rates, any quantum circuit for which error mitigation succeeds in polynomial-time on most input states can also be classically simulated in polynomial-time on most input states. Our algorithms scale exponentially in the inverse noise rate, which is fundamental and makes them impractical for current quantum devices.

decoherence↗

Randomized Algorithms for Scientific Computing (RASC)

Randomized algorithms have propelled advances in artificial intelligence (AI) and represent a foundational research area in advancing AI for Science. Future advancements in DOE Office of Science priority areas such as climate science, astrophysics, fusion, advanced materials, combustion, and quantum computing all require randomized algorithms for surmounting challenges of complexity, robustness, and scalability. Advances in data collection and numerical simulation have changed the dynamics of scientific research and motivate the need for randomized algorithms. For instance, advances in imaging technologies such as X-ray ptychography, electron microscopy, electron energy loss spectroscopy, or adaptive optics lattice light-sheet microscopy collect hyperspectral imaging and scattering data in terabytes, at breakneck speed enabled by state-of-the-art detectors. The data collection is exceptionally fast compared with its analysis. Likewise, advances in high-performance architectures have made exascale computing a reality and changed the economies of scientific computing in the process. Floating-point operations that create data are essentially free in comparison with data movement. Thus far, most approaches have focused on creating faster hardware. Ironically, this faster hardware has exacerbated the problem by making data still easier to create. Under such an onslaught, scientists often resort to heuristic deterministic sampling schemes (e.g., low-precision arithmetic, sampling every nth element) and sacrifice potentially valuable accuracy. Dramatically better results can be achieved via randomized algorithms, reducing the data size as much as or more than naive deterministic subsampling can achieve, while retaining the high accuracy of computing on the full data set. By randomized algorithms we mean those algorithms that employ some form of randomness in internal algorithmic decisions to accelerate time to solution, increase scalability, or improve reliability. Examples include matrix sketching for solving large-scale least-squares problems (see Figure 1) and stochastic gradient descent for training machine learning models. We are not recommending heuristic methods but rather randomized algorithms that have certificates of correctness and probabilistic guarantees of optimality and near-optimality. Such approaches can be useful beyond acceleration, for example, in understanding how to avoid measure zero worst-case scenarios that plague methods such as QR matrix factorization.

97 MATHEMATICS AND COMPUTING↗