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arXiv:2002.04620 · Physical Review Letters

Identification of Symmetry-Protected Topological States on Noisy Quantum Computers

Daniel Azses, Rafael Haenel, Y. Naveh, Robert Raussendorf, Eran Sela, Emanuele G. Dalla Torre

Published Sep 14, 2020• 108 citations
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Abstract

Identifying topological properties is a major challenge because, by definition, topological states do not have a local order parameter. While a generic solution to this challenge is not available yet, a broad class of topological states, namely, symmetry-protected topological (SPT) states, can be identified by distinctive degeneracies in their entanglement spectrum. Here, we propose and realize two complementary protocols to probe these degeneracies based on, respectively, symmetry-resolved entanglement entropies and measurement-based computational algorithms. The two protocols link quantum information processing to the classification of SPT phases of matter. They invoke the creation of a cluster state and are implemented on an IBM quantum computer. The experimental findings are compared to noisy simulations, allowing us to study the stability of topological states to perturbations and noise.

DOI
10.1103/physrevlett.125.120502

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Identifying topological properties is a major challenge because, by definition, topological states do not have a local order parameter. While a generic solution to this challenge is not available yet, a broad class of topological states, namely, symmetry-protected topological (SPT) states, can be identified by distinctive degeneracies in their entanglement spectrum. Here, we propose and realize two complementary protocols to probe these degeneracies based on, respectively, symmetry-resolved entanglement entropies and measurement-based computational algorithms. The two protocols link quantum information processing to the classification of SPT phases of matter. They invoke the creation of a cluster state and are implemented on an IBM quantum computer. The experimental findings are compared to noisy simulations, allowing us to study the stability of topological states to perturbations and noise.
Symmetry-protected topological (SPT) phases extend the Landau paradigm of quantum matter by admitting distinct symmetry-preserving phases that lack any local order parameter. Demonstrating these phases at scale on programmable quantum processors is a key milestone in using quantum hardware to probe emergent many-body phenomena, yet it is impeded by the circuit depth normally required to capture non-trivial entanglement. Here we use a tensor network based approximate quantum compiling (AQC) protocol to construct shallow quantum circuits (18-39 CNOT depth), which prepare 100-site ground states of the spin-1/2 bond-alternating Heisenberg chain in both SPT phases, to 97.9-99.0% fidelity. Upon executing the circuits on IBM quantum hardware, the resulting states exhibit all defining signatures of SPT order including non-local string order for strings of up to length 20, characteristic degeneracies in the entanglement spectrum and clear evidence of symmetry-protected edge modes. The simultaneous observation of these independent diagnostics establishes current quantum computers as versatile platforms for large-scale studies of symmetry-protected quantum matter. More broadly, our results establish a practical foundation for probing non-equilibrium quench dynamics of such systems in regimes that challenge classical computational methods.
It is a challenging problem to correctly characterize the symmetry-protected topological (SPT) phases in open quantum systems. As the measurement-based quantum computation (MBQC) utilizes non-trivial edge states of the SPT phases as the logical qubit, its computational power is closely tied to the non-trivial topological nature of the phases. In this paper, we propose to use the gate fidelity which is a measure of the computational power of the MBQC to identify the SPT phases in mixed-state settings. Specifically, we investigate the robustness of the Haldane phase by considering the MBQC on the Affleck-Kennedy-Lieb-Tasaki state subject to different types of noises. To illustrate how our criterion works, we analytically and numerically calculated the gate fidelity to find that its behavior depends crucially on whether the noises satisfy a certain symmetry condition with respect to the on-site Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ symmetry. In particular, the fidelity for the identity gate, which is given by the sum of the non-local string order parameters, plays an important role. Furthermore, we demonstrate that a stronger symmetry conditions are required to be able to perform other (e.g., the ZZZ-rotation gate) gates with high fidelity. By examining which unitary gates can be implemented with the MBQC on the decohered states, we can gain a useful insight into the richer structure of noisy SPT states that cannot be captured solely by the string order parameters.
Topology and symmetry play critical roles in characterizing quantum phases of matter. Recent advancements have unveiled symmetry-protected topological (SPT) phases in many-body systems as a unique class of short-range entangled states, notable for their nontrivial edge modes and characteristic ground-state entanglement gap. In this study, we demonstrate the robust simulation of many-body ground states of an Ising-cluster model on a quantum computer. By employing the method of quantum imaginary-time evolution (QITE) combined with enhanced zero-noise extrapolation techniques, we achieve accurate measurements of the transition between trivial and cluster SPT phases. Furthermore, we measured the characteristic edge modes and their associated topological entanglement properties, such as the second Rényi entropy, reduced density matrix, and entanglement spectral gap. Our work demonstrates the potential of using QITE in investigating sophisticated quantum phase transitions and critical phenomena on quantum computers.
Topological order offers possibilities for processing quantum information that can be immune to imperfections. However, the question of its stability out of equilibrium is relevant for experiments, where coupling to an environment is unavoidable. In this work, we demonstrate the robustness of certain aspects of Z 2 × Z 2 symmetry-protected topological (SPT) order against a wide class of dissipation channels in the Lindblad and quantum trajectory formalisms of an open quantum system. This is illustrated using the one-dimensional Z X Z cluster Hamiltonian along with Pauli-string jump operators. We show that certain choices of dissipation retaining strong symmetries support a steady-state manifold consisting of two nonlocal qubits and for Hamiltonian perturbations preserving the global symmetry, states in this manifold remain metastable. In contrast, this metastability is destroyed upon breaking the above-mentioned symmetry. While the localized qubits of the cluster Hamiltonian are not conserved by the Lindbladian evolution, they do correspond to weak symmetries and thus retain a memory of their initial state at all times in the quantum trajectories. We utilize this feature to construct protocols to retrieve the quantum information either by monitoring jumps or error mitigation. Our work thus proposes a novel framework to study the dynamics of dissipative SPT phases and opens up the possibility of engineering entangled states relevant to quantum information processing. Published by the American Physical Society 2024
Quantum entanglement is widely used as a diagnostic of topological phases of matter. Beyond entanglement, non-stabilizerness captures a distinct aspect of quantum many-body states by quantifying their distance from the manifold of stabilizer states. In this work, we study the stabilizer Rényi entropy in symmetry protected topological (SPT) phases, including both gapped SPT, non-intrinsically gapless SPT, and intrinsically gapless SPT phases. Under symmetry preserving perturbations, we find numerically that the stabilizer Rényi entropy exhibits an extremum near the phase transition. However, the stabilizer Rényi entropy alone cannot distinguish different SPT phases. In contrast, the Pauli spectrum reveals a characteristic crossing structure at the transition point. This crossing reflects the exchange of dominant Pauli-string correlations associated with the non-local string order parameters of the two topological distinct phases. For gapped SPT and non-intrinsically gapless SPT phases, the crossing structure can be understood from a local-unitary duality that maps the Pauli spectrum between the two phases. For intrinsically gapless SPT phases, such a local-unitary mapping is absent. Instead, we find that the Pauli spectrum mapping is generated by a non-invertible duality transformation. These results show that although the stabilizer Rényi entropy provides only a coarse diagnostic of phase transitions, the Pauli spectrum contains finer information about the exchange of string order sectors. Our findings demonstrate that quantum magic offers a complementary perspective for characterizing both gapped and gapless SPT phases.
We systematically investigate the robustness of symmetry protected topological (SPT) order in open quantum systems by studying the evolution of string order parameters and other probes under noisy channels. We find that one-dimensional SPT order is robust against noisy couplings to the environment that satisfy a strong symmetry condition, while it is destabilized by noise that satisfies only a weak symmetry condition, which generalizes the notion of symmetry for closed systems. We also discuss "transmutation" of SPT phases into other SPT phases of equal or lesser complexity, under noisy channels that satisfy twisted versions of the strong symmetry condition.