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arXiv:2412.06202

Quantum Discrete Adiabatic Linear Solver based on Block Encoding and Eigenvalue Separator

Guojian Wu, Gao, Fang, Qing Gao, Yu Pan

Published Dec 9, 2024• 0 citations
arXivHTMLPDF

Abstract

Linear system solvers are widely used in scientific computing, with the primary goal of solving linear system problems. Classical iterative algorithms typically rely on the conjugate gradient method. The rise of quantum computing has spurred interest in quantum linear system problems (QLSP), particularly following the introduction of the HHL algorithm by Harrow et al. in 2009, which demonstrated the potential for exponential speedup compared to classical algorithms. However, the performance of the HHL algorithm is constrained by its dependence on the square of the condition number. To address this limitation, alternative approaches based on adiabatic quantum computing (AQC) have been proposed, which exhibit complexity scaling linearly with the condition number. AQC solves QLSP by smoothly varying the parameters of the Hamiltonian. However, this method suffers from high Hamiltonian simulation complexity. In response, this work designs new Hamiltonians and proposes a quantum discrete adiabatic linear solver based on block encoding and eigenvalue separation techniques (BEES-QDALS). This approach bypasses Hamiltonian simulation through a first-order approximation and leverages block encoding to achieve equivalent non-unitary operations on qubits. By comparing the fidelity of the original algorithm and BEES-QDALS when solving QLSP with a fixed number of steps, and the number of steps required to reach a target fidelity, it is found that BEES-QDALS significantly outperforms the original algorithm. Specifically, BEES-QDALS achieves higher fidelity with the same number of steps and requires fewer steps to reach the same target fidelity.

DOI
10.48550/arxiv.2412.06202

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Linear system solvers are widely used in scientific computing, with the primary goal of solving linear system problems. Classical iterative algorithms typically rely on the conjugate gradient method. The rise of quantum computing has spurred interest in quantum linear system problems (QLSP), particularly following the introduction of the HHL algorithm by Harrow et al. in 2009, which demonstrated the potential for exponential speedup compared to classical algorithms. However, the performance of the HHL algorithm is constrained by its dependence on the square of the condition number. To address this limitation, alternative approaches based on adiabatic quantum computing (AQC) have been proposed, which exhibit complexity scaling linearly with the condition number. AQC solves QLSP by smoothly varying the parameters of the Hamiltonian. However, this method suffers from high Hamiltonian simulation complexity. In response, this work designs new Hamiltonians and proposes a quantum discrete adiabatic linear solver based on block encoding and eigenvalue separation techniques (BEES-QDALS). This approach bypasses Hamiltonian simulation through a first-order approximation and leverages block encoding to achieve equivalent non-unitary operations on qubits. By comparing the fidelity of the original algorithm and BEES-QDALS when solving QLSP with a fixed number of steps, and the number of steps required to reach a target fidelity, it is found that BEES-QDALS significantly outperforms the original algorithm. Specifically, BEES-QDALS achieves higher fidelity with the same number of steps and requires fewer steps to reach the same target fidelity.
Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. Subaşı . [Phys. Rev. Lett. 122, 060504 (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number κ of the linear system and the target error ϵ . Here we go beyond these results in several ways. Firstly, using filtering [Lin and Tong, Quantum 4, 361 (2020)] and Poissonization techniques [Cunningham and Roland, ArXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling O ( κ log  ( 1 / ϵ ) ) —an exponential improvement in ϵ , and a shaving of a log  κ scaling factor in κ . Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation—which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is 837 κ at ϵ = 10 − 10 for Hermitian matrices.
We consider the quantum implementations of two classical iterative solvers for a system of linear equations, including the row (Kaczmarz) method which utilizes a row of the coefficient matrix in each iteration step, and the column (coordinate descent) method which uses a column instead. These two methods are widely applied in big data science due to their simple iteration schemes. We propose fast quantum algorithms for these two approaches by constructing efficient unitary operators in each iteration step based on the block-encoding technique. The construction is based on the unitaries to prepare the quantum states of rows or columns of the coefficient matrix. If the quantum states are efficiently prepared, for example, by qRAM, then the quantum iterative methods achieve an exponential speedup in the problem size nnn over the classical versions. Meanwhile the dependence on the number of steps is linear, which is the same as the classical counterparts. The complexity of the quantum iterative linear solvers is $O({\ensuremath{\kappa}}_{s}^{2}(logn)\left(log\frac{1}{\ensuremath{\epsilon}}\right))$, where ${\ensuremath{\kappa}}_{s}$ is the scaled condition number, and $\ensuremath{\epsilon}$ is the error. The quantum iterative linear solvers do not depend on Hamiltonian simulations and quantum phase estimation, and also work for the dense case.
We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with O (κ poly(log (κ ε))) runtime, where κ is the condition number, and ε is the target accuracy. This is near optimal with respect to both κ and ε, and is achieved without relying on complicated amplitude amplification procedures that are difficult to implement. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.
In adiabatic quantum computing the aim is to track an eigenstate as the Hamiltonian changes. In the usual setup this is achieved using the natural time-dependent Hamiltonian evolution of the system and the main technical tool is the adiabatic theorem. We propose several alternative processes that achieve the same goal, but can easily be implemented on a gate-based quantum computer without the overhead of simulating time-dependent Hamiltonian evolution. We give a general framework for deriving `adiabatic' theorems for these processes. As an application, we give various algorithms for solving the Quantum Linear Systems Problem (QLSP) with optimal scaling in the condition number. One of these algorithms was previously developed in [Cunningham, Roland 2024] and another can be seen as a randomised version of the discrete adiabatic algorithm of [Costa et al. 2022]. We also describe versions of Trotterisation in our framework, which allows several results from [An et al. 2025] to be reproduced in a randomised setting. In particular, bounds on the Trotter error in terms of the fidelity are obtained that are asymptotically better than the standard bounds.
Existing quantum discrete adiabatic approaches are hindered by circuit depth that increases linearly with the number of evolution steps, a significant challenge for current quantum hardware with limited coherence times. To address this, we propose a co-designed framework that synergistically integrates dynamic circuit capabilities with real-time classical processing. This framework reformulates the quantum adiabatic evolution into discrete, dynamically adjustable segments. The unitary operator for each segment is optimized on-the-fly using classical computation, and circuit multiplexing techniques are leveraged to reduce the overall circuit depth scaling from O(steps×depth(U))O(\text{steps}\times\text{depth}(U))O(steps×depth(U)) to O(depth(U))O(\text{depth}(U))O(depth(U)). We implement and benchmark a quantum discrete adiabatic linear solver based on this framework for linear systems of W∈{2,4,8,16}W \in \{2,4,8,16\}W∈{2,4,8,16} dimensions with condition numbers κ∈{10,20,30,40,50}κ\in \{10,20,30,40,50\}κ∈{10,20,30,40,50}. Our solver successfully overcomes previous depth limitations, maintaining over 80% solution fidelity even under realistic noise models. Key algorithmic optimizations contributing to this performance include a first-order approximation of the discrete evolution operator, a tailored dynamic circuit design exploiting real-imaginary component separation, and noise-resilient post-processing techniques.
Adiabatic Quantum Computing (AQC) is a powerful framework for designing quantum algorithms, yet its practical implementation is often hampered by the difficulty of engineering precise, continuous time-dependent Hamiltonians or the substantial computational overhead of time discretisation. This thesis develops alternative adiabatic frameworks based on dissipative and stochastic quantum evolution to circumvent these challenges.The core methodology introduces novel dynamics where discrete quantum operations, such as time-independent Hamiltonian evolution with Poisson-distributed phase randomisation or randomised unitary applications (qubitization), replace continuous evolution. This stochastic approach allows the system’s average behaviour to be rigorously analysed using new, generalised adiabatic theorems, including the development of novel quantitative bounds for unbounded Hamiltonians and specialised dissipative generators.Algorithmically, this work demonstrates that these discrete, stochastic procedures can achieve the optimal asymptotic time complexity of continuous AQC while mitigating discretisation costs. Specifically, adapted-schedule implementations are shown to yield optimal scaling for the Quantum Linear Systems Problem (QLSP), achieving complexity proportional to the condition number O(κ)O(\kappa)O(κ), and O(N/M)O(\sqrt{N/M})O(N/M​) scaling for unstructured search. By establishing highly efficient, implementable alternatives to continuous AQC, this research offers a significant pathway toward robust and scalable quantum computation. The study also investigates the general family of diagonal Hamiltonians for optimisation problems, revealing that finding the optimal scheduling required for peak performance is, in general, an NP-hard problem.