arXiv:2610.10454

Resolving Spectral Gaps of the Frustrated Spin-1/2 $J_1$-$J_2$ Chain via Dual Spin-Sector Variational Quantum Eigensolver

Sk Saniur Rahaman

Published Oct 7, 2026• 0 citations

Abstract

Finding excited states and resolving spectral gaps in frustrated quantum spin systems using Variational Quantum Eigensolvers is complicated by state-ordering inversions. In the frustrated spin-1/2 $J_1$-$J_2$ Heisenberg chain, standard single-sector variational quantum deflation collapses onto intermediate triplet excitation, preventing the isolation of excited singlets and the singlet-singlet gap. To resolve these gaps, a Dual Spin-Sector Variational Quantum Eigensolver framework is introduced for near-term quantum architectures. By partitioning the search across magnetization sectors $S^z=0$ and $S^z=1$, the lowest triplet is resolved as an unconstrained ground state in $S^z=1$ with zero deflation penalty. Leveraging exact global spin-rotation symmetry, this state is mapped analytically into $S^z=0$ without ancilla overhead, unlocking dual deflation via shallow compute-uncompute circuits. Tracking both singlet-triplet and singlet-singlet gaps across coupling ratios $J_2/J_1$ from $0$ to $1$ for chains up to $16$ qubits faithfully captures all critical phenomena: the continuous Berezinskii-Kosterlitz-Thouless transition, exact Majumdar-Ghosh dimer degeneracy, level-ordering inversion in the gapless liquid phase, and incommensurate spiral excitations. Circuit depth scaling up to twenty layers provides an overparameterization mechanism overcoming classical local minima in strongly frustrated regimes, reducing relative energy errors below $1\%$ and elevating state fidelities above $97\%$ on $16$ qubits. Projective measurement sampling demonstrates resilience against shot noise, providing a practical blueprint for exploring frustrated quantum magnetism on near-term quantum devices.

DOI: 10.48550/arxiv.2610.10454

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Decoupled phase of frustrated spin- 1 2 antiferromagnetic chains with and without long-range order in the ground state

Manoranjan Kumar, Z. G. Soos

arXiv:1309.7542Physical Review BOct 14, 201317 citations

Machine learning approach to study quantum phases of a frustrated one dimensional spin-1/2 system

Sk Saniur Rahaman, Sumit Haldar, Manoranjan Kumar

arXiv:2202.00564Feb 1, 20220 citations

Low-energy singlet sector in spin- 1 J1-J2 Heisenberg model on square lattice

A. Yu. Aktersky, A. V. Syromyatnikov

Jan 1, 20160 citations

Near-Term Quantum Spin Simulation of the Spin-$\frac{1}{2}$ Square $J_{1}-J_{2}$ Heisenberg Model

Dylan Sheils, Trevor David Rhone

arXiv:2406.18474Jun 26, 20241 citation

Equivariant Variational Quantum Eigensolver to detect phase transitions through energy level crossings

Giulio Crognaletti, Giovanni Di Bartolomeo, Michele Vischi +1 more

arXiv:2403.07100Quantum Science and TechnologyDec 9, 20247 citations

Dynamical Spectral Function of the Kagome Quantum Spin Liquid

Jiahang Hu, Runze Chi, Yibin Guo +3 more

cond-mat.str-elCross-listedarXiv:2512.18831Physical Review LettersDec 23, 20250 citations

arXiv:2610.10454

Resolving Spectral Gaps of the Frustrated Spin-1/2 $J_1$ - $J_2$ Chain via Dual Spin-Sector Variational Quantum EigensolverResolving Spectral Gaps of the Frustrated Spin-1/2 $J_1$-$J_2$ Chain via Dual Spin-Sector Variational Quantum Eigensolver

Sk Saniur Rahaman

Published Oct 7, 2026• 0 citations

arXiv HTML PDF

Abstract

Finding excited states and resolving spectral gaps in frustrated quantum spin systems using Variational Quantum Eigensolvers is complicated by state-ordering inversions. In the frustrated spin-1/2

J_1

J_2

Heisenberg chain, standard single-sector variational quantum deflation collapses onto intermediate triplet excitation, preventing the isolation of excited singlets and the singlet-singlet gap. To resolve these gaps, a Dual Spin-Sector Variational Quantum Eigensolver framework is introduced for near-term quantum architectures. By partitioning the search across magnetization sectors

S^z=0

and

S^z=1

, the lowest triplet is resolved as an unconstrained ground state in

S^z=1

with zero deflation penalty. Leveraging exact global spin-rotation symmetry, this state is mapped analytically into

S^z=0

without ancilla overhead, unlocking dual deflation via shallow compute-uncompute circuits. Tracking both singlet-triplet and singlet-singlet gaps across coupling ratios

J_2/J_1

from

0

1

for chains up to

16

qubits faithfully captures all critical phenomena: the continuous Berezinskii-Kosterlitz-Thouless transition, exact Majumdar-Ghosh dimer degeneracy, level-ordering inversion in the gapless liquid phase, and incommensurate spiral excitations. Circuit depth scaling up to twenty layers provides an overparameterization mechanism overcoming classical local minima in strongly frustrated regimes, reducing relative energy errors below

1\%

and elevating state fidelities above

97\%

16

qubits. Projective measurement sampling demonstrates resilience against shot noise, providing a practical blueprint for exploring frustrated quantum magnetism on near-term quantum devices.

DOI: 10.48550/arxiv.2610.10454

Related papers

Decoupled phase of frustrated spin- 1 2 antiferromagnetic chains with and without long-range order in the ground state

Manoranjan Kumar, Z. G. Soos

The quantum phases of one-dimensional spin $s=1/2$ chains are discussed for models with two parameters, frustrating exchange $g={J}_{2}>0$ between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at $g=0$ has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin $S={S}_{A}+{S}_{B}$ , have singlet GS for any $g$ , $\ensuremath{\alpha}\ensuremath{\ge}0$ and decouple at $1/g=0$ to linear Heisenberg antiferromagnets on sublattices $A$ and $B$ of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap ${E}_{m}$ to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at ${g}_{c}=1/4\mathrm{ln}2$ from a GS with perfect LRO to a decoupled phase with ${S}_{A}={S}_{B}=0$ for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.The quantum phases of one-dimensional spin $s=1/2$ chains are discussed for models with two parameters, frustrating exchange $g={J}_{2}>0$ between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at $g=0$ has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin $S={S}_{A}+{S}_{B}$, have singlet GS for any $g$, $\ensuremath{\alpha}\ensuremath{\ge}0$ and decouple at $1/g=0$ to linear Heisenberg antiferromagnets on sublattices $A$ and $B$ of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap ${E}_{m}$ to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at ${g}_{c}=1/4\mathrm{ln}2$ from a GS with perfect LRO to a decoupled phase with ${S}_{A}={S}_{B}=0$ for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.

arXiv:1309.7542Physical Review BOct 14, 201317 citations

Machine learning approach to study quantum phases of a frustrated one dimensional spin-1/2 system

Sk Saniur Rahaman, Sumit Haldar, Manoranjan Kumar

Frustration driven quantum fluctuation leads to many exotic phases in the ground state and study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. Here, a frustrated Heisenberg $J_1-J_2$ model of spin-1/2 chain with nearest exchange interaction $J_1$ and next nearest exchange interaction $J_2$ is studied using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSC) of ground-state (GS) and first excited state (FES) for different $J_2/J_1$ are used as the input in PCA to construct the co-variance matrix. The `quantified principal component' of the largest eigenvalue of co-variance matrix $p_1(J_2/J_1)$ is calculated and it is shown that the nature and variation of $p_1(J_2/J_1)$ can accurately predict the phase transitions and degeneracies in the GS. The $p_1(J_2/J_1)$ calculated from the MPSC of GS can only exhibit the signature of degeneracies in the GS, whereas, $p_1(J_2/J_1)$ calculated from MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that jump in $p_1(J_2/J_1)$ of FES at $J_2/J_1 \approx 0.241$ , indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of first two principal components of FES shows distinct band formation for different phases.Frustration driven quantum fluctuation leads to many exotic phases in the ground state and study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. Here, a frustrated Heisenberg $J_1-J_2$ model of spin-1/2 chain with nearest exchange interaction $J_1$ and next nearest exchange interaction $J_2$ is studied using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSC) of ground-state (GS) and first excited state (FES) for different $J_2/J_1$ are used as the input in PCA to construct the co-variance matrix. The `quantified principal component' of the largest eigenvalue of co-variance matrix $p_1(J_2/J_1)$ is calculated and it is shown that the nature and variation of $p_1(J_2/J_1)$ can accurately predict the phase transitions and degeneracies in the GS. The $p_1(J_2/J_1)$ calculated from the MPSC of GS can only exhibit the signature of degeneracies in the GS, whereas, $p_1(J_2/J_1)$ calculated from MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that jump in $p_1(J_2/J_1)$ of FES at $J_2/J_1 \approx 0.241$, indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of first two principal components of FES shows distinct band formation for different phases.

arXiv:2202.00564Feb 1, 20220 citations

Low-energy singlet sector in spin- 1 J1-J2 Heisenberg model on square lattice

A. Yu. Aktersky, A. V. Syromyatnikov

Based on a special variant of plaquette expansion, an operator is constructed whose eigenvalues give the low-energy singlet spectrum of spin- $\frac12$ Heisenberg antiferromagnet on square lattice with nearest- and frustrating next-nearest-neighbor exchange couplings $J_1$ and $J_2$ . It is well known that a non-magnetic phase arises in this model at $0.4\lesssim J_2/J_1\lesssim 0.6$ sandwiched by two Neel ordered phases. In agreement with previous results, we observe a first-order quantum phase transition (QPT) at $J_2\approx 0.64J_1$ from the non-magnetic phase to the Neel one. Large gap ( $\gtrsim0.4J_1$ ) is found in the singlet spectrum at $J_2<0.64J_1$ that excludes a gapless spin-liquid state at $0.4\lesssim J_2/J_1\lesssim 0.6$ and the deconfined quantum criticality scenario for the QPT to another Neel phase. We observe a first-order QPT at $J_2\approx 0.55J_1$ presumably between two non-magnetic phases.Based on a special variant of plaquette expansion, an operator is constructed whose eigenvalues give the low-energy singlet spectrum of spin-$\frac12$ Heisenberg antiferromagnet on square lattice with nearest- and frustrating next-nearest-neighbor exchange couplings $J_1$ and $J_2$. It is well known that a non-magnetic phase arises in this model at $0.4\lesssim J_2/J_1\lesssim 0.6$ sandwiched by two Neel ordered phases. In agreement with previous results, we observe a first-order quantum phase transition (QPT) at $J_2\approx 0.64J_1$ from the non-magnetic phase to the Neel one. Large gap ($\gtrsim0.4J_1$) is found in the singlet spectrum at $J_2<0.64J_1$ that excludes a gapless spin-liquid state at $0.4\lesssim J_2/J_1\lesssim 0.6$ and the deconfined quantum criticality scenario for the QPT to another Neel phase. We observe a first-order QPT at $J_2\approx 0.55J_1$ presumably between two non-magnetic phases.

Jan 1, 20160 citations

Near-Term Quantum Spin Simulation of the Spin- $\frac{1}{2}$ Square $J_{1}-J_{2}$ Heisenberg ModelNear-Term Quantum Spin Simulation of the Spin-$\frac{1}{2}$ Square $J_{1}-J_{2}$ Heisenberg Model

Dylan Sheils, Trevor David Rhone

Simulating complex spin systems, known for high frustration and entanglement, presents significant challenges due to their intricate energy landscapes. This study focuses on the $J_{1}-J_{2}$ Heisenberg model, renowned for its rich phase behavior on the square lattice, to investigate strongly correlated spin systems. We conducted the first experimental quantum computing study of this model using the 127-qubit IBM Rensselear Eagle processor and the Variational Quantum Eigensolver (VQE) algorithm. By employing classical warm-starting ( $+40\%$ ground state energy approximation) and a newly developed ansatz ( $+9.31\%$ improvement compared to prior best), we improved ground state approximation accuracy on the 16-site variant, achieving usable results with approximately $10^{3}$ iterations, significantly fewer than the $10^{4}-10^{5}$ steps proposed by previous theoretical studies. We utilized existing error mitigation strategies and introduced a novel Classically-Reinforced VQE error mitigation scheme, achieving $93\%$ ground state accuracy, compared to $83.7\%$ with the Quantum Moments algorithm and $60\%$ with standard error mitigation. These strategies reduced the average error of observable prediction from $\approx 20\%$ to $5\%$ , enhancing phase prediction from qualitative to quantitative alignment. Additionally, we explored an experimental implementation of the Quantum Lanczos (QLanczos) algorithm using Variational-Fast Forwarding (VFF) on a 4-qubit site, achieving $\approx 97\%$ ground state approximation. Theoretical simulations indicated that Krylov-based methods outperform VQE, with the Lanczos basis converging faster than the real-time basis. Our study demonstrates that near-term quantum devices can predict phase-relevant observables for the $J_1-J_2$ Heisenberg model, transitioning focus from theoretical to experimental, and suggesting general improvements to VQE-based methods.Simulating complex spin systems, known for high frustration and entanglement, presents significant challenges due to their intricate energy landscapes. This study focuses on the $J_{1}-J_{2}$ Heisenberg model, renowned for its rich phase behavior on the square lattice, to investigate strongly correlated spin systems. We conducted the first experimental quantum computing study of this model using the 127-qubit IBM Rensselear Eagle processor and the Variational Quantum Eigensolver (VQE) algorithm. By employing classical warm-starting ($+40\%$ ground state energy approximation) and a newly developed ansatz ($+9.31\%$ improvement compared to prior best), we improved ground state approximation accuracy on the 16-site variant, achieving usable results with approximately $10^{3}$ iterations, significantly fewer than the $10^{4}-10^{5}$ steps proposed by previous theoretical studies. We utilized existing error mitigation strategies and introduced a novel Classically-Reinforced VQE error mitigation scheme, achieving $93\%$ ground state accuracy, compared to $83.7\%$ with the Quantum Moments algorithm and $60\%$ with standard error mitigation. These strategies reduced the average error of observable prediction from $\approx 20\%$ to $5\%$, enhancing phase prediction from qualitative to quantitative alignment. Additionally, we explored an experimental implementation of the Quantum Lanczos (QLanczos) algorithm using Variational-Fast Forwarding (VFF) on a 4-qubit site, achieving $\approx 97\%$ ground state approximation. Theoretical simulations indicated that Krylov-based methods outperform VQE, with the Lanczos basis converging faster than the real-time basis. Our study demonstrates that near-term quantum devices can predict phase-relevant observables for the $J_1-J_2$ Heisenberg model, transitioning focus from theoretical to experimental, and suggesting general improvements to VQE-based methods.

arXiv:2406.18474Jun 26, 20241 citation

Equivariant Variational Quantum Eigensolver to detect phase transitions through energy level crossings

Giulio Crognaletti, Giovanni Di Bartolomeo, Michele Vischi +1 more

Abstract Level spectroscopy stands as a powerful method for identifying the transition point that delineates distinct quantum phases. Since each quantum phase exhibits a characteristic sequence of excited states, the crossing of energy levels between low-lying excited states offers a reliable mean to estimate the phase transition point. While approaches like the Variational Quantum Eigensolver are useful for approximating ground states of interacting systems using quantum computing, capturing low-energy excitations remains challenging. In our study, we introduce an equivariant quantum circuit that preserves the total spin and the translational symmetry to accurately describe singlet and triplet excited states in the J 1 – J 2 Heisenberg model on a chain, which are crucial for characterizing its transition point. Additionally, we assess the impact of noise on the variational state, showing that conventional mitigation techniques like Zero Noise Extrapolation reliably restore its physical properties.Abstract Level spectroscopy stands as a powerful method for identifying the transition point that delineates distinct quantum phases. Since each quantum phase exhibits a characteristic sequence of excited states, the crossing of energy levels between low-lying excited states offers a reliable mean to estimate the phase transition point. While approaches like the Variational Quantum Eigensolver are useful for approximating ground states of interacting systems using quantum computing, capturing low-energy excitations remains challenging. In our study, we introduce an equivariant quantum circuit that preserves the total spin and the translational symmetry to accurately describe singlet and triplet excited states in the J 1 – J 2 Heisenberg model on a chain, which are crucial for characterizing its transition point. Additionally, we assess the impact of noise on the variational state, showing that conventional mitigation techniques like Zero Noise Extrapolation reliably restore its physical properties.

arXiv:2403.07100Quantum Science and TechnologyDec 9, 20247 citations

Dynamical Spectral Function of the Kagome Quantum Spin Liquid

Jiahang Hu, Runze Chi, Yibin Guo +3 more

Quantum spin liquids (QSLs) host exotic fractionalized magnetic and gauge-field excitations whose microscopic origins and experimental verification remain frustratingly elusive. In the absence of static magnetic order, the spin excitation spectrum constitutes the crucial probe of QSL behavior, but its theoretical computation is a serious challenge. Here we employ state-of-the-art tensor-network methods to obtain the full dynamical spectral function of the $J_1$ - $J_2$ kagome Heisenberg model and benchmark our results by tracking their evolution across the magnetically ordered and QSL phases. Reducing $|J_2|/J_1$ causes increasingly strong spin-wave renormalization, flattening these modes then merging them into a continuum characteristic of deconfined spinons at all finite energies in the QSL. The low-energy continuum and the occurrence of gap closure at multiple high-symmetry points identify this gapless QSL as the U(1) Dirac spin liquid. These results establish a unified understanding of spin excitations in highly frustrated quantum magnets and provide clear spectral fingerprints for experimental detection in candidate kagome QSL materials.Quantum spin liquids (QSLs) host exotic fractionalized magnetic and gauge-field excitations whose microscopic origins and experimental verification remain frustratingly elusive. In the absence of static magnetic order, the spin excitation spectrum constitutes the crucial probe of QSL behavior, but its theoretical computation is a serious challenge. Here we employ state-of-the-art tensor-network methods to obtain the full dynamical spectral function of the $J_1$-$J_2$ kagome Heisenberg model and benchmark our results by tracking their evolution across the magnetically ordered and QSL phases. Reducing $|J_2|/J_1$ causes increasingly strong spin-wave renormalization, flattening these modes then merging them into a continuum characteristic of deconfined spinons at all finite energies in the QSL. The low-energy continuum and the occurrence of gap closure at multiple high-symmetry points identify this gapless QSL as the U(1) Dirac spin liquid. These results establish a unified understanding of spin excitations in highly frustrated quantum magnets and provide clear spectral fingerprints for experimental detection in candidate kagome QSL materials.

cond-mat.str-elCross-listedarXiv:2512.18831Physical Review LettersDec 23, 20250 citations

Resolving Spectral Gaps of the Frustrated Spin-1/2

J_1

J_2

Chain via Dual Spin-Sector Variational Quantum Eigensolver

Finding excited states and resolving spectral gaps in frustrated quantum spin systems using Variational Quantum Eigensolvers is complicated by state-ordering inversions. In the frustrated spin-1/2

J_1

J_2

S^z=0

and

S^z=1

, the lowest triplet is resolved as an unconstrained ground state in

S^z=1

with zero deflation penalty. Leveraging exact global spin-rotation symmetry, this state is mapped analytically into

S^z=0

without ancilla overhead, unlocking dual deflation via shallow compute-uncompute circuits. Tracking both singlet-triplet and singlet-singlet gaps across coupling ratios

J_2/J_1

from

0

1

for chains up to

16

1\%

and elevating state fidelities above

97\%

16

qubits. Projective measurement sampling demonstrates resilience against shot noise, providing a practical blueprint for exploring frustrated quantum magnetism on near-term quantum devices.

The quantum phases of one-dimensional spin

s=1/2

chains are discussed for models with two parameters, frustrating exchange

g={J}_{2}>0

between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at

g=0

has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin

S={S}_{A}+{S}_{B}

, have singlet GS for any

g

, $\ensuremath{\alpha}\ensuremath{\ge}0$ and decouple at

1/g=0

to linear Heisenberg antiferromagnets on sublattices

A

and

B

of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap

{E}_{m}

to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at

{g}_{c}=1/4\mathrm{ln}2

from a GS with perfect LRO to a decoupled phase with

{S}_{A}={S}_{B}=0

for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.

J_1-J_2

model of spin-1/2 chain with nearest exchange interaction

J_1

and next nearest exchange interaction

J_2

is studied using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSC) of ground-state (GS) and first excited state (FES) for different

J_2/J_1

are used as the input in PCA to construct the co-variance matrix. The `quantified principal component' of the largest eigenvalue of co-variance matrix

p_1(J_2/J_1)

is calculated and it is shown that the nature and variation of

p_1(J_2/J_1)

can accurately predict the phase transitions and degeneracies in the GS. The

p_1(J_2/J_1)

calculated from the MPSC of GS can only exhibit the signature of degeneracies in the GS, whereas,

p_1(J_2/J_1)

calculated from MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that jump in

p_1(J_2/J_1)

of FES at

J_2/J_1 \approx 0.241

, indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of first two principal components of FES shows distinct band formation for different phases.

Based on a special variant of plaquette expansion, an operator is constructed whose eigenvalues give the low-energy singlet spectrum of spin-

\frac12

Heisenberg antiferromagnet on square lattice with nearest- and frustrating next-nearest-neighbor exchange couplings

J_1

and

J_2

. It is well known that a non-magnetic phase arises in this model at

0.4\lesssim J_2/J_1\lesssim 0.6

sandwiched by two Neel ordered phases. In agreement with previous results, we observe a first-order quantum phase transition (QPT) at

J_2\approx 0.64J_1

from the non-magnetic phase to the Neel one. Large gap (

\gtrsim0.4J_1

) is found in the singlet spectrum at

J_2<0.64J_1

that excludes a gapless spin-liquid state at

0.4\lesssim J_2/J_1\lesssim 0.6

and the deconfined quantum criticality scenario for the QPT to another Neel phase. We observe a first-order QPT at

J_2\approx 0.55J_1

presumably between two non-magnetic phases.

Near-Term Quantum Spin Simulation of the Spin-

\frac{1}{2}

Square

J_{1}-J_{2}

Heisenberg Model

Simulating complex spin systems, known for high frustration and entanglement, presents significant challenges due to their intricate energy landscapes. This study focuses on the

J_{1}-J_{2}

Heisenberg model, renowned for its rich phase behavior on the square lattice, to investigate strongly correlated spin systems. We conducted the first experimental quantum computing study of this model using the 127-qubit IBM Rensselear Eagle processor and the Variational Quantum Eigensolver (VQE) algorithm. By employing classical warm-starting (

+40\%

ground state energy approximation) and a newly developed ansatz (

+9.31\%

improvement compared to prior best), we improved ground state approximation accuracy on the 16-site variant, achieving usable results with approximately

10^{3}

iterations, significantly fewer than the

10^{4}-10^{5}

steps proposed by previous theoretical studies. We utilized existing error mitigation strategies and introduced a novel Classically-Reinforced VQE error mitigation scheme, achieving

93\%

ground state accuracy, compared to

83.7\%

with the Quantum Moments algorithm and

60\%

with standard error mitigation. These strategies reduced the average error of observable prediction from

\approx 20\%

5\%

, enhancing phase prediction from qualitative to quantitative alignment. Additionally, we explored an experimental implementation of the Quantum Lanczos (QLanczos) algorithm using Variational-Fast Forwarding (VFF) on a 4-qubit site, achieving

\approx 97\%

ground state approximation. Theoretical simulations indicated that Krylov-based methods outperform VQE, with the Lanczos basis converging faster than the real-time basis. Our study demonstrates that near-term quantum devices can predict phase-relevant observables for the

J_1-J_2

Heisenberg model, transitioning focus from theoretical to experimental, and suggesting general improvements to VQE-based methods.

J_1

J_2

kagome Heisenberg model and benchmark our results by tracking their evolution across the magnetically ordered and QSL phases. Reducing

|J_2|/J_1

causes increasingly strong spin-wave renormalization, flattening these modes then merging them into a continuum characteristic of deconfined spinons at all finite energies in the QSL. The low-energy continuum and the occurrence of gap closure at multiple high-symmetry points identify this gapless QSL as the U(1) Dirac spin liquid. These results establish a unified understanding of spin excitations in highly frustrated quantum magnets and provide clear spectral fingerprints for experimental detection in candidate kagome QSL materials.