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arXiv:2309.00759

Nuclear Spectra from Quantum Lanczos Algorithm with Real-Time Evolution and Multiple Reference States

Bowman, Amanda

Published Sep 1, 2023• 0 citations
arXivHTMLPDF

Abstract

Models of quantum systems scale exponentially with the addition of single-particle states, which can present computationally intractable problems. Alternatively, quantum computers can store a many-body basis of $2^n$ dimensions on $n$ qubits. This motivated the quantum eigensolver algorithms developed in recent years, such as the quantum Lanczos algorithm based on the classical, iterative Lanczos algorithm. I performed numerical simulations to find the low-lying eigenstates of $^{20}$Ne, $^{22}$Na, and $^{29}$Na to compare imaginary- and real-time evolution. Though imaginary-time evolution leads to faster convergence, real-time evolution still converges within tens of iterations and satisfies the requirement for unitary operators on quantum computers. Additionally, using multiple reference states leads to faster convergences or higher accuracy for a fixed number of real-time iterations. I performed quantum circuit prototype numerical simulations on a classical computer of the QLanczos algorithm with real-time evolution and multiple reference states to find the low-lying eigenstates of $^{8}$Be. These simulations were run in both the spherical basis and Hartree-Fock basis, demonstrating that an M-scheme spherical basis leads to lower depth circuits than the Hartree-Fock basis. Finally, I present the quantum circuits for the QLanczos algorithm with real-time evolution and multiple references.

DOI
10.48550/arxiv.2309.00759

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Models of quantum systems scale exponentially with the addition of single-particle states, which can present computationally intractable problems. Alternatively, quantum computers can store a many-body basis of 2n2^n2n dimensions on nnn qubits. This motivated the quantum eigensolver algorithms developed in recent years, such as the quantum Lanczos algorithm based on the classical, iterative Lanczos algorithm. I performed numerical simulations to find the low-lying eigenstates of 20^{20}20Ne, 22^{22}22Na, and 29^{29}29Na to compare imaginary- and real-time evolution. Though imaginary-time evolution leads to faster convergence, real-time evolution still converges within tens of iterations and satisfies the requirement for unitary operators on quantum computers. Additionally, using multiple reference states leads to faster convergences or higher accuracy for a fixed number of real-time iterations. I performed quantum circuit prototype numerical simulations on a classical computer of the QLanczos algorithm with real-time evolution and multiple reference states to find the low-lying eigenstates of 8^{8}8Be. These simulations were run in both the spherical basis and Hartree-Fock basis, demonstrating that an M-scheme spherical basis leads to lower depth circuits than the Hartree-Fock basis. Finally, I present the quantum circuits for the QLanczos algorithm with real-time evolution and multiple references.
Abstract Various methods have been developed for the quantum computation of the ground and excited states of physical and chemical systems, but many of them require either large numbers of ancilla qubits or high-dimensional optimization in the presence of noise. The quantum imaginary-time evolution (QITE) and quantum Lanczos (QLanczos) methods proposed in Motta et al. (2020) eschew the aforementioned issues. In this study, we demonstrate the practical application of these algorithms to challenging quantum computations of relevance for chemistry and nuclear physics, using the deuteron-binding energy and molecular hydrogen binding and excited state energies as examples. With the correct choice of initial and final states, we show that the number of timesteps in QITE and QLanczos can be reduced significantly, which commensurately simplifies the required quantum circuit and improves compatibility with NISQ devices. We have performed these calculations on cloud-accessible IBM Q quantum computers. With the application of readout-error mitigation and Richardson error extrapolation, we have obtained ground and excited state energies that agree well with exact results obtained from diagonalization.
Computing eigenvalues is a computationally intensive task central to many applications in the natural sciences, particularly in quantum physics and chemistry, where determining low-lying energy spectra is key to characterizing physical and chemical properties. Toward this end, we present and investigate the quantum block Krylov subspace projector (QBKSP) algorithm---a multireference quantum Lanczos method---to accurately compute low-lying eigenenergies, including degenerate ones. We propose three compact quantum circuits, each suited to different problem settings, for evaluating the necessary expectation values. In contrast to previous multireference quantum Krylov methods, QBKSP achieves linear scaling with respect to the number of Krylov iterations instead of a quadratic one, and furthermore reduces the scaling with respect to the number of references by half for real Hamiltonians and reference states. To assess the impact of the number and fidelity of the initial reference states, as well as the QBKSP performance in different parameter regimes, we perform both error-free and sampling noise-limited simulations. Our results demonstrate that employing multiple reference states significantly improves convergence, particularly in noisy scenarios or when a single reference fails to capture all target eigenstates. Moreover, the QBKSP algorithm allows for the determination of degenerate eigenstates and their multiplicities through appropriate convergence conditions.
There is increasing interest in quantum algorithms (QAs) that are based on the imaginary time evolution (ITE), a successful classical numerical approach to obtain ground states. However, most of the proposals so far require heavy postprocessing computational steps on a classical computer, such as solving linear equations. Here we provide an alternative approach to implement ITE. A key feature in our approach is the use of an orthogonal basis set: the propagated state is efficiently expressed in terms of orthogonal basis states at every step of the evolution. We argue that the number of basis states needed at those steps to achieve an accurate solution can be kept on the order of nnn, the number of qubits, by controlling the precision (number of significant digits) and the imaginary time increment. The number of quantum gates per imaginary time step is estimated to be polynomial in nnn. Additionally, while in many QAs the locality of the Hamiltonian is a key assumption, in our algorithm this restriction is not required. This characteristic of our algorithm renders it useful for studying highly nonlocal systems, such as the occupation-representation nuclear shell model. We illustrate our algorithm through numerical implementation on an IBM quantum simulator.
The nuclear shell model is a general way of expressing the many-body nuclear Hamiltonian and deciphering the underlying nuclear structure. In today's era of modern and high-power computation, the primary limitation of the nuclear shell model is the enormous dimensionality of its Hilbert space, which far exceeds available storage capacity and prevents the diagonalization of the full Hamiltonian matrix in that space. Quantum computing offers a scalable solution to bypass this curse of dimensionality. In this work, we introduce a single-run quantum simulation capable of obtaining multiple shell-model eigenstates simultaneously. The nuclear Hamiltonian is transformed from a bit to a qubit basis using the Jordan-Wigner transformation, explicitly preserving fermionic anti-commutation. We employ a Subspace Search Variational Quantum Eigensolver (SSVQE) along with an Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT) ansatz to construct the quantum circuit required to solve the shell-model problem. The ADAPT-SSVQE algorithm uses a symmetry-preserving single and double-excitation operator pool and optimizes a weighted energy sum to obtain the simultaneous convergence of all eigenstates within a targeted MJ subspace, eliminating the need for post-processing efforts to extract excited spectra. We benchmark this approach by solving the problem for two and three identical nucleons in a j = 9/2 orbital, successfully extracting five and ten mutually orthogonal states, respectively, within a 10-qubit active space. The algorithm achieves spectroscopic accuracy, in simulation, relative to exact diagonalization and intrinsically restores total angular momentum (\hat{J}^2) symmetry.
Developing efficient quantum computing algorithms is essential for tackling computationally challenging problems across various fields. This paper presents a novel quantum algorithm, XZ24, for efficiently computing the eigen-energy spectra of arbitrary quantum systems. Given a Hamiltonian H^\hat{H}H^ and an initial reference state ∣ψref⟩|ψ_{\text{ref}} \rangle∣ψref​⟩, the algorithm extracts information about ⟨ψref∣cos⁡(H^t)∣ψref⟩\langle ψ_{\text{ref}} | \cos(\hat{H} t) | ψ_{\text{ref}} \rangle⟨ψref​∣cos(H^t)∣ψref​⟩ from an auxiliary qubit's state. By applying a Fourier transform, the algorithm resolves the energies of eigenstates of the Hamiltonian with significant overlap with the reference wavefunction. We provide a theoretical analysis and numerical simulations, showing XZ24's superior efficiency and accuracy compared to existing algorithms. XZ24 has three key advantages: 1. It removes the need for eigenstate preparation, requiring only a reference state with non-negligible overlap, improving upon methods like the Variational Quantum Eigensolver. 2. It reduces measurement overhead, measuring only one auxiliary qubit. For a system of size LLL with precision εεε, the sampling complexity scales as O(L⋅ε−1)O(L \cdot ε^{-1})O(L⋅ε−1). When relative precision εεε is sufficient, the complexity scales as O(ε−1)O(ε^{-1})O(ε−1), making measurements independent of system size. 3. It enables simultaneous computation of multiple eigen-energies, depending on the reference state. We anticipate that XZ24 will advance quantum system simulations and enhance applications in quantum computing.
Efficient low-energy state preparation is a key objective in quantum computation and quantum simulation. Quantum imaginary-time evolution replaces real-time dynamics with imaginary-time dynamics, exponentially suppressing higher-energy eigenstates. We introduce deterministic unitary protocols that approximate imaginary-time evolution for ground state preparation. The protocols require multiple copies of the system, real-time evolution under the system Hamiltonian, and controlled-SWAP operations (or more general SWAP-generated unitaries). Our analysis focuses on two concrete circuit families: a tree architecture with provable polynomial-in-depth convergence but rapidly growing width, and a compact "hedge" architecture that achieves comparable accuracy with only polynomial width in a heuristic construction supported by numerics. Numerical evidence indicates that mid-circuit post-selection can accelerate convergence with practical success probabilities. Separately, we demonstrate that circuit volume can be traded for the shot complexity of post-circuit observable estimation in the ground state preparation setting. Finally, we outline concrete platform-specific implementations in which multi-copy registers and SWAP-mediated couplings are natural, illustrating how these hybrid analog-digital circuits can complement existing state-preparation methods in the near term.