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arXiv:2406.03222 · Physical Review Letters

Counting the Ground State Degeneracy by Evolution Methods

Guo Zhen, You Li

Published Sep 5, 2025• 1 citation
arXivHTMLPDF

Abstract

Counting ground state degeneracy of a k-local Hamiltonian is important in many fields of physics. Its complexity class is harder than that of finding the ground state of a k-local Hamiltonian. Very few methods can efficiently count the degeneracy of ground state while many are known for finding the ground state. This Letter proposes an algorithm for mapping the problem of counting ground state degeneracy into one for finding a special ground state of a k-local super Hamiltonian in an enlarged system with twice as many qubits or spins. We show that all traditional methods, designed for finding the ground state by evolution under a function of a Hamiltonian, can be employed to find the special ground state of the super Hamiltonian from a well-designed initial state and thus can be used to count the ground degeneracy. We combine our algorithm with the power method, the Lanczos method, and the quantum imaginary time evolution method for selective examples to illustrate the detection of phase boundaries, competition between frustration and quantum fluctuation, and potential implementation with quantum circuits.

DOI
10.1103/38gb-h7wv

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Counting ground state degeneracy of a k-local Hamiltonian is important in many fields of physics. Its complexity class is harder than that of finding the ground state of a k-local Hamiltonian. Very few methods can efficiently count the degeneracy of ground state while many are known for finding the ground state. This Letter proposes an algorithm for mapping the problem of counting ground state degeneracy into one for finding a special ground state of a k-local super Hamiltonian in an enlarged system with twice as many qubits or spins. We show that all traditional methods, designed for finding the ground state by evolution under a function of a Hamiltonian, can be employed to find the special ground state of the super Hamiltonian from a well-designed initial state and thus can be used to count the ground degeneracy. We combine our algorithm with the power method, the Lanczos method, and the quantum imaginary time evolution method for selective examples to illustrate the detection of phase boundaries, competition between frustration and quantum fluctuation, and potential implementation with quantum circuits.
We study the computational difficulty of computing the ground state degeneracy and the density of states for local Hamiltonians. We show that the difficulty of both problems is exactly captured by a class which we call #BQP, which is the counting version of the quantum complexity class quantum Merlin Arthur. We show that #BQP is not harder than its classical counting counterpart #P, which in turn implies that computing the ground state degeneracy or the density of states for classical Hamiltonians is just as hard as it is for quantum Hamiltonians.
Ground state counting plays an important role in several applications in science and engineering, from estimating residual entropy in physical systems, to bounding engineering reliability and solving combinatorial counting problems. While quantum algorithms such as adiabatic quantum optimization (AQO) and quantum approximate optimization (QAOA) can minimize Hamiltonians, they are inadequate for counting ground states. We modify AQO and QAOA to count the ground states of arbitrary classical spin Hamiltonians, including counting ground states with arbitrary nonnegative weights attached to them. As a concrete example, we show how our method can be used to count the weighted fraction of edge covers on graphs, with user-specified confidence on the relative error of the weighted count, in the asymptotic limit of large graphs. We find the asymptotic computational time complexity of our algorithms, via analytical predictions for AQO and numerical calculations for QAOA, and compare with the classical optimal Monte Carlo algorithm (OMCS), as well as a modified Grover's algorithm. We show that for large problem instances with small weights on the ground states, AQO does not have a quantum speedup over OMCS for a fixed error and confidence, but QAOA has a sub-quadratic speedup on a broad class of numerically simulated problems. Our work is an important step in approaching general ground-state counting problems beyond those that can be solved with Grover's algorithm. It offers algorithms that can employ noisy intermediate-scale quantum devices for solving ground state counting problems on small instances, which can help in identifying more problem classes with quantum speedups.
We describe a protocol for preparing the ground state of a Hamiltonian HHH on a quantum computer. This is done by designing a quantum algorithm that implements the imaginary time evolution operator: e−τHe^{-τH}e−τH. The method relies on the so-called ``qubitization'' procedure of Low and Chuang which, assuming the existence of a unitary encoding of the Hamiltonian H=⟨G∣UH∣G⟩H = \langle G| U_H |G\rangleH=⟨G∣UH​∣G⟩, produces a new operator WHW_HWH​ whose moments are the Chebyshev polynomials of HHH when projected on ∣G⟩|G\rangle∣G⟩. Using this result and the expansion of e−τHe^{-τH}e−τH in terms of Chebyshev polynomials we construct a circuit that implements an approximation of the imaginary time evolution operator which, at large time, projects any state on the ground state, provided a non-trivial initial overlap between the two. We illustrate our method on two models: the transverse field Ising model and a single qubit toy model.
Abstract Most quantum algorithms designed to generate or probe properties of the ground state of a quantum many-body system require as input an initial state with a large overlap with the desired ground state. One approach for preparing such a ground state is Imaginary Time Evolution (ITE). Recent work by [Motta, M. , Sun, C. , Tan, A.T.K. et al (2020)] introduced an algorithm—which we will refer to as Quantum Imaginary Time Evolution (QITE)—that shows how ITE can be approximated by a sequence of unitary operators, making QITE potentially implementable on early fault-tolerant quantum computers. In this work, we provide a heuristic study of the capabilities of the QITE algorithm in approximating the ITE of lattice and molecular electronic structure Hamiltonians. We numerically study the performance of the QITE algorithm when provided with a good classical initial state for a large class of systems, some of which are of interest to industrial applications, and check if QITE is able to qualitatively replicate the ITE behavior and improve over a classical mean-field solution. The systems we consider in this work range from one- and two-dimensional lattice systems of various lattice geometries displaying short- and long-range interactions, to active spaces of molecular electronic structure Hamiltonians. In addition to the comparison of QITE and ITE, we explicitly show how imaginary time evolved fermionic Gaussian states can serve as initial states which can be efficiently computed on classical computers and efficiently implemented on quantum computers for generic spin Hamiltonians in arbitrary lattice geometries and dimensions, which can be of independent interest.
Many computationally hard problems can be encoded in quantum Hamiltonians. The solution to these problems is given by the ground states of these Hamiltonians. A state-of-the-art algorithm for finding the ground state of a Hamiltonian is the so-called Quantum Imaginary Time Evolution (QITE) which approximates imaginary time evolution by a unitary evolution that can be implemented in quantum hardware. In this paper, we review the original algorithm together with a comprehensive computer program, as well as, the variational version of it.
Efficiently preparing approximate ground states of large, strongly correlated systems on quantum hardware is challenging, and yet, nature is innately adept at this. This has motivated the study of thermodynamically inspired approaches to ground-state preparation that aim to replicate cooling processes via imaginary-time evolution. However, synthesizing quantum circuits that efficiently implement imaginary-time evolution is itself difficult, with prior proposals generally adopting heuristic variational approaches or using deep block encodings. Here, we use the insight that quantum imaginary-time evolution is a solution of Brockett's double-bracket flow and synthesize circuits that implement double-bracket flows coherently on the quantum computer. We prove that our double-bracket quantum imaginary-time evolution (DB-QITE) algorithm inherits the cooling guarantees of imaginary-time evolution. Concretely, each step is guaranteed to (i) decrease the energy of an initial approximate ground state by an amount proportional to the energy fluctuations of the initial state and (ii) increase the fidelity with the ground state. We provide gate counts for DB-QITE through numerical simulations in qrisp that demonstrate scenarios where DB-QITE outperforms quantum phase estimation. Thus, DB-QITE provides a means to systematically improve the approximation of a ground state using shallow circuits.