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Finding solutions to systems of linear equations is a common prob-lem in many areas of science and engineering, with much potential for a speedup on quantum devices. While the Harrow-Hassidim-Lloyd (HHL) quantum algorithm yields up to an exponential speed-up over classical algorithms in some cases, it requires a fault-tolerant quantum computer, which is unlikely to be available in the near term. Thus, attention has turned to the investigation of quantum algorithms for noisy intermediate-scale quantum (NISQ) devices where several near-term approaches to solving systems of linear equations have been proposed. This paper focuses on the Variational Quantum Linear Solvers (VQLS), and other closely related methods. This paper makes several contributions that include: the first application of the Evolutionary Ansatz to the VQLS (EAVQLS), the first implementation of the Logical Ansatz VQLS (LAVQLS), based on the Classical Combination of Quantum States (CQS) method, the first proof of principle demonstration of the CQS method on real quantum hardware and a method for the implementation of the Adiabatic Ansatz (AAVQLS). These approaches are implemented and contrasted.
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Solving linear systems of equations is essential for many problems in science and technology, including problems in machine learning. Existing quantum algorithms have demonstrated the potential for large speedups, but the required quantum resources are not immediately available on near-term quantum devices. In this work, we study near-term quantum algorithms for linear systems of equations of the form $Ax = b$. We investigate the use of variational algorithms and analyze their optimization landscapes. There exist types of linear systems for which variational algorithms designed to avoid barren plateaus, such as properly-initialized imaginary time evolution and adiabatic-inspired optimization, suffer from a different plateau problem. To circumvent this issue, we design near-term algorithms based on a core idea: the classical combination of variational quantum states (CQS). We exhibit several provable guarantees for these algorithms, supported by the representation of the linear system on a so-called Ansatz tree. The CQS approach and the Ansatz tree also admit the systematic application of heuristic approaches, including a gradient-based search. We have conducted numerical experiments solving linear systems as large as $2^{300} times 2^{300}$ by considering cases where we can simulate the quantum algorithm efficiently on a classical computer. These experiments demonstrate the algorithms ability to scale to system sizes within reach in near-term quantum devices of about $100$-$300$ qubits.
Efficient sampling from a classical Gibbs distribution is an important computational problem with applications ranging from statistical physics over Monte Carlo and optimization algorithms to machine learning. We introduce a family of quantum algorithms that provide unbiased samples by preparing a state encoding the entire Gibbs distribution. We show that this approach leads to a speedup over a classical Markov chain algorithm for several examples including the Ising model and sampling from weighted independent sets of two different graphs. Our approach connects computational complexity with phase transitions, providing a physical interpretation of quantum speedup. Moreover, it opens the door to exploring potentially useful sampling algorithms on near-term quantum devices as the algorithm for sampling from independent sets on certain graphs can be naturally implemented using Rydberg atom arrays.
The solution of linear systems of equations is a very frequent operation and thus important in many fields. The complexity using classical methods increases linearly with the size of equations. The HHL algorithm proposed by Harrow et al. achieves exponential acceleration compared with the best classical algorithm. However, it has a relatively high demand for qubit resources and the solution $left| x rightrangle $ is in a normalized form. Assuming that the eigenvalues of the coefficient matrix of the linear systems of equations can be represented perfectly by finite binary number strings, three hybrid iterative phase estimation algorithms (HIPEA) are designed based on the iterative phase estimation algorithm in this paper. The complexity is transferred to the measurement operation in an iterative way, and thus the demand of qubit resources is reduced in our hybrid algorithms. Moreover, the solution is stored in a classical register instead of a quantum register, so the exact unnormalized solution can be obtained. The required qubit resources in the three HIPEA algorithms are different. HIPEA-1 only needs one single ancillary qubit. The number of ancillary qubits in HIPEA-2 is equal to the number of nondegenerate eigenvalues of the coefficient matrix of linear systems of equations. HIPEA-3 is designed with a flexible number of ancillary qubits. The HIPEA algorithms proposed in this paper broadens the application range of quantum computation in solving linear systems of equations by avoiding the problem that quantum programs may not be used to solve linear systems of equations due to the lack of qubit resources.
Recent computations involving quantum processing units (QPUs) have demonstrated a series of challenges inherent to hybrid classical-quantum programming, compilation, execution, and verification and validation. Despite considerable progress, system-level noise, limited low-level instructions sets, remote access models, and an overall lack of portability and classical integration presents near-term programming challenges that must be overcome in order to enable reliable scientific quantum computing and support robust hardware benchmarking. In this work, we draw on our experience in programming QPUs to identify common concerns and challenges, and detail best practices for mitigating these challenge within the current hybrid classical-quantum computing paradigm. Following this discussion, we introduce the XACC quantum compilation and execution framework as a hardware and language agnostic solution that addresses many of these hybrid programming challenges. XACC supports extensible methodologies for managing a variety of programming, compilation, and execution concerns across the increasingly diverse set of QPUs. We use recent nuclear physics simulations to illustrate how the framework mitigates programming, compilation, and execution challenges and manages the complex workflow present in QPU-enhanced scientific applications. Finally, we codify the resulting hybrid scientific computing workflow in order to identify key areas requiring future improvement.
We establish an improved classical algorithm for solving linear systems in a model analogous to the QRAM that is used by quantum linear solvers. Precisely, for the linear system $Ax = b$, we show that there is a classical algorithm that outputs a data structure for $x$ allowing sampling and querying to the entries, where $x$ is such that $|x - A^{-1}b|leq epsilon |A^{-1}b|$. This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm is $widetilde{O}(kappa_F^6 kappa^2/epsilon^2 )$, where $kappa_F = |A|_F|A^{-1}|$ and $kappa = |A||A^{-1}|$. This improves the previous best algorithm [Gily{e}n, Song and Tang, arXiv:2009.07268] of complexity $widetilde{O}(kappa_F^6 kappa^6/epsilon^4)$. Our algorithm is based on the randomized Kaczmarz method, which is a particular case of stochastic gradient descent. We also find that when $A$ is row sparse, this method already returns an approximate solution $x$ in time $widetilde{O}(kappa_F^2)$, while the best quantum algorithm known returns $ket{x}$ in time $widetilde{O}(kappa_F)$ when $A$ is stored in the QRAM data structure. As a result, assuming access to QRAM and if $A$ is row sparse, the speedup based on current quantum algorithms is quadratic.
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