Abstract Mapping out phase diagrams of quantum systems using classical simulations can be challenging or intractable due to the computational resources required to simulate even small quantum systems far away from the thermodynamic limit. We investigate using quantum computers and the variational quantum eigensolver (VQE) for this task. In contrast to the task of preparing the exact ground state using VQE, sketching phase diagrams might require less quantum resources and accuracy, because low fidelity approximations to the ground state may be enough to correctly identify different phases. We used classical numerical simulations of low-depth VQE circuits to compute order parameters for four well-studied spin and fermion models which represent a mix of 1D and 2D, and exactly-solvable and classically hard systems. We find that it is possible to predict the location of phase transitions up to reasonable accuracy using states produced by VQE even when their overlap with the true ground state is small. Further, we introduce a model-agnostic predictor of phase transitions based on the speed with which the VQE energy improves with respect to the circuit depth, and find that in some cases this is also able to predict phase transitions.
Abstract Mapping out phase diagrams of quantum systems using classical simulations can be challenging or intractable due to the computational resources required to simulate even small quantum systems far away from the thermodynamic limit. We investigate using quantum computers and the variational quantum eigensolver (VQE) for this task. In contrast to the task of preparing the exact ground state using VQE, sketching phase diagrams might require less quantum resources and accuracy, because low fidelity approximations to the ground state may be enough to correctly identify different phases. We used classical numerical simulations of low-depth VQE circuits to compute order parameters for four well-studied spin and fermion models which represent a mix of 1D and 2D, and exactly-solvable and classically hard systems. We find that it is possible to predict the location of phase transitions up to reasonable accuracy using states produced by VQE even when their overlap with the true ground state is small. Further, we introduce a model-agnostic predictor of phase transitions based on the speed with which the VQE energy improves with respect to the circuit depth, and find that in some cases this is also able to predict phase transitions.
Mapping quantum phase diagrams through classical simulation is notoriously resource-intensive, as even small systems far from the thermodynamic limit demand prohibitive computational effort. The variational quantum eigensolver (VQE) offers a compelling alternative, exploiting approximate ground states to distinguish phases. An appealing proposal, dubbed as Delta-VQE, determines critical points by contrasting variational energies optimized from reference states of distinct phases. Intriguingly, the diagnostic sharpens as circuit depth decreases, highlighting its promise as a resource-conscious probe of quantum criticality. To probe the broader applicability and underlying mechanisms of this approach, we investigate the one-dimensional transverse-field Ising model with a three-spin cluster interaction, a setting in which the Ising transitions are generally situated beyond the self-dual line. We demonstrate that, whenever dual ansätze are employed, Delta-VQE invariably detects the self-dual points rather than the true criticality. In contrast, when ansätze are carefully tailored to embody the competing phases across the boundary, the genuine Ising critical point can be successfully identified with only minor finite-size effects. Our results establish that, while Delta-VQE provides a resource-efficient probe of quantum criticality without requiring precise ground-state preparation, its diagnostic power is fundamentally contingent upon the judicious selection of physically representative ansätze.
The variational quantum eigen solver (VQE), has been widely used to find the ground state energy of different Hamiltonians with no analytical solutions and are classically difficult to compute. In our work, we have used VQE to identify the phase transition boundary for an infinite order phase transition. We use long-range XXZ (LRXXZ) chain for our study. In order to probe infinite order phase transition, we propose to utilise the ground state energy obtained from VQE. The idea rests on the argument that VQE requires an ansatz circuit; therefore, the accuracy of the VQE will rely on this ansatz circuit. We have designed this circuit such that the estimated ground state energy is sensitive to the phase it is evaluated in. It is achieved by applying the constraint that the net spin remains constant throughout the optimisation process. Consequently, the ansatz works in a certain phase where it gives relatively small random error, as it should, when compared to the error in ground state energy calculations of the other phases, where the ansatz fails. By identifying these changes in the behaviour of the error in ground state energy using VQE, we were able to determine the phase boundaries. Using exact diagonalisation, we also compare the behaviour of the energy gradient and energy gap across both the phase transition boundaries for this model. Further, by increasing the depth of the optimisation circuit, we also accurately evaluate the ground energy of the LRXXZ chain for the value of coupling constant, J equal to -1
Finding and probing the ground states of spin lattices, such as the antiferromagnetic Heisenberg model on the kagome lattice (KAFH), is a very challenging problem on classical computers and only possible for relatively small systems. We propose using the variational quantum eigensolver (VQE) to find the ground state of the KAFH on a quantum computer. We find efficient ansatz circuits and show how physically interesting observables can be measured efficiently. To investigate the expressiveness and scaling of our ansatz circuits we used classical, exact simulations of VQE for the KAFH for different lattices ranging from 8 to 24 qubits. We find that the fidelity with the ground state approaches one exponentially in the circuit depth for all lattices considered, except for a 24-qubit lattice with an almost degenerate ground state. We conclude that VQE circuits that are able to represent the ground state of the KAFH on lattices inaccessible to exact diagonalization techniques may be achievable on near-term quantum hardware. However, for large systems, circuits with many variational parameters are needed to achieve high fidelity with the ground state.
Exponential scaling presents a significant challenge in electronic structure calculations performed on classical computers. This paper explores how quantum computer algorithms can accurately represent quantum systems. Variational Quantum Eigensolver (VQE) algorithm is used to compute the ground state energy of hydrogen and helium sequences by implementing variational principle and quantum gates as trial wavefunction. This technique combines classical optimization with quantum computing calculations to simulate quantum systems on noisy and resource-limited computers. The resulting calculated energy is highly consistent to the corresponding exact values and Hartree-Fock calculations with a trend of when the number of atoms increases the calculated energy becomes more negative, leading to a decrease in the percentage error. Moreover, the convergence of the ground state energy of hydrogen and helium atoms was effectively optimized. The desired energy was reached, proven by adjusting the expectation value, and gradually achieving unity in state overlap. These findings demonstrate the VQE method's accuracy in calculating simple quantum systems and its scalability for larger atomic and molecular system, such as those in quantum chemistry and material science. However, challenges in quantum computer simulations, such as limited in qubit numbers and the presence of noise, require further advancements. Therefore, implementing a larger basis sets, advanced qubit mapping, specific chemistry ansatz, and flexible optimization techniques is one way to improve overall calculation.
The problem of finding the ground state energy of a Hamiltonian using a quantum computer is currently solved using either the quantum phase estimation (QPE) or variational quantum eigensolver (VQE) algorithms. For precision $\ensuremath{\epsilon}$, QPE requires O(1) repetitions of circuits with depth $O(1/\ensuremath{\epsilon})$, whereas each expectation estimation subroutine within VQE requires $O(1/{\ensuremath{\epsilon}}^{2})$ samples from circuits with depth O(1). We propose a generalized VQE algorithm that interpolates between these two regimes via a free parameter $\ensuremath{\alpha}\ensuremath{\in}[0,1]$, which can exploit quantum coherence over a circuit depth of $O(1/{\ensuremath{\epsilon}}^{\ensuremath{\alpha}})$ to reduce the number of samples to $O(1/{\ensuremath{\epsilon}}^{2(1\ensuremath{-}\ensuremath{\alpha})})$. Along the way, we give a new routine for expectation estimation under limited quantum resources that is of independent interest.
Quantum Phase Estimation (QPE) is the quantum algorithmic workhorse for computing ground state energies of quantum Hamiltonians with quantum computers. Ground state energy calculation of physical systems is perhaps the most promising use case for quantum computing in terms of scientific and commercial value with a plausible path to outperformance of classical alternatives. This path, however, hinges on the availability of initial states for QPE with significant overlap with the true ground state. Using extensive (classical) numerical computations, we study whether the NISQ-era algorithm VQE (Variational Quantum Eigensolver) could be used to efficiently prepare high-overlap states of disordered fully-connected anisotropic Heisenberg spin glass quantum Hamiltonians with up to 15 qubits. We find that (i) -- consistent with widely held, but rarely numerically illustrated beliefs -- VQE is generally unable to efficiently converge to the ground state for our Hamiltonians, which is a well-known issue with VQE due to a variety of factors including vanishing gradients and local minima; (ii) low energy states do not necessarily have large ground-state overlap, but there is typically a correlation between the two measures; (iii) adding more than three layers to the VQE ansatz neither improves overlap nor the energies found; and (iv) the best-found overlap scaling as a function of the Hamiltonian system size is not strongly exponentially decreasing, suggesting potential for VQE to be a heuristic state preparation algorithm for QPE.