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Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes

Torsten Karzig, Christina Knapp, Roman M. Lutchyn, Parsa Bonderson, Matthew B. Hastings, Chetan Nayak, Jason Alicea, Karsten Flensberg, Stephan Plugge, Yuval Oreg, Charles M. Marcus, Michael H. Freedman

arXiv:1610.05289v4cond-mat.mes-hallcond-mat.supr-conquant-ph

TL;DR

The paper addresses how to build scalable Majorana quantum computers without quasiparticle braiding, T-junctions, or interferometric topological-charge measurements. It proposes Coulomb-blockaded hexon and tetron architectures operated through quantum-dot-assisted parity measurements, while identifying measurement generality as an open design trade-off.

  • Problem

    Topological quantum computation must overcome the difficulty of engineering non-Abelian phases, moving quasiparticles, and unambiguously measuring topological charge.

  • Method

    The paper designs modular arrays of four- or six-MZM Coulomb-blockaded islands and uses quantum dots and interferometry to perform the parity measurements required for measurement-only computation.

  • Results

    The proposed architectures support measurement-based Clifford operations, two-qubit entangling operations, and preparation of precise magic states while suppressing quasiparticle poisoning through charging energy.

  • Takeaways & Limitations

    The designs provide scalable two-dimensional architectures that leverage MZM-based topological information processing rather than active error correction.

  • Takeaways & Limitations

    The benefit of allowing more general measurements remains an open trade-off because they may reduce total measurement counts but require further investigation.

Abstract

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We present designs for scalable quantum computers composed of qubits encoded in aggregates of four or more Majorana zero modes, realized at the ends of topological superconducting wire segments that are assembled into superconducting islands with significant charging energy. Quantum information can be manipulated according to a measurement-only protocol, which is facilitated by tunable couplings between Majorana zero modes and nearby semiconductor quantum dots. Our proposed architecture designs have the following principal virtues: (1) the magnetic field can be aligned in the direction of all of the topological superconducting wires since they are all parallel; (2) topological $T$-junctions are not used, obviating possible difficulties in their fabrication and utilization; (3) quasiparticle poisoning is abated by the charging energy; (4) Clifford operations are executed by a relatively standard measurement: detection of corrections to quantum dot energy, charge, or differential capacitance induced by quantum fluctuations; (5) it is compatible with strategies for producing good approximate magic states.

I. INTRODUCTION

The paper proposes scalable Majorana-based architectures that replace quasiparticle motion and interferometric charge measurements with measurement-only protocols using Coulomb-blockaded MZM islands and quantum dots. The designs use hexon or tetron modules, support Clifford operations and entanglement, and are intended to suppress poisoning while retaining long coherence for low-depth computations.

  • I. INTRODUCTION: Charging energy suppresses quasiparticle poisoning, while residual quasiparticle-excitation errors are exponentially suppressed by Δ/T.For a hexon, poisoning events are suppressed as exp(−EC/T), and such events constitute leakage from the four-dimensional computation subspace.
  • I. INTRODUCTION: Measurement-only protocols remove the need for coherent topological T-junctions and avoid physically moving computational quasiparticles.A sequence of measurements reproduces braiding operations without transporting quasiparticles.
  • I. INTRODUCTION: The architectures encode qubits in parallel topological superconducting wires assembled into Coulomb-blockaded units, using joint fermion-parity measurements instead of quasiparticle braiding.Five designs are analyzed: three hexon architectures with six MZMs per island and two tetron architectures with four.
  • I. INTRODUCTION: The architectures are evaluated by poisoning-protection time, signal visibility, fabrication simplicity, and computational efficiency.The charging-energy and visibility axes scale approximately with EC and E_C^-1, respectively.
  • I. INTRODUCTION: A charging-energy trade-off sets an optimal wire length because increasing L raises topological protection while increasing capacitance lowers EC.For the one-sided hexon, the combined protection is estimated to maximize roughly when EC/T ≈ 2L/ξ, with an experimentally accessible regime EC/T ∼ L/ξ ≫ 1.
  • I. INTRODUCTION: Arbitrary MZM exchanges within a hexon plus one entangling operation between adjacent hexon pairs provide Clifford completeness, while precise magic states can extend the scheme to universal computation.The hexon encodes a logical qubit in four MZMs and uses an ancillary MZM pair.

III. MAJORANA MEASUREMENTS

The measurement scheme couples MZMs to tunable quantum dots so that virtual electron tunneling shifts dot observables according to MZM parity. Coulomb blockade suppresses real charge transfer and environmental noise during idle periods, but measurement visibility depends on coherent, phase-sensitive tunneling.

  • III. MAJORANA MEASUREMENTS: Selective gate tuning couples two or more MZMs to nearby quantum dots, enabling projective measurements of their fermion parity through parity-dependent energy shifts.The device geometries include one dot for two-MZM parity and two dots for four-MZM parity.
  • III. MAJORANA MEASUREMENTS: At T ≪ EC, electron transfers between the dot and MZM island are virtual because real charged states are exponentially suppressed by the island charging energy.The relevant low-energy analysis assumes a spinless dot and tunneling amplitudes smaller than the superconducting gap and charging energy.
  • III. MAJORANA MEASUREMENTS: Noise coupled to quantum-dot charge does not affect the MZM island when the tunnel couplings are off, so idle periods do not collapse the qubit state.This protection applies outside active measurement intervals; final dot occupancy can still differ with a small, charging-energy-suppressed probability.
  • III. MAJORANA MEASUREMENTS: Parity dependence arises from elastic cotunneling through both coupled MZMs, whereas coupling to only one MZM produces a parity-independent correction.The parity-sensitive correction scales with the imaginary part of t1* t2/EC near charge degeneracy.
  • III. MAJORANA MEASUREMENTS: The measurement requires generic complex tunneling amplitudes because parity dependence disappears when both relevant amplitudes are real.An accidental artificial anti-unitary symmetry can make the parity signal weak even though microscopic time-reversal symmetry is broken.

B. Projective measurement of four-MZM parity

Four-MZM parity measurement uses two quantum dots coupled across two superconducting islands, with virtual tunneling paths forming a loop whose energy spectrum depends only on the joint parity. The same principle generalizes to larger even-MZM measurements, although visibility decreases as more MZM pairs are added.

  • B. Projective measurement of four-MZM parity: The four-MZM spectrum contains parity-independent states and parity-dependent hybridized states associated with dot occupancies (1, 0) and (0, 1).The parity dependence enters through the off-diagonal hybridization and the resulting square-root eigenvalue splitting.
  • B. Projective measurement of four-MZM parity: Two quantum dots coupled to four MZMs on two islands produce parity-dependent hybridization of the singly occupied dot states.The analysis treats the two islands as having potentially different charging energies and induced charges, with mutual dot charging energy included when appreciable.
  • B. Projective measurement of four-MZM parity: The parity-sensitive energy term depends on p = p12p34 because fermions can tunnel around the entire four-MZM loop.Partial paths backtrack and contribute only individual-pair parity factors, while the closed loop yields the joint parity.
  • B. Projective measurement of four-MZM parity: Parity sensitivity is strongest when the two dot states are resonant and charge fluctuations are largest, motivating operation near ng,1 = ng,2.The resonant singly occupied states should lie below the (0, 0) and (1, 1) states.
  • B. Projective measurement of four-MZM parity: A single-dot variant can also encode four-MZM parity dependence, but it sacrifices tunability and may introduce low-lying excited states in the connecting wire.Its practical value depends on whether the simpler geometry is substantially easier to realize.
  • B. Projective measurement of four-MZM parity: The same closed-loop mechanism can measure the joint parity of any even number of MZMs, but each added MZM pair reduces measurement visibility.The architecture therefore favors measurements involving the smallest feasible number of MZMs.

C. Experimental proposals for MZM parity measurements

The paper proposes three parity-measurement approaches based on parity-dependent shifts in quantum-dot energy, charge, and differential capacitance. Charge and capacitance detection are presented as experimentally accessible and compatible with scaling, while resonator spectroscopy is fast but faces two-dimensional-layout challenges.

  • Measurement approaches: Parity-dependent shifts in quantum-dot energy, average charge, and differential capacitance enable projective measurements of two- or four-MZM parity.The proposed readouts detect changes in ground-state properties induced by joint MZM parity.
  • Energy spectroscopy: In the double-dot scheme, the hybridized one-electron states have parity-dependent energies, whereas the (0,0) and (1,1) states remain parity independent.The relevant regime uses one electron shared between the two dots and weakly occupied excited states.
  • Energy spectroscopy: Resonator spectroscopy converts the parity-dependent ground-state energy into a frequency shift estimated at Δω ∼100 MHz, with measurement times on the order of 1 µs.The estimate assumes comparable tunneling amplitudes and realistic parameters; operation in large magnetic fields requires technological adaptation.
  • Scaling considerations: Resonator spectroscopy may become problematic in a two-dimensional qubit array because coupling required out-of-plane resonators to planar qubits remains an open experimental problem.The technique is described as suitable for a small number of qubits but potentially difficult to scale in the proposed planar layout.
  • Charge sensing: Quantum-dot charge measurements distinguish parity because quantum charge fluctuations broaden the charge step differently for even and odd joint fermion parity.The charge can be measured accurately at low temperatures, reaching roughly 10^-3 e/√τint, and the approach is compatible with large magnetic fields and two-dimensional scaling.
  • Differential-capacitance sensing: Differential capacitance encodes parity through the curvature of the ground-state energy and can be measured by rf-reflectometry coupled to an LC circuit.For εC ∼ 1–10 K, the predicted parity-dependent change is δCdiff ∼10^2–10^3 aF, compared with measured differential capacitances of order 10 aF in 40 µs.

IV. CLIFFORD-COMPLETE MAJORANA ARCHITECTURES

The architectures use measurement-only protocols to implement topologically protected multi-qubit Clifford operations without physically braiding Majorana zero modes. Hexons combine a computational qubit with an ancillary Majorana pair, enabling single-qubit and entangling Clifford gates through parity measurements.

  • Measurement-only protocols implement the complete set of multi-qubit Clifford gates in a topologically protected manner.The protocol replaces physical exchanges with sequences of projective fermion-parity measurements.
  • A hexon uses six Majorana zero modes: four encode the computational qubit and two form an ancillary pair.The ancillary pair has fixed even fermion parity, while the outer pairs define the computational basis.
  • Quantum dots are tunably coupled to Majorana modes to perform the parity measurements required by the protocols.Forced-measurement procedures can repeat alternating measurements until the desired outcome is obtained.
  • The full single-qubit Clifford group follows from intra-hexon braiding transformations generated by parity-measurement sequences.The transformations R(12) and R(25) suffice, with the Hadamard gate given by H = R(12)R(25)R(12).
  • Two-hexon parity measurements provide entangling Clifford gates while preserving each island’s fermion parity.This preserves compatibility with quasiparticle-poisoning protection from Coulomb charging energy.

2. One-sided Hexon

The one-sided hexon uses quantum-dot-mediated parity measurements and a long, narrow geometry to combine Clifford-complete control with topological protection. Its protection depends on strong hybridization at the backbone side and coherent coupling across the device width.

  • The one-sided hexon supports arbitrary intra-hexon pair-parity measurements and four-Majorana measurements between neighboring hexons.Together, these measurements provide more than enough operations for Clifford completeness.
  • The device should satisfy L ≫ w while the width remains below the effective quantum-dot coherence length.The long geometry supports topological protection and charging-energy-based suppression of quasiparticle-poisoning errors, while quantum-dot coherence limits vertical separation.
  • When backbone-side Majorana modes strongly hybridize, the one-sided hexon has a protection scale corresponding to separation 2L.This requires the vertical separation of the relevant topological superconducting wires to be less than ξ.
  • If backbone-side hybridization splittings are below temperature, the effective protection length is reduced from 2L to L.Weak symmetry breaking or disorder can produce low-energy backbone degrees of freedom that cause this reduction.

3. Two-sided hexon

The alternative hexon architectures trade geometry and fabrication constraints against charging energy and topological protection. Two-sided and linear hexons retain measurement-based Clifford capabilities, while tetrons reduce the Majorana count but require additional operational resources.

  • 3. Two-sided hexon: Two-sided hexons require long coherent links between their left and right sides to perform cross-side pair-parity measurements.A single link of length L per hexon can suffice for arbitrary two-Majorana measurements, given the side connectivity.
  • 3. Two-sided hexon: Two-sided hexons have rectangular connectivity and protection scale L, so they are generally more elongated than one-sided hexons.For comparable protection, their charging energy is expected to be roughly half as large, and disorder can reduce the scale to L/2.
  • 4. Linear hexon: Linear hexons use one wire divided into three topological and two normal-superconducting segments, avoiding additional superconducting backbones.All intra-hexon two-Majorana measurements remain possible, providing single-qubit Clifford completeness.
  • 4. Linear hexon: Linear hexons have protection scale L/5 and therefore require much larger parent-wire length than the other hexon designs.The construction also leads to the smallest expected hexon charging energy among the compared hexon architectures.
  • B. Tetron architectures: Tetrons use four Majorana modes, but their missing ancillary pair prevents topologically protected single-qubit Clifford gates on an isolated tetron.Their more limited measurements require additional resources and operations, creating a trade-off against minimizing measurement types.

2. Linear tetron

Linear tetrons use one wire divided into two topological segments, yielding four MZMs, and can be arranged with links and quantum dots for measurement-only operations.

  • Linear tetron: A linear tetron divides one 1DTS wire with a normal superconducting middle segment, leaving two topological segments and four MZMs.The MZMs lie at the wire ends and at the two topological–normal boundaries.
  • Architecture: Linear tetrons are arranged in a rectangular array with vertical rows of coherent links between neighboring tetron rows.Three links span the length of one tetron.
  • Measurements: Vertical neighboring tetrons support joint Pauli measurements through quantum-dot couplings to corresponding MZMs and capacitance or charge readout.The protocol includes Z(j,k)Z(j,k+1), X(j,k)X(j,k+1), and Y(j,k)Y(j,k+1).
  • Measurements: Horizontal neighboring tetrons require coherent links to couple more distant MZMs through effective quantum dots.Two coherent links and three quantum dots can form an effective quantum dot whose energy depends on joint parity.
  • Operations: The available two-qubit measurements are sufficient for Clifford-complete operations, while additional links enable single-qubit and longer-range entangling measurements.Without links, the two-sided tetron architecture permits only single-qubit X measurement, but standards still enable Clifford-complete operations.
  • Two-sided tetron: A two-sided tetron joins two 1DTSs with a superconducting backbone positioned away from the four MZMs, facilitating horizontal measurements.Semiconductor structures and links support additional measurement ranges, subject to quantum-dot distance constraints.

C. Design summary

The designs share protections based on field alignment, avoiding T-junctions, charging energy, long wires, and sufficient measurements, but differ in visibility, fabrication, and efficiency trade-offs.

  • Common design principles: The architectures align magnetic fields with parallel 1DTSs, avoid topological T-junctions, use charging energy to suppress QPP, lengthen wires, and provide sufficient measurements.These principles are intended to protect encoded quantum information across the designs.
  • Protection: Error rates from QPP and thermal quasiparticles are exponentially suppressed by EC/T and Δ/T, while MZM-hybridization errors are exponentially small in L/ξ.Measurement fidelity also scales exponentially with the integration time of the measurement.
  • Comparison axes: The comparison considers QPP time, signal visibility, fabrication simplicity, and computational efficiency rather than a definitive quantitative ranking.The authors state that precise performance estimates and rankings are difficult.
  • QPP and visibility: Larger EC improves QPP suppression but reduces measurement visibility, creating a trade-off between shorter and longer qubit-unit lengths.Shorter L favors QPP protection, whereas longer L favors protection against MZM hybridization.
  • Fabrication: Fabrication differences remain qualitative and experimentally unresolved: tetrons are slightly easier than corresponding hexons, while linear designs need more links and larger tuned non-topological regions.Backbone deposition is an additional challenge for one-sided and two-sided hexons.
  • Computational efficiency: Hexons are more computationally efficient than tetrons because tetrons require ancillary units for the full Clifford gate set.In the two-sided tetron without links, standards require 3/4 of tetrons to be ancillary, leaving 1/4 as computational data qubits.

V. UNIVERSAL QUANTUM COMPUTING

The paper extends measurement-only Majorana computation toward universal quantum computing by combining topologically protected Clifford operations with magic-state preparation and distillation. It also identifies architectural trade-offs and experimental directions needed to realize scalable implementations.

  • A. T gate: Clifford operations alone are classically simulable, but adding a single non-Clifford gate such as the T gate enables universal quantum computation.The T gate can be implemented through preparation of an ancillary magic state.
  • A. T gate: Magic states can be distilled using only Clifford operations, converting modest-fidelity inputs into higher-fidelity states for universal computation.The original protocol uses 15 inputs with fidelity 1 − ε, where ε ≲ 0.14, and asymptotically yields fidelity 1 − const × ε^3.
  • A. T gate: The five planar layouts support efficient magic-state-distillation circuits using the measurement combinations permitted by each architecture.For tetron designs, Clifford completeness requires at least half of the tetrons to be available.
  • A. T gate: Approximate magic states can be produced by adapting the same classical control electronics used for Clifford operations and by adiabatically evolving a Majorana qubit around a Bloch-sphere loop.The loop produces a geometric phase equal to the enclosed solid angle, while protected great-circle segments arise when one or two couplings vanish.
  • B. Quantum Error Correction: The proposed architectures combine high-fidelity Clifford gates with magic-state preparation and distillation, potentially supporting many gate operations before decoherence.Low-depth computations with magic-state distillation might work without quantum error correction, whereas large-scale computation still requires an error-correcting superstructure.
  • VI. CONCLUSIONS AND NEAR-TERM DIRECTIONS: Large-scale implementations require further work on error-correction hardware, broader measurement sets, circuit calibration, and algorithms tailored to measurement-based layouts.The paper notes that more general measurements may reduce total measurement counts and that layout-specific algorithms can improve efficiency.
  • VI. CONCLUSIONS AND NEAR-TERM DIRECTIONS: Experiments on floating one-dimensional topological superconductors can test coherent-link behavior and provide evidence for topological superconductivity through π shifts in interference patterns.Two floating nanowires form the arms of an Aharonov–Bohm interferometer, although fixed island parities prevent direct access to topological-qubit properties.

Appendix A: Parity-dependence of MZM island-quantum dot energies

This appendix derives the low-energy Hamiltonian and parity-dependent spectrum of coupled Majorana-island and quantum-dot systems. It shows which states depend on joint Majorana parity, validates perturbation theory against exact diagonalization, and identifies conditions for usable parity crossings.

  • Appendix A: Parity-dependence of MZM island-quantum dot energies: The appendix models two Majorana islands, defines their fermion parities, and derives tunneling Hamiltonians for coupling the islands to quantum dots.The low-energy description projects onto Majorana operators while retaining charging-energy and quasiparticle-excitation contributions.
  • Appendix A: Parity-dependence of MZM island-quantum dot energies: For fixed island parities, four low-energy states occur near the quantum-dot degeneracy point: two parity-independent states and two parity-dependent states.The parity-independent energies are insensitive to parity through second order in t/EC, while the other pair depends on the joint parity.
  • Appendix A: Parity-dependence of MZM island-quantum dot energies: The perturbative parity-dependent energies agree well with numerical exact diagonalization of the full Hamiltonian.The comparison is shown in Fig. 16 for the stated charging energies, Majorana splitting, and tunneling amplitudes.
  • Appendix A: Parity-dependence of MZM island-quantum dot energies: Offset charges and Majorana hybridization shift and asymmetrize the parity-dependent energy crossings.When εM = 0, the parity-dependent crossing shifts so both crossings occur at the same energies, subject to the stated state-accessibility condition.
  • Appendix A: Parity-dependence of MZM island-quantum dot energies: To use the crossing for parity readout, the ground state must correspond to a parity-dependent energy or the parity-independent states must be inaccessible.This condition is especially relevant for four-MZM measurements involving MZMs from two different hexons.

Appendix B: Transmon measurement

The appendix estimates whether a transmon-type dispersive readout can resolve the parity-dependent energy shifts. Using representative device parameters, it finds a frequency shift within transmon sensitivity.

  • Appendix B: Transmon measurement: Approximately 100 MHz, the estimated dispersive frequency shift for the representative parameters, falls within the range of transmon sensitivity.The estimate uses g/2π ≈ 40 MHz, δω/2π ≈ 200 MHz, EC ≈ 160 µeV, and t = 0.2EC.

Appendix C: Measurement procedure and dephasing in a single module with four MZMs

This appendix introduces the noise analysis for charge-sensing measurements in a single four-MZM module. The supplied passage only establishes that the noiseless case is considered first.

  • Appendix C: Measurement procedure and dephasing in a single module with four MZMs: The analysis begins with charge-sensing measurements without noise before addressing the general effect of noise.The appendix focuses on measuring MZM hybridization in the devices shown in Fig. 15(b) and (c).

1. Measurement of the MZM hybridization

The protocol extracts the MZM hybridization Bx from quantum-dot charge measurements after controlled tunneling and free-evolution intervals. At a symmetric tunneling point, the two parity sectors respond differently, producing a Bx-dependent charge expectation.

  • System and protocol: Bx = δE12 + δE34 drives oscillations between the qubit’s two low-energy states, with each hybridization energy exponentially suppressed by MZM separation.The direct γ2–γ3 overlap is neglected in the simplified model.
  • System and protocol: The protocol initializes iγ2γ3, disables tunneling for τ0, enables it for τ1, and remeasures the quantum-dot charge.The measured charge expectation depends on Bx.
  • Charge readout: The charge expectation is obtained by evolving the initialized state through the hybridization and tunneling Hamiltonians, then evaluating the quantum-dot charge matrix elements.The calculation uses leading-order expansions in t/EC.
  • Charge readout: The state’s amplitudes after the initial interval contain cos(Bxτ0) and sin(Bxτ0), so repeated measurements can reveal the hybridization dependence.The two resulting parity sectors acquire different charge responses at the symmetric tunneling point.
  • Parity-selective coupling: At t2 = t3 = t, the primed sector is unaffected by tunneling, while the unprimed sector couples to an excited state.This parity-selective coupling underlies the charge readout.

2. The effect of noise

Charge noise affects the hybridized island–dot system through relaxation and dephasing, but the Bx-dependent charge signal can persist. A suitable measurement-time window suppresses decaying terms while preserving parity resolution.

  • Noise sources: The relevant noise arises from electrostatic coupling between quantum-dot charge and fluctuations in substrate background charges.Turning on tunneling effectively couples the MZM island to a charge qubit.
  • Noise sources: T1 and T2 describe energy and phase relaxation of the hybridized island–dot system, not the topological qubit’s expected longer coherence times.Their values depend on the noise spectral density.
  • Noise robustness: At zero temperature, the charge expectation retains a Bx-dependent term that does not decay with τ1.This persistence distinguishes the readout signal from noise-induced relaxation and dephasing terms.
  • Measurement window: Choose τ1 so that max[T1, T2] ≪ τ1 ≪ Bx^-1, suppressing decaying terms while allowing Hhyb to be neglected during readout.Exceeding the upper bound would average over the two MZM parity states.
  • Measurement window: Varying τ0 reveals a cos^2(Bxτ0) dependence, enabling extraction of the MZM hybridization from repeated charge measurements.The charge measurement projects onto the primed or unprimed sector, whose responses differ.
  • Measurement window: For nearly adiabatic tunneling ramps, the system contains no excited-state component and the charge expectation has no decaying terms.This extends the noise-robust readout beyond ideal step-function switching.

Appendix D: Hexon Details

The appendix derives a two-qubit entangling gate using projections and diagrammatic Majorana relations. It also gives an operator-projector derivation and describes shuttling computational MZMs through the qubit.

  • MZM shuttling: The appendix also illustrates shuttling computational MZMs through the qubit using anyonic teleportation.This appears alongside the projector-based gate construction.
  • Hexon entangling gate: The derivation begins with two-qubit basis states of two hexons, whose outermost-pair fermion parities are labeled a, b ∈ {0, 1}.These labels distinguish even and odd fermion-parity states.
  • Hexon entangling gate: Projecting MZMs 4 and 5 to the vacuum channel transforms the initial two-hexon state into the intermediate superposition used for the gate.The projection can be implemented using forced measurement.
  • Hexon entangling gate: The construction uses the diagrammatic braiding relation of MZMs or Ising anyons, followed by projections involving MZMs 3 and 5 and then 3 and 4.The final projection produces the desired entangling gate.
  • Operator derivation: An alternative derivation multiplies projectors written in terms of Majorana operators and uses their projection properties.This provides an algebraic counterpart to the diagrammatic derivation.
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