Experimental realization of quantum algorithms for a linear system inspired by adiabatic quantum computing

Wen, Jingwei; Kong, Xiangyu; Wei, Shijie; Wang, Bi-Xue; Xin, Tao; Long, Gui Lu

doi:10.1103/physreva.99.012320

Cited by 47 publications

(27 citation statements)

References 37 publications

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“…For example, a quantum algorithm for solving systems of linear equations was established for gate-based quantum computers 1 and demonstrated with small-scale problem instances 2 . Additionally, an algorithm for solving linear systems within the adiabatic quantum computing model 3 was experimentally demonstrated 4 , followed by a more recent proposal 5 .…”

Section: Introductionmentioning

confidence: 99%

Quantum annealing for systems of polynomial equations

Chang

Gambhir

Humble

et al. 2019

Sci Rep

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Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10 −8 .

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Section: Introductionmentioning

confidence: 99%

Quantum annealing for systems of polynomial equations

Chang

Gambhir

Humble

et al. 2019

Sci Rep

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“…Example applications of quantum computers include factorization, 3 database search, 4,5 solving linear equations, 6,7 query‐based eigensolver, 8 and full quantum eigensolver 9 . Proof‐of‐principle experiments of these quantum algorithms have been demonstrated on different physical platforms, such as optical system, 10‐13 ion‐trap system, 14,15 superconducting circuits, 16,17 NMR system, 18‐25 NV center, 26 and so on. However, it remains a major challenge in solving practical problems with these quantum algorithms at present, which is related to the limited number of qubits and the relative low gates fidelity under the current noisy intermediate‐scale quantum (NISQ) devices 27 …”

Section: Introductionmentioning

confidence: 99%

Variational quantum packaged deflation for arbitrary excited states

Wen

Yung

et al. 2021

Quantum Engineering

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Determining the spectral structure of molecular Hamiltonians is one of the most important and challenging research fields in quantum chemistry. In the near future, such a problem may be solvable by means of quantum simulation. However, constrained by a handful of qubits and noisy quantum gates available in the near‐term quantum devices, hybrid quantum‐classical algorithms based on variational methods such as variational quantum eigensolver have become a preferable choice. Here we propose an interesting variational quantum algorithm for obtaining the excited states of many‐body Hamiltonians. To obtain the kth excited states, the previous variational quantum deflation algorithm requires the use of k different variational circuits, which is now reduced to only two circuits with our algorithm. The feasibility of the algorithm was numerically tested on the hydrogen molecule with and without random noise, where the chemical accuracy is reached for both cases. This algorithm is expected to significantly reduce the resource requirements of quantum simulation.

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“…In subsequent works, several new algorithms have been put forward that solve QLS with further increased efficiency in comparison to the original HHL algorithm, improving the runtime dependence on the condition number [4], on the precision [5] and on the sparsity [6]. Recently, a new approach inspired by adiabatic quantum computation has introduced a significantly simpler quasi-optimal solving algorithm [7,8], significantly narrowing the gap with experimental implementations [9], making the algorithm compatible with Near-term Intermediate Scale Quantum (NISQ) devices [10,11] and leading to the development of the presently most efficient QLS solvers [12].…”

Section: Introductionmentioning

confidence: 99%

On solving classes of positive-definite quantum linear systems with quadratically improved runtime in the condition number

Orsucci

Dunjko

2021

Quantum

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Quantum algorithms for solving the Quantum Linear System (QLS) problem are among the most investigated quantum algorithms of recent times, with potential applications including the solution of computationally intractable differential equations and speed-ups in machine learning. A fundamental parameter governing the efficiency of QLS solvers is κ, the condition number of the coefficient matrix A, as it has been known since the inception of the QLS problem that for worst-case instances the runtime scales at least linearly in κ [Harrow, Hassidim and Lloyd, PRL 103, 150502 (2009)]. However, for the case of positive-definite matrices classical algorithms can solve linear systems with a runtime scaling as κ, a quadratic improvement compared to the the indefinite case. It is then natural to ask whether QLS solvers may hold an analogous improvement. In this work we answer the question in the negative, showing that solving a QLS entails a runtime linear in κ also when A is positive definite. We then identify broad classes of positive-definite QLS where this lower bound can be circumvented and present two new quantum algorithms featuring a quadratic speed-up in κ: the first is based on efficiently implementing a matrix-block-encoding of A−1, the second constructs a decomposition of the form A=LL† to precondition the system. These methods are widely applicable and both allow to efficiently solve BQP-complete problems.

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Experimental realization of quantum algorithms for a linear system inspired by adiabatic quantum computing

Cited by 47 publications

References 37 publications

Quantum annealing for systems of polynomial equations

Quantum annealing for systems of polynomial equations

Variational quantum packaged deflation for arbitrary excited states

On solving classes of positive-definite quantum linear systems with quadratically improved runtime in the condition number

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