Topological Order from Measurements and Feed-Forward on a Trapped Ion Quantum Computer

TL;DR

This work demonstrates deterministic creation of topological order on a trapped-ion quantum computer by leveraging mid-circuit measurements and feed-forward to implement non-unitary dynamics in constant depth. The authors realize a toric-code ground state on a 4x4 torus with high stabilizer fidelity and measure a negative energy density consistent with the ground state, while also enabling two non-Abelian defect dynamics and anyon braiding interferometry. They validate topological order via topological entanglement entropy measurements and showcase an anyon transmutation experiment enabled by a defect, confirming fermionic braiding statistics with a minimal circuit footprint. The results establish a practical pathway to study complex topological phases and deterministic non-unitary dynamics in the lab, with implications for quantum error correction and future simulations of lattice gauge theories; scalability to larger qubit counts and improved noise control will be key for broader applicability.

Abstract

Quantum systems evolve in time in one of two ways: through the Schrödinger equation or wavefunction collapse. So far, deterministic control of quantum many-body systems in the lab has focused on the former, due to the probabilistic nature of measurements. This imposes serious limitations: preparing long-range entangled states, for example, requires extensive circuit depth if restricted to unitary dynamics. In this work, we use mid-circuit measurement and feed-forward to implement deterministic non-unitary dynamics on Quantinuum's H1 programmable ion-trap quantum computer. Enabled by these capabilities, we demonstrate for the first time a constant-depth procedure for creating a toric code ground state in real-time. In addition to reaching high stabilizer fidelities, we create a non-Abelian defect whose presence is confirmed by transmuting anyons via braiding. This work clears the way towards creating complex topological orders in the lab and exploring deterministic non-unitary dynamics via measurement and feed-forward.

Figures (8)

• Figure 1: Schematic representation of the the toric code ground state from wavefunction collapse. We initialize a system of trapped-ion qubits (encoded in hyperfine states of $^{171}{\rm Yb}^{+}$) in a product state where all stabilizers $B_p = Z^{\otimes 4}=1$ (blue) are satisfied. We measure $A_p = X^{\otimes 4}$ on every other plaquette, randomly leading to $A_p=1$ (gold) or $A_p=-1$ (black, denoting an $e$-anyon). We use feed-forward to pair up and annihilate the $e$-anyons in real time, deterministically producing a clean toric code wavefunction using a finite-depth circuit and nonlocal classical processing.
• Figure 2: Toric code ground state preparation. (a) Definition of the stabilizer operators (\ref{['eq_toric_code']}) on the unraveled torus. Numbers denote the different ions and specify the boundary conditions. Plaquettes are labeled by their upper left qubits. The state comprises 4 $\times$ 4 qubits and periodic boundary conditions. (b) Logical $Z$ string operators are $Z^\text{hori} = Z_0 Z_1 Z_2 Z_3$ ($Z^\text{vert} = Z_0 Z_4 Z_8 Z_{12}$) and their vertical (horizontal) translations. $\overline{Z^\text{hori}}$ and $\overline{Z^\text{vert}}$ denote expectation values of the logical string operators, averaged over translations. (c) Expectation values of the stabilizers obtained from the measurement described in the main text. Error bars denote one standard error on the mean. (d) Entanglement entropy measurement on $2 \times 2$ (top) and $2 \times 3$ regions (bottom). Colored bars denote $S^{(2)}_X$ for different subsystems of a region with shapes as shown in the inset. Dashed lines show exact values. The maximum error in the estimates of $S^{(2)}_X$ for $2\times 2$ ($2 \times 3$) regions is $\pm0.056$ ($\pm0.091$). Hatched white bars denote average topological entanglement entropies.
• Figure 3: Anyon dynamics on a state with two non-Abelian defects. (a) Geometry. The lack of the central qubit and the redefinition of the stabilizers leads to two defective plaquettes. (b) Expectation values of the stabilizers obtained from the state preparation and measurement routine described in the main text. Error bars denote one standard error on the mean. (c) Anyon transmutation. A pair of magnetic anyons is created and one partner is transmuted into an electric anyon by moving it across the line connecting the two defects. The maximum and minimum error in the expectation values of stabilizers are $\pm0.023$ and $\pm0.0066$ respectively. (d) Anyon Interferometry. A fermionic $e-m$ composite anyon is created next to the defect and a controlled-$Z_{10} Z_{8} Z_{4} Z_{7}$ braiding operations is applied with the help of an ancilla.
• Figure 4: Additional data confirming the gate noise bias towards $Z$-errors. A single $Z$-flip occuring during the $(\mathbb{I}+X^{\otimes 4})/2$ projection flips two adjacent $Z$-plaquettes but leaves $X$-plaquettes invariant. (a) Covariances of plaquettes $\braket{P_p P_q} - \braket{P_p}\braket{P_q}$, $X$-plaquettes on the left ($P=A$), $Z$-plaquettes on the right ($P=B$). Neighbouring (non-neighbouring) plaquettes are marked by a red (white) dot. (b) Average $Z$-plaquette correlation functions $\braket{B_p B_q}- \braket{B_p}\braket{B_q}$ over all nearest-neighbour ('adjacent') and non-nearest-neighbour ('distant') plaquettes.
• Figure 5: Additional data for the entanglement entropy measurements. (a, b) $2\times2$ and $2\times3$ regions (solid rectangles) and their labels on $4 \times 4$ torus. Each $2\times 3$ region consists of two vertically adjacent plaquettes. (c, d) Entanglement entropy correction $\gamma$ for each region using the protocol discussed in the main text and sec. \ref{['methods_entropy']}. The maximum value of the error bars is $\pm 0.35\ (\pm 0.54)$ for $2\times2$ ($2\times3$) regions. In the $2\times2$ case, the value of $\gamma$ is averaged over all rotations.