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arXiv:2301.13353 · Quantum

Measurement-efficient quantum Krylov subspace diagonalisation

Zongkang Zhang, Anbang Wang, Xiaosi Xu, Ying Li

Published Aug 13, 2024• 15 citations
arXivHTMLPDF

Abstract

The Krylov subspace methods, being one category of the most important classical numerical methods for linear algebra problems, can be much more powerful when generalised to quantum computing. However, quantum Krylov subspace algorithms are prone to errors due to inevitable statistical fluctuations in quantum measurements. To address this problem, we develop a general theoretical framework to analyse the statistical error and measurement cost. Based on the framework, we propose a quantum algorithm to construct the Hamiltonian-power Krylov subspace that can minimise the measurement cost. In our algorithm, the product of power and Gaussian functions of the Hamiltonian is expressed as an integral of the real-time evolution, such that it can be evaluated on a quantum computer. We compare our algorithm with other established quantum Krylov subspace algorithms in solving two prominent examples. To achieve an error comparable to that of the classical Lanczos algorithm at the same subspace dimension, our algorithm typically requires orders of magnitude fewer measurements than others. Such an improvement can be attributed to the reduced cost of composing projectors onto the ground state. These results show that our algorithm is exceptionally robust to statistical fluctuations and promising for practical applications.

DOI
10.22331/q-2024-08-13-1438

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The Krylov subspace methods, being one category of the most important classical numerical methods for linear algebra problems, can be much more powerful when generalised to quantum computing. However, quantum Krylov subspace algorithms are prone to errors due to inevitable statistical fluctuations in quantum measurements. To address this problem, we develop a general theoretical framework to analyse the statistical error and measurement cost. Based on the framework, we propose a quantum algorithm to construct the Hamiltonian-power Krylov subspace that can minimise the measurement cost. In our algorithm, the product of power and Gaussian functions of the Hamiltonian is expressed as an integral of the real-time evolution, such that it can be evaluated on a quantum computer. We compare our algorithm with other established quantum Krylov subspace algorithms in solving two prominent examples. To achieve an error comparable to that of the classical Lanczos algorithm at the same subspace dimension, our algorithm typically requires orders of magnitude fewer measurements than others. Such an improvement can be attributed to the reduced cost of composing projectors onto the ground state. These results show that our algorithm is exceptionally robust to statistical fluctuations and promising for practical applications.
Quantum Krylov algorithms are a promising approach for ground-state energy estimation on near-term quantum computers. However, their sampling cost is inherently high because each Hamiltonian term is measured individually. Despite various mitigation techniques, this overhead remains a bottleneck for large-scale electronic structure problems. We introduce mirror subspace diagonalization (MSD), which approaches the theoretical lower bound on the sampling cost for quantum Krylov algorithms built upon real-time evolution. MSD uses a finite-difference formula to express the Hamiltonian as a linear combination of time-evolution unitaries with symmetrically shifted timesteps, enabling efficient estimation of the Hamiltonian matrix within the Krylov subspace. By optimizing the timestep and shifting the energy spectrum, finite-difference and statistical errors are minimized simultaneously, allowing MSD to attain the sampling-cost lower bound up to a logarithmic factor. We further apply classical post-processing to infer Hamiltonian moments, which mitigate the ground-state energy error via the Lanczos scheme. Our analysis shows MSD is particularly effective when the Hamiltonian's spectral norm is much smaller than its 1-norm, as in high-accuracy molecular simulations with large basis sets and strong electronic correlations. Numerical results for various molecular models show sampling-cost reductions of roughly 10 to 10,000 times over the conventional quantum Krylov algorithm.
Quantum computers provide new avenues to access ground and excited state properties of systems otherwise difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues on quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, which applies to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.
We present an algorithm that uses block encoding on a quantum computer to exactly construct a Krylov space, which can be used as the basis for the Lanczos method to estimate extremal eigenvalues of Hamiltonians. While the classical Lanczos method has exponential cost in the system size to represent the Krylov states for quantum systems, our efficient quantum algorithm achieves this in polynomial time and memory. The construction presented is exact in the sense that the resulting Krylov space is identical to that of the Lanczos method, so the only approximation with respect to the exact method is due to finite sample noise. This is possible because, unlike previous quantum Krylov methods, our algorithm does not require simulating real or imaginary time evolution. We provide an explicit error bound for the resulting ground state energy estimate in the presence of noise. For our method to be successful efficiently, the only requirement on the input problem is that the overlap of the initial state with the true ground state must be Ω ( 1 / poly ( n ) ) for n qubits.
Quantum subspace-diagonalization methods, particularly Quantum Krylov Diagonalization (QKD), provide a promising route for computing low-energy spectra of quantum many-body Hamiltonians. However, existing quantum Krylov approaches rely on non-orthogonal Krylov bases, requiring overlap-matrix regularization that limits numerical stability and accuracy. In this work, we introduce an Orthogonal Quantum Krylov Diagonalization (OQKD) framework that reformulates the classical Lanczos recursion at the operator level, enabling an orthogonal quantum implementation of Krylov-subspace diagonalization. By expressing Lanczos vectors as polynomial transformations of the Hamiltonian, OQKD reproduces the orthogonality, tridiagonal structure, and convergence behavior of the classical Lanczos algorithm thus eliminating the need for overlap-matrix regularization. We further show that the required Lanczos polynomials can be implemented using block encoding and Generalized Quantum Signal Processing with the same asymptotic query complexity as Chebyshev-based QKD methods. Numerical simulations of the J1J_1J1​--J2J_2J2​ Heisenberg model confirm the classical Lanczos convergence and numerical stability of the proposed method, while the measurement-complexity scaling is established analytically. Building upon the OQKD framework, we then introduce a restarted state-preparation protocol that replaces a single high-degree polynomial transformation with a sequence of fixed low-degree transformations, maintaining an affordable block encoding success probability while retaining comparable convergence. These results establish OQKD as an orthogonal quantum analog of the classical Lanczos algorithm and identify the restarted protocol as a promising state-preparation strategy for Quantum Phase Estimation.
We present an iterative generalisation of the quantum subspace expansion algorithm used with a Krylov basis. The iterative construction connects a sequence of subspaces via their lowest energy states. Diagonalising a Hamiltonian in a given Krylov subspace requires the same quantum resources in both the single step and sequential cases. We propose a variance-based criterion for determining a good iterative sequence and provide numerical evidence that these good sequences display improved numerical stability over a single step in the presence of finite sampling noise. Implementing the generalisation requires additional classical processing with a polynomial overhead in the subspace dimension. By exchanging quantum circuit depth for additional measurements the quantum subspace expansion algorithm appears to be an approach suited to near term or early error-corrected quantum hardware. Our work suggests that the numerical instability limiting the accuracy of this approach can be substantially alleviated in a parameter-free way.
Krylov subspace methods are among the most extensively studied early fault-tolerant quantum algorithms for estimating ground-state energies of quantum systems. However, the rapid onset of ill-conditioning might make accurate energies difficult or even impossible to retrieve. In this communication, we analyse the numerical stability and statistical problems of these methods using numerical simulations both in the presence and absence of sampling noise. While in ideal numerical simulations the generalized eigenvalue problem indeed becomes unstable with increased Krylov subspace size, we find that, in realistic noisy settings, these methods do not primarily suffer from ill-conditioning. Instead, statistical fluctuations dominate and can prevent reliable solution extraction unless appropriate regularization or filtering techniques are employed. We consequently introduce two new metrics, the imaginary and unitary filters, that successfully assess the reliability of the obtained solutions without any knowledge of the true eigenspectrum.