Quantum Linear System Solver Based on Time-optimal Adiabatic Quantum Computing and Quantum Approximate Optimization Algorithm

Significant effort in applied quantum computing has been devoted to the problem of ground state energy estimation for molecules and materials. Yet, for many applications of practical value, additional properties of the ground state must be estimated. These include Green's functions used to compute electron transport in materials and the one-particle reduced density matrices used to compute electric dipoles of molecules. In this paper, we propose a quantum-classical hybrid algorithm to efficiently estimate such ground state properties with high accuracy using low-depth quantum circuits. We provide an analysis of various costs (circuit repetitions, maximal evolution time, and expected total runtime) as a function of target accuracy, spectral gap, and initial ground state overlap. This algorithm suggests a concrete approach to using early fault tolerant quantum computers for carrying out industry-relevant molecular and materials calculations.

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“…Run a quantum linear system algorithm (e.g. [46], [47], or [48]) with constant precision to prepare an initial state |φ 0 such that…”

Section: Quantum Linear System Solvermentioning

confidence: 99%

Computing Ground State Properties with Early Fault-Tolerant Quantum Computers

Zhang

Wang

Johnson

2022

Quantum

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“…Another future direction of research would be to recast our algorithms in the framework of adiabatic quantum computing (AQC) following the works of [LT20b,AL22]. Quantum algorithms for linear systems in this framework have the advantage that a linear dependence on κ can be obtained without using complicated subroutines like variable-time amplitude amplification.…”

Section: Improved Quantum Weighted Least Squares Algorithmsmentioning

confidence: 99%

Quantum Regularized Least Squares

Chakraborty¹,

Morolia²,

Peduri³

2022

Preprint

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Linear regression is a widely used technique to fit linear models and finds widespread applications across different areas such as machine learning and statistics. In most real-world scenarios, however, linear regression problems are often ill-posed or the underlying model suffers from overfitting, leading to erroneous or trivial solutions. This is often dealt with by adding extra constraints, known as regularization. In this paper, we use the frameworks of block-encoding and quantum singular value transformation (QSVT) to design the first quantum algorithms for quantum least squares with general 2 -regularization. These include regularized versions of quantum ordinary least squares, quantum weighted least squares, and quantum generalized least squares. Our quantum algorithms substantially improve upon prior results on quantum ridge regression (polynomial improvement in the condition number and an exponential improvement in accuracy), which is a particular case of our result.To this end, we assume approximate block-encodings of the underlying matrices as input and use robust QSVT algorithms for various linear algebra operations. In particular, we develop a variabletime quantum algorithm for matrix inversion using QSVT, where we use quantum eigenvalue discrimination as a subroutine instead of gapped phase estimation. This ensures that substantially fewer ancilla qubits are required for this procedure than prior results. Owing to the generality of the block-encoding framework, our algorithms are applicable to a variety of input models and can also be seen as improved and generalized versions of prior results on standard (non-regularized) quantum least squares algorithms.

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“…The prevailing strategy for designing quantum differential equation solvers to date is to construct a large linear system recording states during the entire history of the evolution, and then to apply the quantum linear systems algorithms (QLSA) [32][33][34][35][36][37][38] to solve the resulting linear systems of equations. One may perform certain amplification procedures to boost the success probability of getting the final solution.…”

Section: Related Workmentioning

confidence: 99%

“…Moreover, the near optimal quantum algorithms also need to query the initial state for O(κ) times. Such a dependence is most clearly seen from the perspective of adiabatic based near-optimal quantum linear system solvers [36,37,53]. This is because the construction of the adiabatic Hamiltonian corresponding to the linear system uses the initial state, and thus each query to the adiabatic Hamiltonian also queries the initial state.…”

Section: Contributionmentioning

confidence: 99%

Time-marching based quantum solvers for time-dependent linear differential equations

Fang¹,

Lin²,

Tao³

2022

Preprint

Self Cite

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The time-marching strategy, which propagates the solution from one time step to the next, is a natural strategy for solving time-dependent differential equations on classical computers, as well as for solving the Hamiltonian simulation problem on quantum computers. For more general linear differential equationsa time-marching based quantum solver can suffer from exponentially vanishing success probability with respect to the number of time steps and is thus considered impractical. We solve this problem by repeatedly invoking a technique called the uniform singular value amplification, and the overall success probability can be lower bounded by a quantity that is independent of the number of time steps. The success probability can be further improved using a compression gadget lemma. This provides a path of designing quantum differential equation solvers that is alternative to those based on quantum linear systems algorithms (QLSA). We demonstrate the performance of the time-marching strategy with a high-order integrator based on the truncated Dyson series. The complexity of the algorithm depends linearly on the amplification ratio, which quantifies the deviation from a unitary dynamics. We prove that the linear dependence on the amplification ratio attains the query complexity lower bound and thus cannot be improved in general. This algorithm also surpasses existing QLSA based solvers in three aspects: (1) A(t) does not need to be diagonalizable. ( 2) A(t) can be non-smooth, and is only of bounded variation. (3) It can use fewer queries to the initial state |ψ 0 . Finally, we demonstrate the timemarching strategy with a first-order truncated Magnus series, which simplifies the implementation compared to high-order truncated Dyson series approach, while retaining the aforementioned benefits. Our analysis also raises some open questions concerning the differences between time-marching and QLSA based methods for solving differential equations.

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Quantum Linear System Solver Based on Time-optimal Adiabatic Quantum Computing and Quantum Approximate Optimization Algorithm

Cited by 52 publications

References 33 publications

Computing Ground State Properties with Early Fault-Tolerant Quantum Computers

Computing Ground State Properties with Early Fault-Tolerant Quantum Computers

Quantum Regularized Least Squares

Time-marching based quantum solvers for time-dependent linear differential equations

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