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Register a new userLarge-scale quantum hybrid solution for linear systems of equations
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Added by
Michael Perelshtein R.
Publication date
2020
fields
Physics
and research's language is
English
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State-of-the-art noisy intermediate-scale quantum devices (NISQ), although imperfect, enable computational tasks that are manifestly beyond the capabilities of modern classical supercomputers. However, present quantum computations are restricted to exploring specific simplified protocols, whereas the implementation of full-scale quantum algorithms aimed at solving concrete large scale problems arising in data analysis and numerical modelling remains a challenge. Here we introduce and implement a hybrid quantum algorithm for solving linear systems of equations with exponential speedup, utilizing quantum phase estimation, one of the exemplary core protocols for quantum computing. We introduce theoretically classes of linear systems that are suitable for current generation quantum machines and solve experimentally a $2^{17}$-dimensional problem on superconducting IBMQ devices, a record for linear system solution on quantum computers. The considered large-scale algorithm shows superiority over conventional solutions, demonstrates advantages of quantum data processing via phase estimation and holds high promise for meeting practically relevant challenges.
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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.
Recently, it is shown that quantum computers can be used for obtaining certain information about the solution of a linear system Ax=b exponentially faster than what is possible with classical computation. Here we first review some key aspects of the algorithm from the standpoint of finding its efficient quantum circuit implementation using only elementary quantum operations, which is important for determining the potential usefulness of the algorithm in practical settings. Then we present a small-scale quantum circuit that solves a 2x2 linear system. The quantum circuit uses only 4 qubits, implying a tempting possibility for experimental realization. Furthermore, the circuit is numerically simulated and its performance under different circuit parameter settings is demonstrated.
Quantum computers have the potential of solving certain problems exponentially faster than classical computers. Recently, Harrow, Hassidim and Lloyd proposed a quantum algorithm for solving linear systems of equations: given an $Ntimes{N}$ matrix $A$ and a vector $vec b$, find the vector $vec x$ that satisfies $Avec x = vec b$. It has been shown that using the algorithm one could obtain the solution encoded in a quantum state $|x$ using $O(log{N})$ quantum operations, while classical algorithms require at least O(N) steps. If one is not interested in the solution $vec{x}$ itself but certain statistical feature of the solution ${x}|M|x$ ($M$ is some quantum mechanical operator), the quantum algorithm will be able to achieve exponential speedup over the best classical algorithm as $N$ grows. Here we report a proof-of-concept experimental demonstration of the quantum algorithm using a 4-qubit nuclear magnetic resonance (NMR) quantum information processor. For all the three sets of experiments with different choices of $vec b$, we obtain the solutions with over 96% fidelity. This experiment is a first implementation of the algorithm. Because solving linear systems is a common problem in nearly all fields of science and engineering, we will also discuss the implication of our results on the potential of using quantum computers for solving practical linear systems.
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.
Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time proportional to the number of variables N. A recently proposed quantum algorithm shows that quantum computers could solve linear systems in a time scale of order log(N), giving an exponential speedup over classical computers. Here we realize the simplest instance of this algorithm, solving 2*2 linear equations for various input vectors on a quantum computer. We use four quantum bits and four controlled logic gates to implement every subroutine required, demonstrating the working principle of this algorithm.
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