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arXiv:2401.09091 · Quantum

Enhancing Scalability of Quantum Eigenvalue Transformation of Unitary Matrices for Ground State Preparation through Adaptive Finer Filtering

Erenay Karacan, Yanbin Chen, Christian B. Mendl

Published Feb 6, 2025• 1 citation
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Abstract

Hamiltonian simulation is a domain where quantum computers have the potential to outperform their classical counterparts. One of the main challenges of such quantum algorithms is increasing the system size, which is necessary to achieve meaningful quantum advantage. In this work, we present an approach to improve the scalability of eigenspace filtering for the ground state preparation of a given Hamiltonian. Our method aims to tackle limitations introduced by a small spectral gap and high degeneracy of low energy states. It is based on an adaptive sequence of eigenspace filtering through Quantum Eigenvalue Transformation of Unitary Matrices (QETU) combined with spectrum profiling. By combining our proposed algorithm with state-of-the-art phase estimation methods, we achieved good approximations for the ground state energy with local, two-qubit gate depolarizing probability up to 10−4 . To demonstrate the key results in this work, we ran simulations with the transverse-field Ising Model on classical computers using Qiskit . We compare the performance of our approach with the static implementation of QETU and show that we can consistently achieve three to four orders of magnitude improvement in the absolute error rate.

DOI
10.22331/q-2025-02-06-1624

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Hamiltonian simulation is a domain where quantum computers have the potential to outperform their classical counterparts. One of the main challenges of such quantum algorithms is increasing the system size, which is necessary to achieve meaningful quantum advantage. In this work, we present an approach to improve the scalability of eigenspace filtering for the ground state preparation of a given Hamiltonian. Our method aims to tackle limitations introduced by a small spectral gap and high degeneracy of low energy states. It is based on an adaptive sequence of eigenspace filtering through Quantum Eigenvalue Transformation of Unitary Matrices (QETU) combined with spectrum profiling. By combining our proposed algorithm with state-of-the-art phase estimation methods, we achieved good approximations for the ground state energy with local, two-qubit gate depolarizing probability up to 10−4 . To demonstrate the key results in this work, we ran simulations with the transverse-field Ising Model on classical computers using Qiskit . We compare the performance of our approach with the static implementation of QETU and show that we can consistently achieve three to four orders of magnitude improvement in the absolute error rate.
We present several improvements to the recently developed ground-state preparation algorithm based on the quantum eigenvalue transformation for unitary matrices (QETU), apply this algorithm to a lattice formulation of U(1) gauge theory in (2+1)(2+1)(2+1) dimensions, as well as propose an alternative application of QETU, a highly efficient preparation of Gaussian distributions. The QETU technique was originally proposed as an algorithm for nearly optimal ground-state preparation and ground-state energy estimation on early fault-tolerant devices. It uses the time-evolution input model, which can potentially overcome the large overall prefactor in the asymptotic gate cost arising in similar algorithms based on the Hamiltonian input model. We present modifications to the original QETU algorithm that significantly reduce the cost for the cases of both exact and Trotterized implementation of the time evolution circuit. We use QETU to prepare the ground state of a U(1) lattice gauge theory in two spatial dimensions, explore the dependence of computational resources on the desired precision and system parameters, and discuss the applicability of our results to general lattice gauge theories. We also demonstrate how the QETU technique can be utilized for preparing Gaussian distributions and wave packets in a way which outperforms existing algorithms for as little as ${n}_{q}\ensuremath{\gtrsim}2--5$ qubits.
Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] can estimate the ground state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block encoding input model of the Hamiltonian, whose implementation is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QET-U), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit and no multi-qubit control operations, and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anti-commutation relations and further removes the need of implementing the controlled Hamiltonian evolution. Coupled with Trotter based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM Qiskit for the transverse field Ising model. If we are further allowed to use multi-qubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground state preparation and energy estimating using a constant number of ancilla qubits (no more than 3).
Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum computing. In this work, we propose a quantum algorithm that given a non-normal matrix, outputs an estimate of an eigenvalue to within additive error εεε with probability at least 1−pfail1-p_{\rm fail}1−pfail​. Our estimation strategy is to sample points on the complex plane and examine the distance between the sampled point and the eigenvalues. We show that the distance is related to the smallest singular value of the shifted matrix, hence reducing the problem to ground state energy estimation via Hermitianization. With the knowledge of an eigenvalue, we are able to prepare the associated eigenvector using ground state preparation. Our estimating scheme can also be modified to approximate the extreme eigenvalue, and in particular the spectral gap. The algorithm is implemented based on the block encoding input model and requires O(κ2ε−(2m−1)log⁡(1/pfail))O(κ^2ε^{-(2m-1)}\log(1/p_{\rm fail}))O(κ2ε−(2m−1)log(1/pfail​)) queries to the block encoding oracle. Our algorithm is the first general eigenvalue algorithm that achieves this scaling. We also perform numerical simulation to validate the algorithms.
We introduce a classical estimator for the post-processing of quantum phase estimation (QPE) data when a single target phase is isolated within a known interval, as is typical of ground state energy estimation of gapped systems. Our estimator filters the QPE signal within this promise region and recovers the phase through a moment-projection routine, which is robust to both external spurious phases and experimental noise. In the noiseless case this achieves an exponential suppression of bias with respect to a naive mean estimator. In the presence of global depolarizing noise the bias is exponentially small in the circuit depth ttt, and the variance is O(t−2F−2)O(t^{-2}F^{-2})O(t−2F−2) for circuit fidelity FFF. This improves by a factor of t2t^2t2 over a naive shifted-and-rescaled-mean approach. To mitigate realistic circuit-level noise, we combine our method with the explicit unbiasing scheme described in [Dutkiewicz et al., 2025]. This yields an overhead interpolating between the F−4F^{-4}F−4 scaling typical of explicitly unbiased error mitigation and a reduced F−2F^{-2}F−2 scaling when the noise samples fall outside the promise interval. We validate our estimators on a small-scale simulation of the Ising model, observing better-than-expected performance for a global depolarizing noise approximation. This robustness to both multiple eigenvalues and realistic noise makes limited-depth phase estimation practical for early fault tolerant quantum experiments.
Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes O~(d 3/2/ε)\tilde{\mathcal{O}}(d^{\,3/2}/ε)O~(d3/2/ε) queries to the time-evolution operator of the system and O~(d 3/2)\tilde{\mathcal{O}}(d^{\,3/2})O~(d3/2) queries to a block-encoding of the perturbation, for ddd cooling steps and an εεε-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.
Quantum computing holds immense promise for simulating quantum systems, a critical task for advancing our understanding of complex quantum phenomena. One of the primary goals in this domain is to accurately approximate the ground state of quantum systems. The Fermi-Hubbard model, particularly, is of profound interest due to its implications for high-temperature superconductivity and strongly correlated electron systems. The quantum eigenvalue transformation of unitary matrices (QETU) algorithm offers a novel approach for ground state estimation by utilizing a controlled Hamiltonian time evolution operator, circumventing the resource-intensive block-encoding required by previous methods. In this work, we apply the QETU algorithm to the 2×22 \times 22×2 Fermi-Hubbard model, presenting circuit simplifications tailored to the model and introducing a mapping to a 9-qubit grid-like hardware architecture inspired by fermionic swap networks. We investigate how the selection of a favorable hardware architecture can benefit the circuit construction. Additionally, we explore the feasibility of this method under the influence of noise, focusing on its robustness and practical applicability.