Topological quantum phases of quantum materials are defined through their topological invariants. These topological invariants are quantities that characterize the global geometrical properties of the quantum wave functions and thus are immune to local noise. Here, we present a strategy to measure topological invariants on quantum computers. We show that our strategy can be easily integrated with the variational quantum eigensolver (VQE) so that the topological properties of generic quantum many-body states can be characterized on current quantum hardware. We demonstrate the robust nature of the method by measuring topological invariants for both non-interacting and interacting models, and map out interacting quantum phase diagrams on quantum simulators and IBM quantum hardware.
Topological quantum phases of quantum materials are defined through their topological invariants. These topological invariants are quantities that characterize the global geometrical properties of the quantum wave functions and thus are immune to local noise. Here, we present a strategy to measure topological invariants on quantum computers. We show that our strategy can be easily integrated with the variational quantum eigensolver (VQE) so that the topological properties of generic quantum many-body states can be characterized on current quantum hardware. We demonstrate the robust nature of the method by measuring topological invariants for both non-interacting and interacting models, and map out interacting quantum phase diagrams on quantum simulators and IBM quantum hardware.
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.
Topological protection is employed in fault-tolerant error correction and in developing quantum algorithms with topological qubits. But, topological protection intrinsic to models being simulated , also robustly protects calculations, even on NISQ hardware. We leverage it by simulating Kitaev-inspired models on IBM quantum computers and accurately determining their phase diagrams. This requires constructing conventional quantum circuits for Majorana braiding to prepare the ground states of Kitaev-inspired models. The entanglement entropy is then measured to calculate the quantum phase boundaries. We show how maintaining particle-hole symmetry when sampling through the Brillouin zone is critical to obtaining high accuracy. This work illustrates how topological protection intrinsic to a quantum model can be employed to perform robust calculations on NISQ hardware, when one measures the appropriate protected quantum properties. It opens the door for further simulation of topological quantum models on quantum hardware available today.
Quantum computers hold the promise of solving computational problems that are intractable for classical computers, by harnessing the principles of quantum mechanics. However, quantum computations are highly susceptible to noise and decoherence, which place strict limits on the size and depth of the circuits that can be reliably executed on current hardware. Topological quantum computing (TQC) has emerged as one of the promising approaches to building fault-tolerant quantum computers. The key idea is to store quantum information in the global properties of a physical system, making it naturally resilient to the local noise and errors that affect conventional qubits. This thesis focuses on synthetic topological quantum matter, the physical platform on which TQC is built, and investigates its electronic structure and topological properties. We study the conditions under which these systems enter a topological phase and host Majorana zero modes (MZMs), exotic zero-energy quasiparticles that appear at the boundaries of the system. MZMs are the key building block of TQC, as their quantum mechanical properties allow them to encode and process quantum information in a way that is protected from errors. We demonstrate the existence of MZMs in two canoni- cal topological systems, the Kitaev chain and the semiconducting-superconducting nanowire. To address the experimental challenge of detecting MZMs, we develop an optical spectroscopy technique based on excitonic interactions, which provides clear and measurable signatures of the topological phase. We also explore the variational quantum eigensolver as a quantum algorithmic approach to approximating the ground-state energy of the many-body spectrum of topological systems. Furthermore, we discuss the practical setups for topological quantum matter and simulate them through numerical analysis. Altogether, we present several theoretical and numerical methods that can be used for numerical simulations and practical purposes of detecting MZMs in topological quantum matter.
Programmable quantum processors are suitable platforms for simulating quantum systems, of which topological phases are of particular interest. We simulate the quench dynamics of a one-dimensional system on IBM Q devices. The topological properties of the dynamics are described by the dynamical winding number and the Berry phase, which are simulated with the quantum circuit model. The results show that despite the noise present in the current quantum computers, the dynamical topological invariants are robust. Moreover, to investigate the influence of open quantum system, we analytically solve the master equation in Lindblad form and show that the dynamical winding number and the change in Berry phase are not affected by the dissipation. This study sheds light on the robustness of topological phases on the noisy intermediate-scale quantum computers.
Abstract In classical and single-particle settings, non-trivial band topology always gives rise to robust boundary modes. For quantum many-body systems, however, multiple topological fermions are not always able to coexist, since Pauli exclusion prevents additional fermions from occupying the limited number of available topological modes. In this work, we show, through IBM quantum computers, how one can robustly stabilize more fermions than the number of topological modes through specially designed 2-fermion interactions. Our demonstration hinges on the realization of BDI- and D-class topological Hamiltonians on transmon-based quantum hardware, and relied on a tensor network-aided circuit recompilation approach. We also achieved the full reconstruction of multiple-fermion topological band structures through iterative quantum phase estimation (IQPE). All in all, our work showcases how advances in quantum algorithm implementation enable noisy intermediate-scale quantum (NISQ) devices to be exploited for topological stabilization beyond the context of single-particle topological invariants.
We introduce the Biorthogonal Variational Quantum Eigensolver (B-VQE), a quantum algorithm for simulating non-Hermitian many-body systems on noisy intermediate-scale quantum (NISQ) hardware. Non-Hermitian quantum matter exhibits exceptional points, parity-time symmetry breaking, and non-Hermitian skin effects, yet existing quantum algorithms often rely on costly post-selection procedures and are not designed to capture biorthogonal eigenstates.
B-VQE employs independent variational circuits to represent the left and right eigenstates of a non-Hermitian Hamiltonian and optimizes a biorthogonal objective function that directly tracks non-Hermitian phase transitions. The framework incorporates an Exceptional-Point Detector (EPD) that identifies exceptional points through a hardware-native coalescence metric and a Non-Hermitian Quantum Geometric Tensor (NH-QGT) readout that distinguishes state-topological and band-topological signatures in interacting many-body systems. To overcome the exponential overhead associated with conventional non-Hermitian simulation, we develop an importance-sampling mitigation strategy that removes the need for ancilla-based post-selection while retaining polynomial computational scaling.
We validate the approach on three representative models: a non-Hermitian Hubbard chain, a non-Hermitian XXZ spin chain, and a two-dimensional non-Hermitian (t)-(J) model. B-VQE achieves relative energy errors below (5\times10^{-3}) and locates exceptional points with high accuracy on noise-free simulations while resolving phase boundaries associated with localization, quantum scars, and skin-effect physics. These results establish B-VQE as a scalable NISQ methodology for constructing non-Hermitian many-body phase diagrams and exploring topological and critical phenomena in open quantum systems.