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arXiv:2208.00271 · Physical review. B./Physical review. B

Bang-bang algorithms for quantum many-body ground states: A tensor network exploration

Ruoshui Wang, Timothy H. Hsieh, Guifré Vidal

Published Nov 17, 2022• 5 citations
arXivHTMLPDF

Abstract

We use matrix product techniques to investigate the performance of two algorithms for obtaining the ground state of a quantum many-body Hamiltonian $H={H}_{A}+{H}_{B}$ in infinite systems. The first algorithm is a generalization of the quantum approximate optimization algorithm and uses a quantum computer to evolve an initial product state into an approximation of the ground state of $H$ by alternating between ${H}_{A}$ and ${H}_{B}$. We show for the one-dimensional (1D) quantum Ising model that the accuracy in representing a gapped ground state improves exponentially with the number of alternations. The second algorithm is the variational imaginary time ansatz, which uses a classical computer to simulate the ground state via alternating imaginary time steps with ${H}_{A}$ and ${H}_{B}$. We find for the 1D quantum Ising model that an accurate approximation to the ground state is obtained with a total imaginary time $\ensuremath{\tau}$ that grows only logarithmically with the inverse energy gap $1/\mathrm{\ensuremath{\Delta}}$ of $H$. This is much faster than imaginary time evolution by $H$, which would require $\ensuremath{\tau}\ensuremath{\sim}1/\mathrm{\ensuremath{\Delta}}$.

DOI
10.1103/physrevb.106.195133

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We use matrix product techniques to investigate the performance of two algorithms for obtaining the ground state of a quantum many-body Hamiltonian H=HA+HBH={H}_{A}+{H}_{B}H=HA​+HB​ in infinite systems. The first algorithm is a generalization of the quantum approximate optimization algorithm and uses a quantum computer to evolve an initial product state into an approximation of the ground state of HHH by alternating between HA{H}_{A}HA​ and HB{H}_{B}HB​. We show for the one-dimensional (1D) quantum Ising model that the accuracy in representing a gapped ground state improves exponentially with the number of alternations. The second algorithm is the variational imaginary time ansatz, which uses a classical computer to simulate the ground state via alternating imaginary time steps with HA{H}_{A}HA​ and HB{H}_{B}HB​. We find for the 1D quantum Ising model that an accurate approximation to the ground state is obtained with a total imaginary time \ensuremathτ\ensuremath{\tau}\ensuremathτ that grows only logarithmically with the inverse energy gap 1/\ensuremathΔ1/\mathrm{\ensuremath{\Delta}}1/\ensuremathΔ of HHH. This is much faster than imaginary time evolution by HHH, which would require \ensuremathτ\ensuremath∼1/\ensuremathΔ\ensuremath{\tau}\ensuremath{\sim}1/\mathrm{\ensuremath{\Delta}}\ensuremathτ\ensuremath∼1/\ensuremathΔ.
A bang-bang (BB) algorithm prepares the ground state of a two-dimensional (2D) quantum many-body Hamiltonian H=H1+H2H={H}_{1}+{H}_{2}H=H1​+H2​ by evolving an initial product state alternating between H1{H}_{1}H1​ and H2{H}_{2}H2​. We use the neighborhood tensor update to simulate the BB evolution with an infinite pair-entangled projected state (iPEPS). The alternating sequence is optimized with the final energy as a cost function. The energy is calculated with a tangent space power method for the sake of its stability. The method is benchmarked in the 2D transverse field quantum Ising model near its quantum critical point against a ground state obtained with variational optimization of the iPEPS. The optimal BB sequence differs nonperturbatively from a sequence simulating quantum annealing or adiabatic preparation (AP) of the ground state. The optimal BB energy converges with the number of bangs much faster than the optimal AP energy.
A bang-bang (BB) algorithm prepares the ground state of a lattice quantum many-body Hamiltonian H=H1+H2H=H_1+H_2H=H1​+H2​ by evolving an initial product state alternating between H1H_1H1​ and H2H_2H2​. We optimize the algorithm with tensor networks in one and two dimensions. The optimization has two stages. In stage one, a shallow translationally-invariant circuit is optimized in an infinite lattice. In stage two, the infinite-lattice gate sequence is used as a starting point for a finite lattice where it remains optimal in the bulk. The prepared state requires optimization only at its boundary, within a healing length from lattice edges, and the gate sequence needs to be modified only within the causal cone of the boundary. We test the procedure in the 1D and 2D quantum Ising model near its quantum critical point employing, respectively, the matrix product state (MPS) and the pair-entangled projected state (PEPS). At the boundary already the infinite-lattice sequence turns out to provide a more accurate state than in the bulk.
Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation.Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated.Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance.It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood.Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods.Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).
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.
Simulation of quantum systems is challenging due to the exponential size of the state space. Tensor networks provide a systematically improvable approximation for quantum states. 2D tensor networks such as Projected Entangled Pair States (PEPS) are well-suited for key classes of physical systems and quantum circuits. However, direct contraction of PEPS networks has exponential cost, while approximate algorithms require computations with large tensors. We propose new scalable algorithms and software abstractions for PEPS-based methods, accelerating the bottleneck operation of contraction and refactorization of a tensor subnetwork. We employ randomized SVD with an implicit matrix to reduce cost and memory footprint asymptotically. Further, we develop a distributed-memory PEPS library and study accuracy and efficiency of alternative algorithms for PEPS contraction and evolution on the Stampede2 supercomputer. We also simulate a popular near-term quantum algorithm, the Variational Quantum Eigensolver (VQE), and benchmark Imaginary Time Evolution (ITE), which compute ground states of Hamiltonians.
Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.