PaperlayerView exact revision on arXiv

Source-linked AI summary

Roadmap to fault tolerant quantum computation using topological qubit arrays

David Aasen, Morteza Aghaee, Zulfi Alam, Mariusz Andrzejczuk, Andrey Antipov, Mikhail Astafev, Lukas Avilovas, Amin Barzegar, Bela Bauer, Jonathan Becker, Juan M. Bello-Rivas, Umesh Bhaskar, Alex Bocharov, Srini Boddapati, David Bohn, Jouri Bommer, Parsa Bonderson, Jan Borovsky, Leo Bourdet, Samuel Boutin, Tom Brown, Gary Campbell, Lucas Casparis, Srivatsa Chakravarthi, Rui Chao, Benjamin J. Chapman, Sohail Chatoor, Anna Wulff Christensen, Patrick Codd, William Cole, Paul Cooper, Fabiano Corsetti, Ajuan Cui, Wim van Dam, Tareq El Dandachi, Sahar Daraeizadeh, Adrian Dumitrascu, Andreas Ekefjärd, Saeed Fallahi, Luca Galletti, Geoff Gardner, Raghu Gatta, Haris Gavranovic, Michael Goulding, Deshan Govender, Flavio Griggio, Ruben Grigoryan, Sebastian Grijalva, Sergei Gronin, Jan Gukelberger, Jeongwan Haah, Marzie Hamdast, Esben Bork Hansen, Matthew Hastings, Sebastian Heedt, Samantha Ho, Justin Hogaboam, Laurens Holgaard, Kevin Van Hoogdalem, Jinnapat Indrapiromkul, Henrik Ingerslev, Lovro Ivancevic, Sarah Jablonski, Thomas Jensen, Jaspreet Jhoja, Jeffrey Jones, Kostya Kalashnikov, Ray Kallaher, Rachpon Kalra, Farhad Karimi, Torsten Karzig, Seth Kimes, Vadym Kliuchnikov, Maren Elisabeth Kloster, Christina Knapp, Derek Knee, Jonne Koski, Pasi Kostamo, Jamie Kuesel, Brad Lackey, Tom Laeven, Jeffrey Lai, Gijs de Lange, Thorvald Larsen, Jason Lee, Kyunghoon Lee, Grant Leum, Kongyi Li, Tyler Lindemann, Marijn Lucas, Roman Lutchyn, Morten Hannibal Madsen, Nash Madulid, Michael Manfra, Signe Brynold Markussen, Esteban Martinez, Marco Mattila, Jake Mattinson, Robert McNeil, Antonio Rodolph Mei, Ryan V. Mishmash, Gopakumar Mohandas, Christian Mollgaard, Michiel de Moor, Trevor Morgan, George Moussa, Anirudh Narla, Chetan Nayak, Jens Hedegaard Nielsen, William Hvidtfelt Padkær Nielsen, Frédéric Nolet, Mike Nystrom, Eoin O'Farrell, Keita Otani, Adam Paetznick, Camille Papon, Andres Paz, Karl Petersson, Luca Petit, Dima Pikulin, Diego Olivier Fernandez Pons, Sam Quinn, Mohana Rajpalke, Alejandro Alcaraz Ramirez, Katrine Rasmussen, David Razmadze, Ben Reichardt, Yuan Ren, Ken Reneris, Roy Riccomini, Ivan Sadovskyy, Lauri Sainiemi, Juan Carlos Estrada Saldaña, Irene Sanlorenzo, Simon Schaal, Emma Schmidgall, Cristina Sfiligoj, Marcus P. da Silva, Shilpi Singh, Sarat Sinha, Mathias Soeken, Patrick Sohr, Tomas Stankevic, Lieuwe Stek, Patrick Strøm-Hansen, Eric Stuppard, Aarthi Sundaram, Henri Suominen, Judith Suter, Satoshi Suzuki, Krysta Svore, Sam Teicher, Nivetha Thiyagarajah, Raj Tholapi, Mason Thomas, Dennis Tom, Emily Toomey, Josh Tracy, Matthias Troyer, Michelle Turley, Matthew D. Turner, Shivendra Upadhyay, Ivan Urban, Alexander Vaschillo, Dmitrii Viazmitinov, Dominik Vogel, Zhenghan Wang, John Watson, Alex Webster, Joseph Weston, Timothy Williamson, Georg W. Winkler, David J. van Woerkom, Brian Paquelet Wütz, Chung Kai Yang, Richard Yu, Emrah Yucelen, Jesús Herranz Zamorano, Roland Zeisel, Guoji Zheng, Justin Zilke, Andrew Zimmerman

arXiv:2502.12252v3quant-phcond-mat.supr-con

TL;DR

The paper addresses how to build fault-tolerant quantum computation with topologically protected Majorana-based qubits. It develops a staged tetron-device roadmap and associated measurement-based protocols, culminating in logical-qubit error-correction demonstrations using lattice surgery. The roadmap connects single-qubit benchmarking, braiding, error detection, and scalable error correction within one architecture.

  • Problem

    The paper asks how tetrons can provide a concrete path from Majorana-based qubit demonstrations to fault-tolerant quantum computation using measurement-based error-correction codes.

  • Method

    The authors specify progressively larger tetron devices and protocols for benchmarking, measurement-based braiding, quantum error detection, and lattice-surgery-based error correction.

  • Results

    The roadmap identifies device generations from single-qubit benchmarking through two-qubit braiding and eight-qubit logical improvement to a tetron array supporting two logical qubits.

  • Takeaways & Limitations

    Measurement-based Pauli operations and tetron-specific codes provide the architecture’s route toward scalable fault-tolerant computation, with non-Clifford operations supplied separately by a T-gate.

Abstract

from arXiv · show

We describe a concrete device roadmap towards a fault-tolerant quantum computing architecture based on noise-resilient, topologically protected Majorana-based qubits. Our roadmap encompasses four generations of devices: a single-qubit device that enables a measurement-based qubit benchmarking protocol; a two-qubit device that uses measurement-based braiding to perform single-qubit Clifford operations; an eight-qubit device that can be used to show an improvement of a two-qubit operation when performed on logical qubits rather than directly on physical qubits; and a topological qubit array supporting lattice surgery demonstrations on two logical qubits. Devices that enable this path require a superconductor-semiconductor heterostructure that supports a topological phase, quantum dots and coupling between those quantum dots that can create the appropriate loops for interferometric measurements, and a microwave readout system that can perform fast, low-error single-shot measurements. We describe the key design components of these qubit devices, along with the associated protocols for demonstrations of single-qubit benchmarking, Clifford gate execution, quantum error detection, and quantum error correction, which differ greatly from those in more conventional qubits. Finally, we comment on implications and advantages of this architecture for utility-scale quantum computation.

I. INTRODUCTION: FAULT-TOLERANT QUANTUM COMPUTATION USING TETRONS

The paper presents a tetron roadmap that incrementally develops measurement-based operations and error-correction capabilities toward fault-tolerant quantum computing. Tetrons use Majorana zero modes, interferometric quantum-dot measurements, and topologically protected Pauli measurements.

  • Motivation: Fault-tolerant quantum-error-correction overhead depends on the number of physical qubits per logical qubit and the depth of the measurement sequence.These overheads are important performance metrics for large-scale fault-tolerant quantum computers.
  • Architecture: Tetrons encode qubits in four Majorana zero modes and use single- and two-qubit Pauli measurements as a native instruction set.This differs from conventional platforms, where such measurements are assembled from multi-qubit Clifford gates and single-qubit measurements.
  • Roadmap: The roadmap scales from a single-qubit benchmarking device through two-qubit braiding and an eight-qubit error-detection demonstration toward a fault-distance-7 tetron array for lattice surgery.The devices progressively build capabilities for scalable quantum error correction.
  • Architecture: Interferometric loops between tetron islands and nearby quantum dots produce measurable quantum-capacitance shifts for microwave parity readout.Two-sided tetrons join parallel topological wires with a trivial superconducting backbone into one island with charging energy.
  • Architecture: Tetrons are predicted to suppress idle and measurement errors exponentially with topological-gap-to-temperature ratio, device-length-to-coherence-length ratio, and measurement signal-to-noise ratio.The resulting low error rates support scalable fault-tolerant computation with topological codes tailored to measurement-based qubits.
  • Protocols: The paper introduces measurement-based qubit benchmarking, measurement-based braiding, tetron-specific error detection, and simulations of logical improvement using pairwise measurement-based codes.The proposed demonstrations use gate-defined 2DEG Majorana nanowires and describe their design elements, operating principles, and dominant error sources.
  • Protocols: Measurement-based benchmarking statistically tests whether X and Z measurements behave as anticommuting projective measurements while assessing both reported outcomes and post-measurement states.The protocol uses sequential same-basis and alternating-basis measurement statistics.

B. Device design

The two-sided tetron uses gate-defined topological nanowires, Majorana zero modes, quantum dots, and tunable tunnel junctions to support interferometric measurements and two-dimensional connectivity.

  • Core architecture: Majorana zero modes occupy the endpoints of the topological wires, while the backbone connects their midpoints into an H-shaped structure.The two-sided geometry places Majorana zero modes at four corners, supporting two-dimensional multi-qubit layouts.
  • Core architecture: The device combines two parallel topological superconducting nanowires, a perpendicular trivial superconducting backbone, five quantum dots, and tunable tunnel junctions.The nanowires can be realized in a 2DEG proximitized by an epitaxial s-wave superconductor such as Al.
  • Topological wires: Gate-defined plunger electrodes tune the superconducting wires, which are designed for single-subband operation and reduced residual Majorana coupling.For similar material stacks, wire lengths of 3–5 µm are identified as meeting the relevant design criteria.
  • Topological wires: The device requires a low-disorder operating range in magnetic field and gate voltage where both horizontal wires remain topological without introducing low-energy states in the trivial backbone.The horizontal-wire separation is approximately 1 µm, set by quantum-dot requirements.
  • Quantum-dot layout: Quantum dots adjacent to each Majorana endpoint and a longer dot parallel to the lower wire provide the coupling paths needed for interferometric measurement loops.The architecture is related to other proposed Majorana platforms, including magnetic-atom chains and superconducting heterostructures.

2. Operating principles and dominant error sources

Tetrons encode qubits in four Majorana zero modes and use interferometric Pauli measurements, whose performance is shaped by readout errors, state changes, and residual unwanted coupling.

  • Qubit encoding: A tetron encodes a qubit in four Majorana zero modes with fixed total parity, using Z = iγ1γ2 and X = iγ1γ3 as Pauli operators.The associated basis choice is conventional because topological qubits have near-degenerate ground states.
  • Measurement principle: X and Z measurements form interferometric loops between Majorana modes and quantum dots, with parity inferred from a microwave-detected quantum-capacitance shift.The state-dependent dot-spectrum shift changes the resonator frequency coupled to a nearby gate.
  • Dominant errors: Classical readout errors are governed by the measurement signal-to-noise ratio, while quantum-state errors increase when longer measurement times allow changes during measurement.An interferometric parity measurement reported SNR 0.52 for a 1 µs measurement time.
  • Control trade-offs: Detuning-based control suppresses the unwanted loop coupling only by approximately t^2/E_QD, allowing charge-noise fluctuations to limit coherence and measurement-basis orthogonality.Cutter-based control can instead make the residual coupling exponentially small in tunnel-barrier width; Y measurement requires cutter-based control in the illustrated device.

III. MEASUREMENT-BASED BRAIDING TRANSFORMATIONS

Measurement-based braiding uses auxiliary qubits and Pauli measurements to realize single-qubit Clifford operations while preserving topological protection and enabling tomography-based validation.

  • Protocol motivation: A two-qubit system, with one computational and one auxiliary qubit, is required for measurement-based execution of the full single-qubit Clifford group.The paper proposes measurement sequences and a device design supporting these operations, then simulates their fidelity under a tetron-motivated noise model.
  • Protocol motivation: The architecture performs topologically protected operations through single- and two-qubit Pauli measurements rather than physically moving or adiabatically coupling anyons.Direct physical Clifford gates are not required for scalable computation in the described architecture, where logical Clifford operations use lattice surgery.
  • Parity constraint: The measurement sequences preserve each tetron’s fixed fermion parity, so every measurement must involve an even number of Majorana operators on each island.This parity constraint follows from the charging energy of the superconducting island.
  • S-gate protocol: The phase gate S can be generated, up to a Pauli correction, by a four-measurement sequence equivalent to braiding Majorana modes 1 and 2 on the computational qubit.The sequence contains only one two-qubit measurement, and its probabilistic Pauli correction can be tracked in software.
  • Validation: Gate-set tomography reconstructs the noisy action of each measurement sequence, separates state-preparation and measurement errors with reference experiments, and compares the resulting superoperator with the ideal gate.The single-qubit Clifford group has 24 elements but six Pauli equivalence classes, for which optimized sequences are considered.

B. Two-qubit device design

The two-qubit device extends the tetron design with a coherent link supporting cross-tetron measurement loops, while its simulated gate fidelity depends on single-qubit and readout errors and faces correlated-error risks.

  • Device architecture: Two vertically stacked tetrons support all single-qubit Pauli measurements plus the two-qubit measurements MZZ, MYY, MYZ, and MZY.These operations are implemented through two-qubit measurement loops in the device layout.
  • Device architecture: The single-qubit device’s longer quantum dot is replaced by a coherent floating topological-wire link between the tetrons.The link facilitates measurements involving Majorana modes on opposite sides of a tetron, including single-qubit Y and Z measurements.
  • Noise analysis: The simulated S-gate fidelity is evaluated versus single-qubit Pauli error probability p1 and classical assignment error probability pa at fixed two-qubit Pauli error probability p2 = 0.1.Because the sequence contains only one two-qubit measurement, the fidelity is only weakly dependent on p2.
  • Design rationale: Replacing the long quantum dot with a coherent link permits longer topological wires and smaller Majorana energy splitting, which helps increase qubit lifetime.The coherent link is governed by the topological gap rather than the decreasing level spacing of a longer quantum dot.
  • Noise analysis: Two-qubit measurements can introduce correlated errors when tetrons exchange an electron through Majorana modes during measurement, changing the final island charge states.These state errors are modeled as Pauli errors in the noise analysis.
  • Array extension: The 4 × 2 array schematic connects qubit columns with a double rail of quantum dots, while the idle-ladder syndrome circuit alternates XX and ZZ measurement steps.The instantaneous stabilizer group includes the just-measured two-qubit operators and the inferred Y Y Y Y product.

IV. QUANTUM ERROR DETECTION IN A 4 × 2 ARRAY OF TETRONS

The ladder code provides a measurement-based route to quantum error detection using one- and two-qubit Pauli measurements. An eight-qubit 4 × 2 tetron device extends this circuit to measure ZZ between two logical qubits.

  • Code choice: Hastings-Haah Floquet codes use one- and two-body measurements to extract error syndromes, matching tetron capabilities.The ladder code uses the same two-qubit measurements as scalable Hastings-Haah Floquet codes.
  • Ladder code: The ladder code is defined on an N × 2 tetron array with XX measurements between columns and YY and ZZ measurements between rows.For N = 2, it is equivalent to the d = 2 Bacon-Shor error-detecting code.
  • Error detection: The 2 × 2 idle circuit alternates horizontal XX and vertical ZZ measurements, while changes in stabilizer outcomes indicate errors.The instantaneous stabilizer group combines the all-qubit Y product with the measurements performed at each step.
  • Logical measurement: The 4 × 2 device stacks two logical qubits and adds measurement steps that entangle them for a logical ZZ measurement.The circuit performs the idle X and Z steps, followed by an additional X step and a new Y step.

C. Decay experiment and simulations

The decay experiment compares physical and ladder-code-encoded ZZ measurements by fitting logical-error decay under a parameterized noise model. Simulations identify when encoding yields logical improvement and outline requirements for scalable implementations.

  • Decay experiment: The experiment compares repetition-code decay using direct physical ZZ measurements with decay using repetition code concatenated with the ladder code.Runs initialize a logical state, repeat the ZZ circuit, post-select on unchanged stabilizers, and characterize the final state.
  • Performance metric: Average logical improvement Λ is extracted from the ratio of average physical-circuit decay rate to logical-circuit decay rate.Decay rates are obtained by varying the number of rounds and fitting the results to exponentials.
  • Noise simulations: The simulations vary single-qubit Pauli error probability p1, two-qubit Pauli error probability p2, and assignment error pa.Figure 8 maps the parameter region in which quantum error detection produces logical improvement.
  • Noise conditions: There is generally no improvement when p1 = p2 = p because the circuits are not fault-tolerant for circuit-level noise.The authors expect p2 < p1 for tetron qubits in practice.
  • Scalable outlook: The scalable proposal uses a 2 × 1 logical-qubit tetron array with fault distance 7, requiring 13 × 13 tetrons per patch and supporting sparse-failure mitigation.The equivalent 4.8.8 Hastings-Haah implementation contains 196 tetrons.
  • Utility-scale implications: The architecture targets high density, microsecond physical operations, exponentially reduced error mechanisms, and simplified digital control.A tetron is estimated to occupy roughly 5 µm × 3 µm, while protected operations benefit from ratios such as Δ/kBT and L/ξ.

Appendix A: Additional aspects of MBQB

The MBQB appendix formalizes measurement instruments and operational tests for characterizing tetron measurements. It uses random measurement sequences, reset operations, conditional probabilities, and restricted tomography to quantify behavior.

  • Measurement formalism: Measurement instruments are non-deterministic linear transformations whose compositions describe sequences of Pauli measurements.The outcome probability is obtained from the trace of the corresponding instrument acting on the input state.
  • Operational metrics: Conditional probabilities quantify the outcome of one Pauli measurement immediately after another measurement and its recorded outcome.The formalism considers X and Z instruments with outcomes ±1.
  • Reset operation: A reset operation is formed by applying approximately non-commuting measurements without conditioning on outcomes, producing an outcome-averaged state independent of the input.This operation approximates the maximally mixed state in the ideal case.
  • Experimental protocol: Random instrument sequences can estimate operational errors by averaging over state preparations and maximizing over measurement bases.Fair-sampling sequences, including de Bruijn constructions, allow compact sequences to be repeatedly cycled.
  • Tomography: Gate-set tomography reconstructs only the rebit-preserving part when experiments are restricted to X and Z measurements without ancillas.Complete-positivity constraints can bound remaining matrix elements and broader error metrics.

Appendix B: Noise model for tetrons

The tetron noise model combines assignment, single-qubit Pauli, and two-qubit Pauli errors with mechanisms tied to measurement, residual coupling, charge noise, and pulse dynamics. It also relates material and device parameters to target effective error rates.

  • Noise channels: The effective tetron noise model includes assignment errors, single-qubit Pauli errors, and two-qubit Pauli errors.These channels represent measurement bit flips and errors associated with the qubit state and two-qubit operations.
  • Circuit model: Measurement and idle noise channels are applied during single- and two-qubit measurement steps, with measurement outcomes affected by state flips.The two-qubit channel implicitly applies the single-qubit channel to both qubits.
  • Physical mechanisms: Residual Majorana coupling produces Pauli errors through charge noise, while coupling through quantum dots is exponentially suppressed by tunnel-barrier width.The relevant error basis depends on whether coupling occurs through the topological wire or quantum dots.
  • Correlated errors: Two-qubit Pauli errors can arise from electron exchange during two-qubit measurements and can be tracked using heralding measurements.With heralding, p2 is approximated by heralding-measurement error and is expected to be smaller than p1.
  • Coherent errors: The Pauli-twirl approximation maps small coherent rotations into effective single-qubit Pauli errors of order O(Θ^2).The approximation is used for quantum-error-correction syndrome-extraction circuits.
  • Parameter requirements: Target effective error rates of 10^-4 require approximately L/ξ ≳ 20 and Δ/kBT ≳ 12 under the stated assumptions.The lifetime scaling is used to derive the gap-to-temperature bound.
  • Pulse constraints: Measurement pulses must be diabatic relative to εres and adiabatic relative to the measurement avoided crossing and topological gap.The practical pulse-duration constraint is that pulses should be slower than approximately 1 ns.

Appendix C: Tetron coherence times

Tetron coherence cannot be characterized by conventional T1 and T2 definitions alone because ideal idle states are degenerate and realistic residual couplings select a basis. Repeated X- or Z-basis measurements with inserted idle times instead extract basis-dependent coherence times, while measurement errors remain distinct from idle lifetimes.

  • Ideal topological qubits have degenerate idle states, making conventional basis-dependent T1 and T2 definitions arbitrary.Residual Majorana coupling in realistic tetrons produces a small energy splitting and selects a diagonalized basis.
  • Residual coupling between Majorana zero modes creates an energy splitting that approximately aligns the diagonalized basis with the same-wire Pauli operator.
  • Repeated X- or Z-basis measurements with intervening idle times extract coherence times in different measurement bases.
  • Tetron measurement errors are not directly related to idle qubit lifetimes, unlike coherence-limited operations in many conventional platforms.

Appendix D: Non-Clifford operations

The appendix describes non-Clifford T-state preparation by dynamically coupling Majorana modes, while measurement sequences and diagrammatic calculus implement protected Clifford transformations. The physical T-gate is timing-sensitive and therefore requires a looser error target than topologically protected operations.

  • Non-Clifford operations: A T-state can be prepared by coupling two Majorana zero modes for a controlled duration, adding a dynamical phase to their Hamiltonian.The coupling uses paths analogous to those employed for Pauli measurement loops.
  • Non-Clifford operations: When φ = π/8, the pulse sequence implements a T-gate.
  • Non-Clifford operations: The T-gate phase is sensitive to the precise pulse sequence, so topological physical error targets do not apply directly to this rotation.
  • Non-Clifford operations: A physical T-state error rate of 0.01–0.05 is sufficient for the overheads discussed in the cited distillation work because T-state injection is infrequent.
  • Non-Clifford operations: Higher-fidelity or less timing-sensitive T-state protocols trade greater protocol complexity against possible spacetime-overhead reductions.
  • Clifford operations: Measurement sequences realize single-qubit Clifford transformations by tracking outcomes and applying corresponding Pauli corrections.

1. Simulation details

The simulations model noisy measurement sequences with full density-matrix trajectories, outcome-dependent corrections, and process tomography. They also describe stabilizer-based ladder-code circuits for idle operation and logical ZZ measurement.

  • Simulation details: Exact simulations propagate full density matrices through noisy quantum-instrument circuits while tracking every measurement-outcome trajectory.The effective noise model is parameterized by pa, p1, and p2.
  • Simulation details: Tracing out the auxiliary qubit and applying outcome-dependent Pauli corrections produces the computational-qubit state associated with the target Clifford operation.
  • Simulation details: Process tomography extracts the noise channel and averages its overlap with the ideal Clifford gate to obtain the fidelity F[S].
  • Ladder-code circuits: In the 2 × 2 ladder code, stabilizer-eigenvalue changes identify errors while horizontal ZZ and vertical XX operators generate the logical qubit.
  • Ladder-code circuits: The ladder-code circuit implements a logical ZZ measurement by inferring products of stabilizers from outcomes across successive measurement steps.

2. Repetition code decay experiment

The repetition-code experiment compares physical and logical ZZ measurements using post-selected decay experiments and exact noisy simulations. Logical improvement is expected over a broader parameter region for the higher-weight X operator than for the Z operator.

  • Repetition-code decay experiment: The comparison runs a repetition code directly on two physical qubits against the same code concatenated with a 2 × 2 ladder code.
  • Repetition-code decay experiment: The concatenated code gives the logical X operator two additional physical qubits and the logical Z operator one additional physical qubit relative to their physical counterparts.
  • Repetition-code decay experiment: Post-selected decay experiments prepare distinct initial states to detect eZ through XX and eX through ZI.
  • Repetition-code decay experiment: Average logical improvement compares repetition-code performance on physical and logical qubits across both initial conditions.
  • Simulation details: Exact eight-qubit density-matrix simulations track noisy measurement trajectories and compute post-selected acceptance rates for the decay protocol.
  • Results: The ratio of physical to logical decay rates is evaluated separately for eX and eZ, with a larger logical-improvement region expected for eX.
Loading 2502.12252v3…