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arXiv:2401.03015 · Quantum

Ground state energy and magnetization curve of a frustrated magnetic system from real-time evolution on a digital quantum processor

Aaron Szasz, Ed Younis, Wibe Albert de Jong

Published Apr 9, 2025• 2 citations
arXivHTMLPDF

Abstract

Models of interacting many-body quantum systems that may realize new exotic phases of matter, notably quantum spin liquids, are challenging to study using even state-of-the-art classical methods such as tensor network simulations. Quantum computing provides a promising route for overcoming these difficulties to find ground states, dynamics, and more. In this paper, we argue that recently developed hybrid quantum-classical algorithms based on real-time evolution are promising methods for solving a particularly important model in the search for spin liquids, the antiferromagnetic Heisenberg model on the two-dimensional kagome lattice. We show how to construct efficient quantum circuits to implement time evolution for the model and to evaluate key observables on the quantum computer, and we argue that the method has favorable scaling with increasing system size. We then restrict to a 12-spin star plaquette from the kagome lattice and a related 8-spin system, and we give an empirical demonstration on these small systems that the hybrid algorithms can efficiently find the ground state energy and the magnetization curve. For these demonstrations, we use four levels of approximation: exact state vectors, exact state vectors with statistical noise from sampling, noisy classical emulators, and (for the 8-spin system only) real quantum hardware, specifically the Quantinuum H1-1 processor; for the noisy simulations and hardware demonstration, we also employ error mitigation strategies based on the symmetries of the Hamiltonian. Our results strongly suggest that these hybrid algorithms present a promising direction for studying quantum spin liquids and more generally for resolving important unsolved problems in condensed matter theory and beyond.

DOI
10.22331/q-2025-04-09-1704

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Theory of Quantum Spin Liquids in Quasi 1-D Systems

Amir Mohammadaghaei

Ever since the introduction of resonating valence bond states by P.\\ W.\\ Anderson in 1970s, the search for exotic many--body quantum behavior in peculiar states of matter known as quantum spin liquids has been a central challenge in the field of condensed matter physics. An important searching ground for such \\emph{quantum--ness} is the low energy physics of frustrated magnets. In a quantum system, frustration---the inability of a physical system to achieve a global state that consistently minimizes its energy locally---enables the unavoidable quantum fluctuations to hybridize an extensive number of classical states into highly entangled quantum states at low temperatures. It can be argued that this is the most practical approach to achieve genuine many--body quantum behavior at large scales. The fundamental source of frustration in many physical systems, is the existence of local non--commuting terms. In the case of antiferromagnet spin theories the simplest examples are triangles of nearest--neighboring sites. Consequently, lattices with triangles as building blocks have long been studied in hopes of discovering spin liquids physics. In particular the kagome lattice, which is composed of corner--sharing triangles, has been the center of at least four decades of spin liquid research. This is due to the underconstrained nature of system made out of corner--sharing triangles. After the first three chapters, introduction in chapter~\\ref{chap:intro}, preliminary physics in chapter~\\ref{chap:prelim}, and a brief introduction to spin--liquids in chapter~\\ref{chap:QSLs}, in chapter~\\ref{chap:kagome} I tackle a simpler version of the long-standing problem of spin--$1/2$ Heisenberg antiferromagnets by considering a quasi--1D lattice consisting entirely of corner--sharing triangles, \\emph{kagome strip}. I illustrate that the standard Heisenberg antiferromagnet Hamiltonian over kagome strip is an extended gapless quantum phase, that is well characterized by two fermionic/bosonic gapless modes and power--law decaying spin and bond--energy correlations. I also demonstrate that the correlation functions oscillate at tunably incommensurate wave vectors. It turns out that this phase can be identified by a particular marginal instability of a two-band spinon Fermi surface coupled to an emergent U(1) gauge field. This interpretation is supported by analytic Abelian bosonization and with extensive numerical large--scale density matrix renormalization group study as well as variational Monte Carlo calculations on Gutzwiller ans\\"atz wave functions. This intriguing result is the first numerical demonstration of emergent fermionic spinons in a simple SU(2) invariant nearest-neighbor Heisenberg model beyond the strictly 1D (Bethe chain) limit. The unexpected success of the fermionic spinons in describing the low energy physics of kagome antiferromagnet, reintroduces the questions about the validity of spinon physics as an effective theory of quantum spin liquids. However, numerical methods are still far behind the theoretical advances. And still even today, an accurate description of the quantum spin liquid states using tensor network methods is notoriously challenging. It is known that for large quasi-1D systems, the density matrix renormalization group and related methods usually require significant computational resources and sometimes fail to converge to a satisfactory state. On the other hand, variational wavefunctions acquired from the Gutzwiller projection of gaussian fermionic theories have long served as both a theoretical starting point for the construction of such spin liquid states and as an inspiration for numerical variational Monte Carlo (VMC) to calculate observables of interest. Noting this observation I examine a different method by exploring the possibility of constructing a matrix product state (MPS) representation for a Gutzwiller--projected state from two given MPS representations of gaussian fermionic theories in the \\ref{chap:gutz} chapter. I investigate the complexity of different approaches to achieve Gutzwiller projection for MPSs and introduce the novel algorithm which we call the Guzwiller zipper method. The performance of the algorithm is tested against two copies of a single half-filled band of spin-1/2 fermionic spinons. In a successful attempt to describe the nature of spin liquid states on quasi-1D strips of triangular and kagome-like lattices, I apply this method to two MPS of multi-band fermionic spinon theories and compare with the complexity of the traditional VMC approach. In particular, we methodically disprove the conjecture of a spinon fermi surface spin liquid for the triangular lattice. Finally, I conclude this thesis by laying out a bird's eye view of the current and future of spin liquid physics in kagome and triangular lattices and pointing out numerous possible applications of the novel Gutzwiller zipper method. In addition, I would also discuss its possible extensions to more complicated tensor networks as well as versions for higher spatial dimensions.

•Jan 1, 2020•0 citations
Models of interacting many-body quantum systems that may realize new exotic phases of matter, notably quantum spin liquids, are challenging to study using even state-of-the-art classical methods such as tensor network simulations. Quantum computing provides a promising route for overcoming these difficulties to find ground states, dynamics, and more. In this paper, we argue that recently developed hybrid quantum-classical algorithms based on real-time evolution are promising methods for solving a particularly important model in the search for spin liquids, the antiferromagnetic Heisenberg model on the two-dimensional kagome lattice. We show how to construct efficient quantum circuits to implement time evolution for the model and to evaluate key observables on the quantum computer, and we argue that the method has favorable scaling with increasing system size. We then restrict to a 12-spin star plaquette from the kagome lattice and a related 8-spin system, and we give an empirical demonstration on these small systems that the hybrid algorithms can efficiently find the ground state energy and the magnetization curve. For these demonstrations, we use four levels of approximation: exact state vectors, exact state vectors with statistical noise from sampling, noisy classical emulators, and (for the 8-spin system only) real quantum hardware, specifically the Quantinuum H1-1 processor; for the noisy simulations and hardware demonstration, we also employ error mitigation strategies based on the symmetries of the Hamiltonian. Our results strongly suggest that these hybrid algorithms present a promising direction for studying quantum spin liquids and more generally for resolving important unsolved problems in condensed matter theory and beyond.
Quantum spin simulations of frustrated lattices remain challenging for both classical and quantum algorithms, particularly in parameter regimes relevant to quantum spin liquid (QSL) phases. In this work, we apply Sample-based Krylov Quantum Diagonalization (SKQD) to estimate the ground state of the antiferromagnetic XXZ Heisenberg model on the J1J_1J1​--J2J_2J2​ square lattice, the Kagome lattice, and a 1D chain, studying system sizes from 12 to 72 spins. In our application of SKQD, we identify a ZZ deformation of Δ=2Δ=2Δ=2 as a sufficiently sparse Hamiltonian and introduce two modifications to the SKQD framework tailored to spin models: a canonical bitstring compression scheme that preserves the effectiveness of configuration recovery under spin-flip degeneracy, and the use of multiple Krylov subspaces to improve ground state coverage without any increase in quantum resources. For the 1D chain and Kagome lattice, SKQD achieves sub-percent ground-state energy errors at system sizes up to 24 spins, including a relative error of 0.002%0.002\%0.002% on the 12-site Kagome lattice, surpassing the best prior VQE result of 0.01%0.01\%0.01% on the same system while requiring no variational optimization. SKQD further extends to system sizes well beyond the reach of prior quantum algorithm studies, reaching 72 spins across all three geometries. Beyond 24 spins, accuracy degrades to relative errors of 19%19\%19%--36%36\%36% at 72 sites, but the gradual scaling of error with system size suggests these limits are set by available shot budgets and circuit depth rather than fundamental algorithmic constraints. Although classical tensor network methods remain state-of-the-art for these models, this work establishes a new benchmark for quantum simulation of the frustrated Heisenberg model and demonstrates SKQD as a scalable, hardware-compatible approach for studying strongly correlated spin systems.
Solving challenging problems in physical, chemical, and materials sciences is one of the most promising applications of quantum utility that can be realized on current noisy hardware, considering (i) the direct map (encoding) from the quantum particles and their interactions to the qubits and their entangling gates and (ii) the rapidly improved quantum hardware and advanced error-mitigation techniques. Understanding quantum spin liquid in frustrated magnetic materials is a longstanding challenge in condensed matter physics and the nature of the ground-state phases is highly debated among researchers. Using IBM quantum computers with superconducting qubits, we implemented a variational quantum eigensolver (VQE) algorithm to prepare the ground states of two 12-site cluster approximations of these highly frustrated magnetic materials. The interaction graphs of the two corresponding Hamiltonians are (a) the six-pointed star graph (a unit cell of the kagome lattice) and (b) the cuboctahedral graph (the kagome on a sphere). These are also two instances of Quantum Max Cut problem. With the VQE based on the Hamiltonian variational ansatz acting on a valence bond solid initial trial state, we prepared the ground states and obtained the exact ground energy on simulator and high accuracy on noisy hardware. The deep ansatz necessary to reach the ground state of the cuboctahedral graph indicates that it is a hard instance of Quantum Max Cut.
In this work we investigate the ground state properties of a candidate quantum spin liquid using a superconducting Noisy Intermediate-Scale Quantum (NISQ) device. Specifically, we study the antiferromagnetic Heisenberg model on a Kagome lattice, a geometrically frustrated structure that gives rise to a highly degenerate energy spectrum. To successfully simulate this system, we employ a qubit reduction strategy leveraging the Contextual Subspace methodology, significantly reducing the problem size prior to execution on the quantum device. We improve the quality of these subspaces by using the wavefunctions obtained from low bond dimension Density Matrix Renormalization Group (DMRG) calculations to bias the subspace stabilizers through a symplectic approximate symmetry generator extraction algorithm. Reducing the Hamiltonian size allows us to implement tiled circuit ensembles and deploy the Variational Quantum Eigensolver (VQE) to estimate the ground state energy. We adopt a hybrid quantum error mitigation strategy combining Readout Error Mitigation (REM), Symmetry Verification (SV) and Zero Noise Extrapolation (ZNE). This approach yields high-accuracy energy estimates, achieving error rates on the order of 0.01% and thus demonstrating the potential of near-term quantum devices for probing frustrated quantum materials.
Finite-temperature properties of strongly correlated quantum matter are central to condensed matter, chemistry, and high-energy physics, yet are often inaccessible to classical methods such as quantum Monte Carlo (QMC). Here, we investigate dissipative thermal state preparation of frustrated spin systems using digital quantum computers. We focus on two paradigmatic models on the kagome lattice: the antiferromagnetic Heisenberg model (AFHM), whose finite-temperature properties are inaccessible to QMC due to a severe sign problem, and the antiferromagnetic Ising model (AFIM), which serves as a sign-problem-free benchmark. Using IBM quantum processors, we prepare approximate thermal states of the AFIM on kagome lattices with up to 79 spins coupled to 60 environment qubits. We observe the emergence of a robust steady state with an adjustable effective temperature that persists in circuits with over 1000 layers of two-qubit gates. We further study the scalability of the dissipative protocol through classical statevector simulations of the AFIM and AFHM. On lattices with up to 27 sites, we find that the circuit depth to reach thermal equilibrium is independent of system size and grows at most linearly with inverse temperature. These results establish engineered dissipation as a promising approach to finite-temperature quantum simulation of frustrated matter, and point toward regimes where quantum devices may outperform classical methods.
Spin liquids─an emergent, exotic collective phase of matter─have garnered enormous attention in recent years. While experimentally many prospective candidates have been proposed and realized, theoretically modeling real materials that display such behavior may pose serious challenges due to the inherently high correlation content of such phases. Over the last few decades, the second-quantum revolution has been the harbinger of a novel computational paradigm capable of initiating a foundational evolution in computational physics. In this report, we strive to use the power of the latter to study a prototypical model, a spin-1/2-unit cell of a Kagome antiferromagnet. Extended lattices of such unit cells are known to possess a magnetically disordered spin-liquid ground state. We employ robust classical numerical techniques such as the density-matrix renormalization group (DMRG) to identify the nature of the ground state through a matrix-product state (MPS) formulation. We subsequently use the gained insight to construct an auxiliary Hamiltonian with reduced measurables and also design an ansatz that is modular and gate-efficient. With robust error-mitigation strategies, we are able to demonstrate that the said ansatz is capable of accurately representing the target ground state even on a real IBMQ backend within 1% accuracy in energy. Since the protocol is linearly scaling O ( n ) in the number of unit cells, gate requirements, and the number of measurements, it is straightforwardly extendable to larger Kagome lattices that can pave the way for efficient construction of spin-liquid ground states on a quantum device.
Ever since the introduction of resonating valence bond states by P.\\ W.\\ Anderson in 1970s, the search for exotic many--body quantum behavior in peculiar states of matter known as quantum spin liquids has been a central challenge in the field of condensed matter physics. An important searching ground for such \\emph{quantum--ness} is the low energy physics of frustrated magnets. In a quantum system, frustration---the inability of a physical system to achieve a global state that consistently minimizes its energy locally---enables the unavoidable quantum fluctuations to hybridize an extensive number of classical states into highly entangled quantum states at low temperatures. It can be argued that this is the most practical approach to achieve genuine many--body quantum behavior at large scales. The fundamental source of frustration in many physical systems, is the existence of local non--commuting terms. In the case of antiferromagnet spin theories the simplest examples are triangles of nearest--neighboring sites. Consequently, lattices with triangles as building blocks have long been studied in hopes of discovering spin liquids physics. In particular the kagome lattice, which is composed of corner--sharing triangles, has been the center of at least four decades of spin liquid research. This is due to the underconstrained nature of system made out of corner--sharing triangles. After the first three chapters, introduction in chapter~\\ref{chap:intro}, preliminary physics in chapter~\\ref{chap:prelim}, and a brief introduction to spin--liquids in chapter~\\ref{chap:QSLs}, in chapter~\\ref{chap:kagome} I tackle a simpler version of the long-standing problem of spin--1/21/21/2 Heisenberg antiferromagnets by considering a quasi--1D lattice consisting entirely of corner--sharing triangles, \\emph{kagome strip}. I illustrate that the standard Heisenberg antiferromagnet Hamiltonian over kagome strip is an extended gapless quantum phase, that is well characterized by two fermionic/bosonic gapless modes and power--law decaying spin and bond--energy correlations. I also demonstrate that the correlation functions oscillate at tunably incommensurate wave vectors. It turns out that this phase can be identified by a particular marginal instability of a two-band spinon Fermi surface coupled to an emergent U(1) gauge field. This interpretation is supported by analytic Abelian bosonization and with extensive numerical large--scale density matrix renormalization group study as well as variational Monte Carlo calculations on Gutzwiller ans\\"atz wave functions. This intriguing result is the first numerical demonstration of emergent fermionic spinons in a simple SU(2) invariant nearest-neighbor Heisenberg model beyond the strictly 1D (Bethe chain) limit. The unexpected success of the fermionic spinons in describing the low energy physics of kagome antiferromagnet, reintroduces the questions about the validity of spinon physics as an effective theory of quantum spin liquids. However, numerical methods are still far behind the theoretical advances. And still even today, an accurate description of the quantum spin liquid states using tensor network methods is notoriously challenging. It is known that for large quasi-1D systems, the density matrix renormalization group and related methods usually require significant computational resources and sometimes fail to converge to a satisfactory state. On the other hand, variational wavefunctions acquired from the Gutzwiller projection of gaussian fermionic theories have long served as both a theoretical starting point for the construction of such spin liquid states and as an inspiration for numerical variational Monte Carlo (VMC) to calculate observables of interest. Noting this observation I examine a different method by exploring the possibility of constructing a matrix product state (MPS) representation for a Gutzwiller--projected state from two given MPS representations of gaussian fermionic theories in the \\ref{chap:gutz} chapter. I investigate the complexity of different approaches to achieve Gutzwiller projection for MPSs and introduce the novel algorithm which we call the Guzwiller zipper method. The performance of the algorithm is tested against two copies of a single half-filled band of spin-1/2 fermionic spinons. In a successful attempt to describe the nature of spin liquid states on quasi-1D strips of triangular and kagome-like lattices, I apply this method to two MPS of multi-band fermionic spinon theories and compare with the complexity of the traditional VMC approach. In particular, we methodically disprove the conjecture of a spinon fermi surface spin liquid for the triangular lattice. Finally, I conclude this thesis by laying out a bird's eye view of the current and future of spin liquid physics in kagome and triangular lattices and pointing out numerous possible applications of the novel Gutzwiller zipper method. In addition, I would also discuss its possible extensions to more complicated tensor networks as well as versions for higher spatial dimensions.