Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations

Scalable quantum algorithms for the simulation of quantum many-body systems in thermal equilibrium are important for predicting properties of quantum matter at finite temperatures. Here we describe and benchmark a quantum computing version of the minimally entangled typical thermal states (METTS) algorithm for which we adopt an adaptive variational approach to perform the required quantum imaginary time evolution. The algorithm, which we name AVQMETTS, dynamically generates compact and problem-specific quantum circuits, which are suitable for noisy intermediate-scale quantum (NISQ) hardware. We benchmark AVQMETTS on statevector simulators and perform thermal energy calculations of integrable and nonintegrable quantum spin models in one and two dimensions and demonstrate an approximately linear system-size scaling of the circuit complexity. We further map out the finite-temperature phase transition line of the two-dimensional transverse field Ising model. Finally, we study the impact of noise on AVQMETTS calculations using a phenomenological noise model.Comment: 13 pages, 6 figure

Adaptive variational ground state preparation for spin-1 models on qubit-based architectures

We apply the adaptive variational quantum imaginary time evolution (AVQITE) method to prepare ground states of one-dimensional spin $S=1$ models. We compare different spin-to-qubit encodings (standard binary, Gray, unary, and multiplet) with regard to the performance and quantum resource cost of the algorithm. Using statevector simulations we study two well-known spin-1 models: the Blume-Capel model of transverse-field Ising spins with single-ion anisotropy, and the XXZ model with single-ion anisotropy. We consider system sizes of up to $20$ qubits, which corresponds to spin-$1$ chains up to length $10$. We determine the dependence of the number of CNOT gates in the AVQITE state preparation circuit on the encoding, the initial state, and the choice of operator pool in the adaptive method. Independent on the choice of encoding, we find that the CNOT gate count scales cubically with the number of spins for the Blume-Capel model and quartically for the anistropic XXZ model. However, the multiplet and Gray encodings present smaller prefactors in the scaling relations. These results provide useful insights for the implementation of AVQITE on quantum hardware.Comment: 11 pages, 6 figure

Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models

We perform a systematic study of preparing ground states of correlated multi-orbital impurity models using variational quantum eigensolvers (VQEs). We consider both fixed and adaptive wavefunction ans\"atze and analyze the resulting gate depths and the performance with and without noise. For the adaptive procedure, we develop an operator pool consisting of pairwise commutators of Hamiltonian terms that allows for a fair comparison between the adaptive and fixed Hamiltonian variational ansatz. Using noiseless statevector simulations, we find that the most compact ans\"atze are obtained in an atomic orbital representation and using parity encoding. Focusing on the adaptive algorithms, which yield the circuits with the least number of CNOTs, we then show that in the presence of sampling noise, high-fidelity state preparation can still be achieved with the Hamiltonian commutator pool. By utilizing Hamiltonian integral factorization and a noise resilient optimizer, we show that this approach requires only a modest number of about $2^{12}$ shots per measurement circuit. We discover a dichotomy of the operator pool complexity in the presence of sampling noise, where a small pool size reduces the adaptive overhead but a larger pool size accelerates convergence to the ground state. When considering realistic gate noise in addition, we observe that the variable optimization can still be performed as long as the two-qubit gate error lies below $10^{-3}$, which is close but below current hardware levels. Finally, we measure the ground state energy of the $e_g$ model on IBM and Quantinuum quantum hardware using the converged adaptive ansatz. We perform a systematic error mitigation analysis on the IBM results and obtain a relative error of 0.7\% using symmetry-based postselection and zero-noise extrapolation (ZNE).Comment: 19 pages, 9 figure

The correlation functions of certain random antiferromagnetic spin-1∕2 critical chains

We study the spin-spin correlations in two distinct random critical XX spin-1/2 chain models via exact diagonalization. For the well-known case of uncorrelated random coupling constants, we study the non-universal numerical prefactors and relate them to the corresponding Lyapunov exponent of the underlying single-parameter scaling theory. We have also obtained the functional form of the correct scaling variables important for describing even the strongest finite-size effects. Finally, with respect to the distribution of the correlations, we have numerically determined that they converge to a universal (disorder-independent) non-trivial and narrow distribution when properly rescaled by the spin-spin separation distance in units of the Lyapunov exponent. With respect to the less known case of correlated coupling constants, we have determined the corresponding exponents and shown that both typical and mean correlations functions decay algebraically with the distance. While the exponents of the transverse typical and mean correlations are nearly equal, implying a narrow distribution of transverse correlations, the longitudinal typical and mean correlations critical exponents are quite distinct implying much broader distributions. Further comparisons between these models are given

Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models

We perform a systematic study of preparing ground states of correlated eg and t2g multi-orbital impurity models using variational quantum eigensolvers (VQEs). Both xed and adaptive wavefunction ans atze are considered and the resulting gate depths and performance with and without quantum sampling noise are analyzed. We investigate the qubit adaptive derivative-assembled pseudo-trotter (ADAPT) VQE approach in the Hartree-Fock orbital basis, as well as the Hamiltonian variational ansatz (HVA) and an adaptive variant of it in the atomic orbital basis. An operator pool composed of pairwise commutators of the Hamiltonian terms is developed to allow a fair comparison between the adaptive and the xed HVA ansatz. Using statevector simulations, we show that the most compact ans atze are obtained in the atomic orbital representation with symmetry-based Pauli tapering in parity encoding. We further perform adaptive VQE calculations including sampling noise, and demonstrate that high- delity state preparation can be achieved with the Hamiltonian commutator pool. By utilizing a doubly decomposed form of the impurity Hamiltonian and a noise resilient optimizer, we show that this approach requires only a modest number of about 212 samples per energy evaluation. We discover a dichotomy of the operator pool complexity in the presence of quantum noise, where a small pool size reduces the adaptive overhead but a larger pool size accelerates the convergence to the ground state. Finally, we measure the ground state energy of the eg model on IBM quantum hardware using the converged qubit-ADAPT ansatz, and obtain a relative error of 0.7% using error mitigation techniques including symmetry- ltering and zero-noise extrapolation.This is a pre-print of the article Mukherjee, Anirban, Noah F. Berthusen, João C. Getelina, Peter P. Orth, and Yong-Xin Yao. "Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models." arXiv preprint arXiv:2203.06745 (2022). DOI: 10.48550/arXiv.2203.06745. Copyright 2022 The Authors. Posted with permission

Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations

Scalable quantum algorithms for the simulation of quantum many-body systems in thermal equilibrium are important for predicting properties of quantum matter at finite temperatures. Here we describe and benchmark a quantum computing version of the minimally entangled typical thermal states (METTS) algorithm for which we adopt an adaptive variational approach to perform the required quantum imaginary time evolution. The algorithm, which we name AVQMETTS, dynamically generates compact and problem-specific quantum circuits, which are suitable for noisy intermediate-scale quantum (NISQ) hardware. We benchmark AVQMETTS on statevector simulators and perform thermal energy calculations of integrable and nonintegrable quantum spin models in one and two dimensions and demonstrate an approximately linear system-size scaling of the circuit complexity. We further map out the finite-temperature phase transition line of the two-dimensional transverse field Ising model. Finally, we study the impact of noise on AVQMETTS calculations using a phenomenological noise model