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Quantum linear systems algorithm with exponentially improved dependence on precision

Andrew M. Childs, Robin Kothari, Rolando D. Somma

Published Nov 7, 2015• 30 citations
OpenAlex

Abstract

Harrow, Hassidim, and Lloyd showed that for a suitably specified $N \times N$ matrix $A$ and $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations $A\vec{x}=\vec{b}$. If $A$ is sparse and well-conditioned, their algorithm runs in time $\mathrm{poly}(\log N, 1/\epsilon)$, where $\epsilon$ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in $\log(1/\epsilon)$, exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on $\epsilon$ is prohibitive.

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Harrow, Hassidim, and Lloyd [ Phys. Rev. Lett., 103 (2009), 150502] showed that for a suitably specified $N \times N$ matrix $A$ and an $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations $A\vec{x} = \vec{b}$. If $A$ is sparse and well-conditioned, their algorithm runs in time ${poly}(\log N, 1/\epsilon)$, where $\epsilon$ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in $\log(1/\epsilon)$, exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on $\epsilon$ is prohibitive.

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Harrow, Hassidim, and Lloyd showed that for a suitably specified N×NN \times NN×N
matrix AAA and NNN-dimensional vector b⃗\vec{b}b, there is a quantum algorithm
that outputs a quantum state proportional to the solution of the linear system
of equations Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b. If AAA is sparse and well-conditioned, their
algorithm runs in time poly(log⁡N,1/ϵ)\mathrm{poly}(\log N, 1/\epsilon)poly(logN,1/ϵ), where ϵ\epsilonϵ is
the desired precision in the output state. We improve this to an algorithm
whose running time is polynomial in log⁡(1/ϵ)\log(1/\epsilon)log(1/ϵ), exponentially improving
the dependence on precision while keeping essentially the same dependence on
other parameters. Our algorithm is based on a general technique for
implementing any operator with a suitable Fourier or Chebyshev series
representation. This allows us to bypass the quantum phase estimation
algorithm, whose dependence on ϵ\epsilonϵ is prohibitive.
Harrow, Hassidim, and Lloyd [ Phys. Rev. Lett., 103 (2009), 150502] showed that for a suitably specified N×NN \times NN×N matrix AAA and an NNN-dimensional vector b⃗\vec{b}b, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations Ax⃗=b⃗A\vec{x} = \vec{b}Ax=b. If AAA is sparse and well-conditioned, their algorithm runs in time poly(log⁡N,1/ϵ){poly}(\log N, 1/\epsilon)poly(logN,1/ϵ), where ϵ\epsilonϵ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in log⁡(1/ϵ)\log(1/\epsilon)log(1/ϵ), exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on ϵ\epsilonϵ is prohibitive.
The Harrow-Hassidim-Lloyd (HHL) quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart. The problem of solving a system of linear equations has a wide scope of applications, and thus HHL constitutes an important algorithmic primitive. In these notes, we present the HHL algorithm and its improved versions in detail, including explanations of the constituent sub- routines. More specifically, we discuss various quantum subroutines such as quantum phase estimation and amplitude amplification, as well as the important question of loading data into a quantum computer, via quantum RAM. The improvements to the original algorithm exploit variable-time amplitude amplification as well as a method for implementing linear combinations of unitary operations (LCUs) based on a decomposition of the operators using Fourier and Chebyshev series. Finally, we discuss a linear solver based on the quantum singular value estimation (QSVE) subroutine.
Solving linear systems of equations is a frequently encountered problem in machine learning and optimization. Given a matrix AAA and a vector b\mathbf{b}b the task is to find the vector x\mathbf{x}x such that Ax=bA\mathbf{x}=\mathbf{b}Ax=b. We describe a quantum algorithm that achieves a sparsity-independent runtime scaling of $\mathcal{O}\mathbf{(}{\ensuremath{\kappa}}^{2}\sqrt{n}\text{polylog}(n)/\ensuremath{\epsilon}\mathbf{)}$ for an $n\ifmmode\times\else\texttimes\fi{}n$ dimensional AAA with bounded spectral norm, where $\ensuremath{\kappa}$ denotes the condition number of AAA, and $\ensuremath{\epsilon}$ is the desired precision parameter. This amounts to a polynomial improvement over known quantum linear system algorithms when applied to dense matrices, and poses a new state of the art for solving dense linear systems on a quantum computer. Furthermore, an exponential improvement is achievable if the rank of AAA is polylogarithmic in the matrix dimension. Our algorithm is built upon a singular value estimation subroutine, which makes use of a memory architecture that allows for efficient preparation of quantum states that correspond to the rows of AAA and the vector of Euclidean norms of the rows of AAA.
In the quantum linear systems problem (QLSP), we are given query access to a ddd-sparse N×NN\times NN×N matrix AAA with condition number κκκ, and the ability to prepare a quantum state proportional to a vector b⃗\vec bb. The goal is to output an εεε-approximation to the quantum state proportional to the solution x⃗\vec xx of Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities O(κdlog⁡(1/ε))O(κd\log(1/ε))O(κdlog(1/ε)) and κd(κd/ε)o(1)κ\sqrt d(κd/ε)^{o(1)}κd​(κd/ε)o(1), while the best known lower bounds were Ω(κlog⁡(1/ε))Ω(κ\log(1/ε))Ω(κlog(1/ε)) and Ω(κd)Ω(κ\sqrt d)Ω(κd​). We improve these bounds and show that the complexity of the QLSP is Θ(κdlog⁡(1/ε))Θ(κ\sqrt d\log(1/ε))Θ(κd​log(1/ε)). We also resolve an open problem of Berry and Childs by showing that any N×NN\times NN×N unitary can be implemented with bounded error using O(N)O(\sqrt N)O(N​) queries to its matrix entries.
Given a matrix AAA of dimension M×NM \times NM×N and a vector b⃗\vec{b}b, the quantum linear system (QLS) problem asks for the preparation of a quantum state ∣y⃗⟩|\vec{y}\rangle∣y​⟩ proportional to the solution of Ay⃗=b⃗A\vec{y} = \vec{b}Ay​=b. Existing QLS algorithms have runtimes that scale linearly with the condition number κ(A)κ(A)κ(A), the sparsity of AAA, and logarithmically with inverse precision, but often overlook structural properties of b⃗\vec{b}b, whose alignment with AAA's eigenspaces can greatly affect performance. In this work, we present a new QLS algorithm that explicitly leverages the structure of the right-hand side vector b⃗\vec{b}b. The runtime of our algorithm depends polynomially on the sparsity of the augmented matrix H=[A,−b⃗]H = [A, -\vec{b}]H=[A,−b], the inverse precision, the ℓ2\ell_2ℓ2​ norm of the solution y⃗=A+b⃗\vec{y} = A^+ \vec{b}y​=A+b, and a new instance-dependent parameter ET=∑i=1Mpi2⋅di, ET= \sum_{i=1}^M p_i^2 \cdot d_i, ET=∑i=1M​pi2​⋅di​, where p⃗=(AA⊤)+b⃗\vec{p} = (AA^{\top})^+ \vec{b}p​=(AA⊤)+b, and did_idi​ denotes the squared ℓ2\ell_2ℓ2​ norm of the iii-th row of HHH. We also introduce a structure-aware rescaling technique tailored to the solution y⃗=A+b⃗\vec{y} = A^+ \vec{b}y​=A+b. Unlike left preconditioning methods, which transform the linear system to DAy⃗=Db⃗DA\vec{y} = D\vec{b}DAy​=Db, our approach applies a right rescaling matrix, reformulating the linear system as ADz⃗=b⃗AD\vec{z} = \vec{b}ADz=b. As an application of our instance-aware QLS algorithm and new rescaling scheme, we develop a quantum algorithm for solving multivariate polynomial systems in regimes where prior QLS-based methods fail. This yields an end-to-end framework applicable to a broad class of problems. In particular, we apply it to the maximum independent set (MIS) problem, formulated as a special case of a polynomial system, and show through detailed analysis that, under certain conditions, our quantum algorithm for MIS runs in polynomial time.
Systems of linear equations are ubiquitous across science, engineering, machine learning, and even finance. While classical methods can be prohibitively slow for large-scale problems, quantum linear systems algorithms offer the potential for exponential speedup in certain parameter regimes. However, a significant gap persists between this theoretical promise and practical implementation, as the advantages are often obscured by the substantial quantum resources and high sensitivity to noise inherent in current quantum hardware. One way to bridge this gap is through the use of Iterative Refinement, a classical post-processing scheme that can exponentially improve the accuracy to which a linear system of equations can be solved using low-precision arithmetic. In the context of quantum linear systems algorithms, such as the HHL algorithm proposed by Harrow, Hassidim, and Loyd, Iterative Refinement can greatly reduce the quantum resources required to calculate an accurate solution in terms of tomography cost, circuit volume, and fault-tolerant overhead. Here, we compute and benchmark highly precise solutions to linear systems of equations of up to eight variables by running HHL with Iterative Refinement on NISQ quantum computers. We also present our open-source implementation, emphasizing that our circuit is not tailored to specific problem instances, as most available implementations are.