Simulating Topological Order on Quantum Processors

TL;DR

The paper surveys progress toward realizing topological order on programmable quantum processors, examining both symmetry-protected and intrinsic topological phases through cluster, AKLT, and toric code paradigms. It analyzes state-preparation strategies on NISQ devices, from constant-depth cluster-state preparation to sequential and measurement-assisted AKLT schemes and adaptive, finite-depth toric-code protocols. Diagnostic tools such as entanglement spectra, non-local string order parameters, topological entanglement entropy, and anyon braiding/fusion measurements are discussed as central to verifying topological order and emergent excitations. The authors also review experimental milestones across superconducting qubits, trapped ions, and Rydberg arrays, and outline challenges in scalability, fidelity, and preserving topological invariants, pointing toward robust, fault-tolerant pathways for quantum information processing.

Abstract

It is an ongoing quest to realize topologically ordered quantum states on different platforms including condensed matter systems, quantum simulators and digital quantum processors. Unlike conventional states characterized by their local order, these exotic states are characterized by their non-local entanglement. The consequences of topological order can be as profound as they are surprising, ranging from the emergence of fractionalized anyonic excitations to potentially providing a scalable platform for quantum error correction. This deep connection to quantum computing naturally motivates the realization and study of topologically ordered quantum states on quantum processors. However, due to the non-local nature of these states, their study presents a challenge for near-term quantum devices. This Perspective aims to review the recent progress towards the experimental realization of topologically ordered quantum states, their potential applications, and promising directions of future research.

Figures (5)

• Figure 1: Quantum many-body phase diagrams split into the trivial phase (white) and different Topologically Ordered (TO) phases (orange). When symmetries are enforced, the phase diagram is additionally split into different Symmetry Protected Topological (SPT) and Symmetry Enriched Topological (SET) phases (separated by dashed lines) and Symmetry Broken (SB) phases (separated by dotted lines).
• Figure 2: Preparation of SPT states on quantum computers: (a) Unitary circuit for preparing the cluster state choo2018measurement. (b) Illustration of the spin-1 AKLT state using two projected spin-1/2 particles. (c) Diagram of the MPS representation of the AKLT state, with boundary conditions denoted by $L$ and $R$, respectively Smith2023. (d) Sequential unitary preparation of the AKLT state using a linear-depth circuit Smith2023. (e) Circuit diagram for measurement-assisted preparation Smith2023.
• Figure 3: Characterizing SPT order on quantum computers: (a) Entanglement spectrum of the cluster state measured on an IBM quantum computer (red) compared to exact values (light blue). The entanglement gap and near-degenerate low-lying levels signal topological order, though statistical noise (blue) and experimental imperfections lift the expected degeneracy.choo2018measurement (b) Detection of the SPT-to-trivial phase transition via a string order parameter measured using interferometry on an IBM quantum computer.Smith2022 (c) Exact QCNN output along a path crossing an SPT phase obtained using a QCNN.Cong2019
• Figure 4: Preparing topological order on quantum processors. (a) Unitary quantum circuit to prepare the toric code ground state, along with measurements of the star and plaquette stabilisers at each step. Satzinger2021 The state can be realised using a single layer of Hadamard gates and a linear sequence of nearest neighbour CNOT gates. (b) Finite depth preparation of the toric code using adaptive quantum circuits with measurement and feedback.Kitaev2003 Starting with a product state that satisfies the stars, measuring the plaquette operators randomly projects onto the $\pm 1$ eigenstates. Any remaining plaquette defects can be removed by a single layer of single qubit gates, which are determined by the measurement outcomes.
• Figure 5: Characterising topological order. (a) Different size partitions for computing the topological entropy on a quantum processor Satzinger2021. (b) Experimental data corresponding to these partitions for the toric code, computed using full state tomography and randomized measurement methods. Satzinger2021 (c) Topological entanglement entropy measurement for the double Fibonacci model on a superconducting quantum processor. xu2024nonabelian (d) Measured anyon statistics in the toric code. Satzinger2021 (e) Measurement of the Fredenhagen-Marcu order parameter for a system of Rydberg atoms indicating $\mathbb{Z}_2$ topologically order. Semeghini2021