Scaling roadmap for modular trapped-ion QEC and lattice-surgery teleportation

TL;DR

The paper develops a physics-informed framework to scale modular trapped-ion QEC using triangular color codes and lattice-surgery teleportation. It combines micr oscopic noise modeling, Pauli-frame simulations, and architecture-aware transpilation to compare beam-deflector linear traps with integrated photonics for memory and logical teleportation tasks. Key findings show that lattice-surgery-based teleportation is feasible in near-term devices, with integrated-photonics layouts offering superior scaling potential, while flag-based dynamic circuits and advanced decoders are crucial for achieving fault-tolerant performance. The work provides concrete QEC and teraquop footprints to guide experimental design and outlines a roadmap toward modular, scalable FT trapped-ion processors.

Abstract

We present a footprint study for the scaling of modular quantum error correction (QEC) protocols designed for triangular color codes, including a lattice-surgery-based logical teleportation gadget, and compare the performance of various possible architectures based on trapped ions. The differences in these architectures arise from the technology that enables the connectivity between physical qubits and the modularity required for the QEC gadgets, which is either based on laser-beam deflectors focused to independent modules hosting mid-size ion crystals, or integrated photonics guided to segmented modules of the trap and allowing for the manipulation of smaller ion crystals. Our approach integrates the transpilation of the QEC gadgets into native trapped-ion primitives and a detailed account of the specific laser addressing and ion transport leading to different amounts of crosstalk errors, motional excitation and idle qubit errors. Combining a microscopically-informed noise model with an efficient Pauli-frame simulator and different scalable decoders, we assess the near-term performance of the color-code memory and teleportation protocols on these architectures. Our analysis demonstrates that modular color-code teleportation is achievable in these near-term trapped-ion architectures, and identifies the integrated-photonics connectivity as the most promising route for longer-term scaling.

Figures (18)

• Figure 1: Hexagonal color code family: Qubit layout of the first two instances of the hexagonal color code. Plaquettes in the lattice are tri-colored such that adjacent plaquettes have different colors. Each plaquette defines both an $X-$type and $Z-$type stabilizer. Chains of $X$ or $Z$ operators along a boundary side define logical $X$ and $Z$ operators, respectively. The [[7, 1, 3]] code, also called the Steane code PhysRevLett.77.793, only contains weight-4 stabilizers. The [[19,1,5]] and bigger color codes include as well hexagonal plaquette stabilizers in the bulk.
• Figure 2: Flag-type verified logical state preparation: preparation of a logical $\ket{0}$ state for the Steane code Goto2016. The circuit uses a non-FT encoding to create the desired state, followed by a verification step. The verification circuit implements a logical $Z$ measurement which detects any fault that propagated to more than one data qubit, causing an uncorrectable error. If the measurement outcome is flipped, the prepared qubit is discarded.
• Figure 3: Color code state preparation fidelity: logical error rates achieved by different state preparation protocols for the color code. Error rates are determined under standard circuit-level noise and lookup table decoding. For any CSS code, a codeword can be prepared by measuring the stabilizers to project to the code space. The green line shows the performance of this approach for the color code. A state preparation based on preparation+verification is also benchmarked. This latter method provides higher fidelities for the X and Z basis eigenstate preparation (blue line), but it is non-FT when preparing a $\ket{{\rm +i}}$ state (orange line). For that state, the syndrome extraction scheme has to be used, incurring in a performance cost.
• Figure 4: Syndrome extraction circuits for the color code. a) Syndrome extraction circuit for a weight-4 $X-$type stabilizer with a single ancilla qubit. A single error happening in the ancilla qubit can propagate to multiple data qubits, causing an uncorrectable error, as indicated by the red wires. b) Flagged syndrome extraction circuit. An error in the ancilla qubit could propagate to two data qubits causing a non-correctable error. By adding a flag qubit, the weight-2 error can be detected and corrected. c) Superdense syndrome extraction circuit: fault tolerant circuit for simultaneous extraction of $X$ and $Z$ stabilizers, using an ancilla Bell pair gidney2023newcircuitsopensource. Any dangerous error happening during the circuit is detected by either ancilla qubit.
• Figure 5: Small color code syndrome extraction performance: performance of different syndrome extraction circuits for the $d=3$ color code. Logical error rates are computed under standard circuit level noise, using a lookup table decoder. Sequential stabilizer readout has a huge performance impact due to idle errors, and it is advisable to measure all stabilizers in parallel if enough ancilla qubits are available. For small $d=3$ color codes, the superdense syndrome extraction scheme provides no advantage to the simultaneous flagged readout, due to the latter protocol switching to an un-flagged syndrome extraction as soon as an error is detected.