Abstract Classical techniques for the simulation of complex many-body systems can be severely limited by the exponentially increasing dimensions of quantum systems, and prior assumptions are often required to efficiently extract information about non-local quantum properties. Short depth quantum algorithms are proposed that efficiently estimate non-local quantum properties such as the quantum coherence and discrete Berry phases to probe the quantum phase transitions in many body systems. The gate complexities of these algorithms scale linearly while the depth complexity scale logarithmically with system size. These algorithms are applied to probe the well-known Haldane to quantum paramagnetic phase transitions in spin-1 Heisenberg antiferromagnetic chains, which are topological phase transitions that cannot be identified via local order parameters. The algorithms are shown to be able to extract non-local information about the system give the correct signatures of topological phase transitions near critical points.
Abstract Classical techniques for the simulation of complex many-body systems can be severely limited by the exponentially increasing dimensions of quantum systems, and prior assumptions are often required to efficiently extract information about non-local quantum properties. Short depth quantum algorithms are proposed that efficiently estimate non-local quantum properties such as the quantum coherence and discrete Berry phases to probe the quantum phase transitions in many body systems. The gate complexities of these algorithms scale linearly while the depth complexity scale logarithmically with system size. These algorithms are applied to probe the well-known Haldane to quantum paramagnetic phase transitions in spin-1 Heisenberg antiferromagnetic chains, which are topological phase transitions that cannot be identified via local order parameters. The algorithms are shown to be able to extract non-local information about the system give the correct signatures of topological phase transitions near critical points.
Efficient preparation of many-body ground states is key to harnessing the power of quantum computers in studying quantum many-body systems. In this work, we propose a simple method to design exact linear-depth parameterized quantum circuits which prepare a family of ground states across topological quantum phase transitions in 2D. We achieve this by constructing ground states represented by isometric tensor networks (isoTNS), which form a subclass of tensor network states that are efficiently preparable. By continuously tuning a parameter in the wavefunction, the many-body ground state undergoes quantum phase transitions, exhibiting distinct 2D quantum phases. We illustrate this by constructing an isoTNS path with bond dimension D = 2 interpolating between distinct symmetry-enriched topological (SET) phases. At the transition point, the wavefunction is related to a gapless point in the classical six-vertex model. Furthermore, the critical wavefunction supports a power-law correlation along one spatial direction while remaining long-range ordered in the other spatial direction. We provide an explicit parametrized local quantum circuit for the path and show that the 2D isoTNS can also be efficiently simulated by a holographic quantum algorithm requiring only an 1D array of qubits. Published by the American Physical Society 2024
Characterizing quantum topological phases requires measuring non-local string order parameters, demanding access to the full system, which is often experimentally unfeasible. In this work, we introduce a data-efficient supervised learning framework that circumvents this limitation by recognizing quantum phases from small subsystems. Our protocol utilizes a quantum kernel constructed from the reduced density matrices of these subsystems, which can be efficiently estimated experimentally. We benchmark our framework with the classification of the phase diagrams of two spin models on one-dimensional lattices, namely the generalized cluster-Ising spin-1/2 chain and the anisotropic Haldane spin-1 chain. Remarkably, our approach achieves high accuracy in phase classification when operations are limited to as few as one to four sites, and it also generalizes to longer chains even when trained on moderate system sizes. These findings demonstrate that local reduced density matrices preserve vital signatures of global topological phases, offering a practical route to characterize rich phase diagrams of quantum many-body systems.
Estimating the eigenstate properties of quantum systems is a long-standing, challenging problem for both classical and quantum computing. Existing universal quantum algorithms typically rely on ideal and efficient query models (e.g., time evolution operator or block encoding of the Hamiltonian), which, however, become suboptimal for actual implementation at the quantum circuit level. Here, we present a full-stack design of quantum algorithms for estimating the eigenenergy and eigenstate properties, which can achieve high precision and good scaling with system size. The gate complexity per circuit for estimating generic Hamiltonians' eigenstate properties is [Formula: see text], which has a logarithmic dependence on the inverse precision ε. For lattice Hamiltonians, the circuit depth of our design achieves near-optimal system-size scaling, even with local qubit connectivity. Our full-stack algorithm has low overhead in circuit compilation, which thus results in a small actual gate count (cnot and non-Clifford gates) for lattice and molecular problems compared to advanced eigenstate algorithms. The algorithm is implemented on IBM quantum devices using up to 2000 two-qubit gates and 20,000 single-qubit gates and achieves high-precision eigenenergy estimation for Heisenberg-type Hamiltonians, demonstrating its noise robustness.
Topological quantum phases emerge from correlated quantum many-body systems containing novel features such as nontrivial entanglement structure and mutual statistics. However, the co-emerged exponential computational complexity strongly hampers the research of these systems using classical computers. Present booming quantum computing techniques offer a new way to investigate these challenging systems: the quantum simulation approach. Combining current available noise intermediate-scale quantum (NISQ) devices with variational quantum eigensolver (VQE) algorithm to solve quantum many-body problems has attracted extensive attention. Although the NISQ devices have up to hundreds of qubits, treating a quantum many-body problem with a similar size on these devices is still impossible. One fundamental reason for this is the lack of scalability in current quantum algorithms (i.e., the required quantum resources for realizing a quantum algorithm increase much faster than the increase of the problem size). In this poster, we introduce how to realize scalable VQE calculation in the topological quantum phases by designing problem-specified scalable parameterized quantum circuits (PQC). We construct a PQC with a two-layer structure to capture both the basic entanglement structure of the target topological phase and finite correlations governed by the details of the many-body Hamiltonian. Utilizing such a PQC, we numerically simulate both symmetry-protected topological (SPT) Haldane phase and trivial dimmer phase in the non-exactly solvable alternating Heisenberg chain using the VQE algorithm. Furthermore, we realize these optimized circuits on real quantum computers provided by IBM Quantum and observe the main characterizations of the SPT Haldane phase.
The characterization of the quantum phase transition in a lowdimensional system has attracted a considerable amount of attention in quantum manybody systems. As one of the fundamental models in quantum magnetism, spin-1 models have richer phase diagrams and show more complex physical phenomena. In the spin-1 antiferromagnetic XXZ model, the Haldane phase and the Nel phase are the gapped topologic phases which cannot be characterized by the local order parameters. To characterize the nature in such phases, one has to calculate the non-local long range order parameters. Normally, the non-local order parameter in the topological phase is obtained from the extrapolation of finite-sized system in numerical study. However, it is difficult to extract the critical exponents with such an extrapolated non-local order parameter due to the numerical accuracy. In a recently developed tensor network representation, i.e., the infinite matrix product state (iMPS) algorithm, it was shown that the non-local order can be directly calculated from a very large lattice distance in an infinite-sized system rather than an extrapolated order parameter in a finite-sized system. Therefore, it is worthwhile using this convenient technique to study the non-local orders in the topological phases and characterize the quantum criticalities in the topological quantum phase transitions. In this paper, by utilizing the infinite matrix product state algorithm based on the tensor network representation and infinite time evolving block decimation method, the quantum entanglement, fidelity, and critical exponents of the topological phase transition are investigated in the one-dimensional infinite spin-1 bond-alternating XXZ Heisenberg model. It is found that there is always a local dimerization order existing in the whole parameter range when the bond-alternative strength parameter changes from 0 to 1. Also, due to the effect of the bond-alternating, there appears a quantum phase transition from the long-rang ordering topological Nel phase to the local ordering dimerization phase. The von Neumann entropy, fidelity per lattice site, and order parameters all give the same phase transition point at c = 0.638. To identify the type of quantum phase transition, the central charge c 0.5 is extracted from the ground state von Neumann entropy and the finite correlation length, which indicates that the phase transition belongs to the two-dimensional Ising universality class. Furthermore, it is found that the Nel order and the susceptibility of Nel order have power-law relations to |-c|. From the numerical fitting of the Nel order and its susceptibility, we obtain the characteristic critical exponents ' = 0.236 and ' = 0.838. It indicates that such critical exponents from our method characterize the nature of the quantum phase transition. Our critical exponents from the iMPS method can provide guidance for studying the properties of the phase transition in quantum spin systems.
In this paper, we analyze the circuit complexity for preparing ground states of quantum many-body systems. In particular, how this complexity grows as the ground state approaches a quantum phase transition. We discuss different definitions of complexity, namely the one following the Fubini-Study metric or the Nielsen complexity. We also explore different models: Ising, ZZXZ or Dicke. In addition, different forms of state preparation are investigated: analytic or exact diagonalization techniques, adiabatic algorithms (with and without shortcuts), and Quantum Variational Eigensolvers. We find that the divergence (or lack thereof) of the complexity near a phase transition depends on the non-local character of the operations used to reach the ground state. For Fubini-Study based complexity, we extract the universal properties and their critical exponents. In practical algorithms, we find that the complexity depends crucially on whether or not the system passes close to a quantum critical point when preparing the state. For both VQE and Adiabatic algorithms, we provide explicit expressions and bound the growth of complexity with respect to the system size and the execution time, respectively.