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Erasure conversion for singlet-triplet spin qubits enables high-performance shuttling-based quantum error correction

Adam Siegel, Simon Benjamin

arXiv:2601.10461v1quant-ph

TL;DR

Shuttling-related noise and leakage pose challenges for spin-qubit quantum error correction. The paper develops a fault-tolerant framework using singlet-triplet qubits, automatic leakage detection, and leakage-aware decoding. With XZZX, it reports a twofold threshold increase and logical error reductions of several orders of magnitude.

  • Problem

    Shuttling-related phase noise motivates more resilient spin-qubit encodings, but singlet-triplet qubits introduce leakage that the framework must address.

    Source: §I · p. 2, §V · p. 10

  • Method

    The paper develops a fault-tolerant framework with singlet-triplet data qubits, automatic leakage detection, stabilizer circuits, and leakage-aware decoding.

    Source: §1 · p. 7, §V · pp. 10–11

  • Results

    The XZZX approach reaches a 1.3% threshold versus 0.45% for LD encoding, with orders-of-magnitude lower logical error rates when ST shuttling noise is a fraction of LD noise.

    Source: §1 · p. 9, §2 · p. 10

  • Takeaways & Limitations

    The authors identify singlet-triplet dual-spin qubits as a natural realization of erasure qubits in semiconductor platforms.

    Source: §V · p. 10

    The LD–ST comparison uses equal electron counts, which the authors describe as likely a worst-case scenario.

    Source: §2 · p. 10

Abstract

from arXiv · show

Fast and high fidelity shuttling of spin qubits has been demonstrated in semiconductor quantum dot devices. Several architectures based on shuttling have been proposed; it has been suggested that singlet-triplet (dual-spin) qubits could be optimal for the highest shuttling fidelities. Here we present a fault-tolerant framework for quantum error correction based on such dual-spin qubits, establishing them as a natural realisation of erasure qubits within semiconductor architectures. We introduce a hardware-efficient leakage-detection protocol that automatically projects leaked qubits back onto the computational subspace, without the need for measurement feedback or increased classical control overheads. When combined with the XZZX surface code and leakage-aware decoding, we demonstrate a twofold increase in the error correction threshold and achieve orders-of-magnitude reductions in logical error rates. This establishes the singlet-triplet encoding as a practical route toward high-fidelity shuttling and erasure-based, fault-tolerant quantum computation in semiconductor devices.

I. INTRODUCTION

Quantum error correction requires scalable, high-fidelity stabilizer measurements, while shuttling offers semiconductor spin qubits longer-range connectivity. Singlet-triplet encoding may reduce shuttling-related phase noise, but its leakage errors require detection beyond standard stabilizers.

I. INTRODUCTION

  • Repeated stabilizer measurements make fast, high-fidelity, scalable physical operations essential for quantum error correction.Each measurement circuit may be repeated billions of times, and physical failure rates must remain below the code threshold.Source: §I · p. 1
  • 99.99% fidelity per shuttling increment has been demonstrated, and shuttling can extend connectivity and reduce compilation overheads.The cited experiments report high-speed shuttling; longer-range connectivity may also support high-rate error-correcting codes.Source: §I · p. 1
  • Shuttling-based architectures move data or ancilla qubits between operation zones, and some arrange logical qubits as one-dimensional objects.Proposals include Crossbar, Spiderweb, SpinBus, Looped Pipeline, and Snakes on a Plane.Source: §I · p. 1
  • Singlet-triplet qubits are expected to resist long-correlation-length phase noise better than single-spin qubits, but can leak outside their odd-parity computational subspace.Their approximate decoherence-free subspace may also reduce the need to calibrate phase accumulated along a shuttling track.Source: §I · p. 2

A. Loss-DiVincenzo qubits

  • The illustrated protocol transfers information to a new data pair before measuring the old pair, addressing leakage that standard stabilizers do not catch.The figure describes the transfer as both detecting leakage and projecting the new pair back into the computational subspace without measurement feedback.Source: Figure 1 · p. 3

B. Singlet-triplet qubits

Singlet-triplet qubits encode information in a two-electron double quantum dot, with singlet and triplet states forming the computational subspace. Their operations use electrically controlled exchange and magnetic-field-gradient terms, and their two-qubit gates extend familiar spin-qubit operations.

B. Singlet-triplet qubits

  • The ST computational basis uses |T0⟩=|+⟩ and |S⟩=|−⟩, within the odd-parity subspace of a two-electron double quantum dot.The equivalent name “dual-spin qubit” is also used.Source: B. Singlet-triplet qubits · pp. 3–4
  • Single-qubit rotations use exchange coupling J and magnetic-field gradient ΔBz, both controlled electrically.Setting J=ΔEZ implements Hadamard gates in the computational subspace.Source: B. Singlet-triplet qubits · p. 4
  • CZ gates act across one spin from each pair, while CNOT gates use one regular and one anti-controlled CNOT.The anti-controlled operation can use the same gate complexity by adjusting drive frequency, without X gates on the control.Source: B. Singlet-triplet qubits · p. 4, Figure 2 · p. 4
  • The model assumes gate noise mainly comes from imperfect gate parameters, so exchange-based Hadamard and CZ operations retain spin-conservation properties.A noisy CZ cannot leak one targeted ST qubit without leaking the other.Source: B. Singlet-triplet qubits · p. 4
  • Initialization can prepare the singlet state, while the modeled measurement distinguishes singlet, T0, and leaked outcomes.The measurement model does not explore variations in assignment reliability among outcomes.Source: B. Singlet-triplet qubits · p. 4

C. Summary

The section summarizes the ST encoding, its leakage states, relevant gate properties, and the initialization and measurement capabilities assumed in the paper. It also points to a table mapping single-spin Pauli errors to ST-qubit operations.

C. Summary

  • ST qubits can leak from the computational subspace into |T−⟩=|↓↓⟩ or |T+⟩=|↑↑⟩.The computational subspace is the singlet-triplet encoding.Source: C. Summary · p. 4
  • The considered Hadamard and CZ gates are spin-conserving, so Hadamard preserves the even states and noisy CZ cannot leak only one target qubit.The CNOT gate does not share this stated property.Source: C. Summary · pp. 4–5
  • ST qubits can be initialized in the singlet state and measured to distinguish singlet, T0, or leakage.The measurement outcomes are ideally distinguishable.Source: C. Summary · p. 5
  • Table I maps Pauli errors on individual spins of an ST pair to the resulting ST-qubit Pauli operation or leakage.The table uses L to indicate leakage outside the computational subspace.Source: Table I · p. 5

III. FAULT-TOLERANT PROTOCOLS FOR SINGLET-TRIPLET SPIN QUBITS

The memory-mode surface-code framework treats singlet-triplet pairs as a bit-flip base code concatenated with the surface code. It combines leakage detection and stabiliser circuits so flagged leakage can inform tailored decoding.

III. FAULT-TOLERANT PROTOCOLS FOR SINGLET-TRIPLET SPIN QUBITS

  • Singlet-triplet pairs act as a bit-flip base code concatenated with the surface code, with detected flips informing a tailored decoder.The section focuses on surface-code memory mode; it notes logical gates can be deduced from the native operations.Source: §III · p. 5

A. Leakage detection

  • Four exchange gates transfer a data state to a fresh singlet pair before the old pair is measured; leaked inputs yield |T±⟩ and the output is projected back into the computational subspace.Computational inputs yield a singlet outcome and a deterministic X frame update; leakage flags enable erasure-aware decoding without feedback to the chip.Source: A. Leakage detection · p. 5, Figure 3 · p. 5

B. Stabiliser circuits

  • The exchange-only stabiliser circuit implements X, or Z when H gates are removed, while the exchange-and-driven circuit implements a bias-preserving XZZX stabiliser.Leakage detection is performed between stabiliser rounds so the state is almost always in the computational subspace.Source: B. Stabiliser circuits · pp. 5–6, Figure 4 · p. 6
  • ST stabilisers convert some Pauli X noise into detectable erasure, and the XZZX circuit removes all first-order Pauli X noise, leaving Pauli Z and leakage noise to first order.Alternating which ancilla spin controls the CNOT gates prevents a single entangling-gate error from producing a Pauli X error on an ST pair.Source: B. Stabiliser circuits · p. 6
  • Alternating the ancilla control spin avoids hook errors, while using the ancilla as control prevents leakage from propagating to it through stabiliser entangling gates.The passage characterises leakage as a single-spin X or Y flip; only Z errors can propagate to the ancilla in this arrangement.Source: B. Stabiliser circuits · p. 6

C. Decoder

The decoder uses leakage-detection information to adjust MWPM edge weights. CSS and XZZX handle leakage differently: CSS exploits paired detections, while XZZX also uses |T0⟩ measurements to identify Pauli X errors.

C. Decoder

  • MWPM builds a spacetime-syndrome matching graph and uses leakage-detection information to adjust edge weights when finding a correction.The decoder is chosen for simplicity in this initial study.Source: C. Decoder · p. 6

1. CSS surface code

  • CSS decoding sets edges through leaked data qubits to weight 0 and pairs first-order leakage detections in spacetime to distinguish leakage from likely measurement errors.The CSS code uses edge weight 1 between neighbouring stabilisers otherwise, a choice described as sub-optimal for the later circuit-level noise models.Source: §1 · pp. 6–7, Figure 5 · p. 7
  • For CSS ancillas, the decoder pastes over leaked stabiliser measurements before computing the regular difference syndrome.When a measurement is |T±⟩, the rule can randomly replace it with S or T0 with equal probability.Source: §1 · p. 7

2. XZZX surface code

  • XZZX decoding uses distinct X, Z, and time-like weights to account for biased noise, with p^2/10 estimating X noise when Z noise is Θ(p).The division by 10 approximately accounts for the circuit's conversion of first-order Pauli X errors into leakage.Source: §2 · p. 7
  • In XZZX, a |T0⟩ measurement flags locations for Pauli X errors, so the corresponding X edges receive weight 0.The described first-order X errors arise from Z or ZZ errors after the first CZ; the caption marks this with orange stars.Source: §2 · pp. 7–8, Figure 5 · p. 7
  • XZZX sets both X and Z edge weights to 0 at |T±⟩ detections, but CNOTs prevent pairing detections and raise the risk of false leakage flags.Leaked ancilla measurements use the same pasting-over strategy as CSS.Source: Figure 5 · p. 7, §2 · p. 8

IV. SURFACE CODE SIMULATIONS

The simulations evaluate CSS and XZZX surface codes in memory mode using noise models intended to reflect implemented device operations. They model gate, shuttling, and measurement noise, with gate parameters sampled and fixed at each location.

IV. SURFACE CODE SIMULATIONS

  • The study simulates square CSS and XZZX surface codes in memory mode to evaluate the proposed protocols.It reports that the simulations confirm high performance of the methods.Source: §IV · p. 8

A. Noise models

  • Gate noise is modeled through fluctuations in exchange coupling, Zeeman energy difference, and gate duration rather than a generic depolarising channel.For each gate location, physical parameters are sampled from independent normal distributions and then held fixed, yielding location-specific fidelities whose average is set to the target value.Source: A. Noise models · p. 8
  • The reported gate-noise results model timing errors, while additional simulations indicate that other parameters have a similar impact on results.The timing-error model uses analytical expressions relating fidelity to timing spread, with separate strengths for gate families.Source: A. Noise models · pp. 8–9
  • Shuttling between two-qubit gates is modeled as dephasing, with ST shuttling assumed no worse than LD shuttling; measurement errors use encoding-specific outcome flips.The setup supposes shuttling both data and ancilla qubits between each two-qubit gate, which halves the shuttling distance and may reduce shuttling times.Source: A. Noise models · p. 9

B. Simulations details

The simulations compare leakage-aware CSS and XZZX schemes with LD encoding and assess their thresholds under a common physical noise level. XZZX+ST achieves the highest reported threshold, while decoder choices and false leakage detections remain relevant limitations.

B. Simulations details

  • The simulations twirl coherent errors only after stabiliser and leakage-detection measurements, then use a custom QEC pipeline because leaked-qubit gates are not Clifford operations.The pipeline samples local gate parameters and runs 10,000–10,000,000 Monte Carlo trials across d rounds, decoding the joint syndrome to estimate logical failure probability.Source: B. Simulations details · p. 9

1. Threshold estimation

  • For CSS, the authors suggest detecting leakage less often—only on data qubits neighbouring leaked ancillas—to reduce introduced physical errors while moderately affecting detection accuracy.This proposed variant further exploits the pairing property of the CSS leakage detections.Source: §1 · pp. 9–10
  • 1.3% threshold for XZZX+ST exceeds 0.49% for CSS+ST and 0.45% for LD when shuttling noise is zero and other noise mechanisms share infidelity p.The XZZX+ST scheme also has lower logical error rates in the reported comparison; Fig. 6 plots logical error rate against physical noise for code distances 7, 9, 11, and 13.Source: §1 · p. 9, Figure 6 · p. 10
  • XZZX performance is partially limited by false leakage detections that cannot be pair-matched, and decoder weights are a simple heuristic rather than mechanism-specific.The authors suggest asymmetric measurement-error modeling may improve leakage, no-error, and Pauli X discrimination; comprehensive weight tuning or alternative decoders may improve all schemes.Source: §1 · p. 10

2. Impact of shuttling errors

The ST+XZZX scheme retains higher performance than LD under the compared shuttling-noise settings, with orders-of-magnitude lower logical error rates when ST shuttling noise is lower. The discussion presents singlet-triplet qubits as an erasure-based QEC route, supported by leakage-detection circuits and leakage-aware decoding.

2. Impact of shuttling errors

  • The comparison sets non-shuttling infidelities to p = 0.1%, fixes psh,LD at 0.5%, and varies psh,ST from 0.1% to 0.5%.Effective distance is d for LD and d/2 for ST because the ST encoding uses twice as many spins, so the comparison uses equal numbers of spins.Source: §2 · p. 10, Figure 7 · p. 11
  • The equal-electron comparison is described as likely worst-case because two ST spins occupy a close double dot and need not represent twice the device resources.The authors note possible adjustments to device size, measurement components, and control electronics, and suggest XZZX may reduce overhead by shrinking the code in the X direction.Source: §2 · p. 10

V. DISCUSSION

  • The proposed fault-tolerant framework uses singlet-triplet (dual-spin) qubits as a natural realization of erasure qubits in semiconductor platforms.The discussion argues that replacing single-spin data qubits with dual-spin encodings can improve resilience to shuttling-induced noise through intrinsic protection within their decoherence-free subspace.Source: §V · p. 10

2. Impact of shuttling errors

  • Orders-of-magnitude lower logical error rates are reported for ST+XZZX when psh,ST is a fraction of psh,LD; ST+XZZX also performs better when the two shuttling-noise levels are equal.Figure 7 plots logical error rate against effective code distance, comparing LD at deff = d with ST+XZZX at deff = d/2 for equal electron counts.Source: §2 · p. 10, Figure 7 · p. 11

V. DISCUSSION

  • The hardware-efficient erasure-detection circuit projects leaked qubits back into the computational subspace without measurement feedback, reducing classical-control overheads and timing constraints.The discussion identifies leakage out of the computational subspace as a new error mechanism introduced by the encoding.Source: §V · pp. 10–11
  • The stabiliser-measurement circuits include an exchange-only protocol with paired erasures and a driven-gate protocol that converts most X-type noise into erasure.The driven-gate protocol creates strongly biased noise at the cost of slightly lower leakage-detection performance.Source: §V · p. 11
  • Combined with XZZX and bespoke leakage-aware decoders, the driven-gate noise tailoring yields a twofold threshold increase and several orders of magnitude improvement in logical error rates.Source: §V · p. 11
  • The discussion frames singlet-triplet encoding as a pathway toward erasure-based quantum error correction, rather than solely as a noise-resilient upgrade to semiconductor architectures.Source: §V · p. 11

Appendix A: Leakage detection

The leakage-detection circuit preserves computational inputs up to a deterministic X and projects leaked inputs back into the computational subspace. The stabiliser circuits avoid distance-reducing hook errors because the resulting leakage is detectable.

Appendix A: Leakage detection

  • Computational inputs yield a singlet measurement outcome and are preserved up to a deterministic X that can be stored in memory.The measurement outcome is always a singlet.Source: Appendix A · p. 15
  • Leaked inputs produce even parity on the measured first pair, projecting the output into the computational subspace in an unknown |0⟩ or |1⟩ state.The derivation uses that Hadamard acts as identity on an even state.Source: Appendix A · p. 15

Appendix B: Hook errors

  • For Z stabilisers, hook errors can leak the first or second data qubit while placing a Z error on the third or fourth, so detectable leakage prevents distance reduction.The relevant propagated errors arise from XX or YY errors on the first or second CZ.Source: Appendix B · p. 15, Figure 8 · p. 16
  • For X stabilisers, Hadamards leave leaked states idle, yielding leakage plus an X error; XZZX hooks can also yield detectable leakage without two Z errors.The XZZX circuit's CNOTs need not create leakage in pairs, but its hook errors still do not reduce code distance.Source: Appendix B · p. 15, Figure 8 · p. 16

Appendix C: Noise models

The noise models assume random gate-parameter fluctuations that preserve gate-matrix structure. For driven single-qubit gates, timing errors provide a practical approximation because they alter the weighting of otherwise general Pauli errors without dramatically changing the study.

Appendix C: Noise models

  • The noise assumption is that random gate-parameter fluctuations preserve the ideal gate matrix's structure, including the exchange gate's zero entries.Charge noise is given as an example source of such fluctuations.Source: Appendix C · p. 16

1. Driven single-qubit gate

  • For a driven single-qubit X gate, a timing error makes all matrix elements nonzero and permits general Pauli X, Y, or Z noise, while changing their weighting is not dramatic for this study.A perfect X gate requires Ω_rT = π; an imperfect one is characterized by Ω_rT = π + δθ.Source: §1 · p. 16
  • Integrating fidelity over Gaussian angular errors yields Eq. (15) when other parameters are perfect and σ_δθ = Ω_rσ_T.The fidelity is extracted for small δθ before integration over a normal distribution.Source: §1 · p. 16

2. Exchange gates

Exchange gates have a structured, spin-conserving matrix, so generic depolarising noise does not capture their noise structure. Timing-error analyses derive the CZ expression, and the Hadamard fidelity has the same form.

2. Exchange gates

  • Exchange-gate matrices contain spin-conservation-related zero entries, and timing errors produce a six-nonzero-element form rather than generic depolarising noise.This structure motivates the focus on timing errors.Source: §2 · p. 17

a. CZ gate √

  • For the CZ gate, choosing Ω_rT_0 = π and applying virtual single-qubit Z gates yields the small-δθ fidelity expression integrated into Eq. (16).The virtual Z gates do not contribute to final infidelity.Source: a. CZ gate √ · p. 17

b. H gate

  • The Hadamard gate uses J = ΔE_Z and Ω_rT_0 = π/2, with virtual Z gates, and its fidelity has the same form as the CZ gate's.Source: b. H gate · p. 17

3. CNOT gate

The CNOT is implemented by driving the |↑↑⟩↔|↑↓⟩ transition in the J ≪ ΔE_Z regime. Synchronised rotations produce the desired gate, and timing errors lead to the expression in Eq. (17).

3. CNOT gate

  • The CNOT targets the |↑↑⟩↔|↑↓⟩ transition with a drive in the J ≪ ΔE_Z, near-resonant regime.Source: §3 · p. 17
  • Synchronising rotations so θ_1(t) = 2θ_2(t), with θ_2(T_0) = π, makes R_1 = I and R_2 = X, producing the desired CNOT.The time-evolution calculation is simplified by the block-diagonal Hamiltonian.Source: §3 · pp. 17–18
  • Gaussian timing errors in the rotation angles lead to the CNOT infidelity expression in Eq. (17).The error condition is θ_2 = θ_1/2 = π + δθ.Source: §3 · p. 18

Appendix D: Simulations details

Appendix D describes simulations that convert stabiliser and leakage-detection circuits into Pauli-error models, then use those models to estimate logical error rates. The workflow includes circuit simulation, twirling, syndrome decoding, and repeated Monte Carlo trials.

a. Stabiliser and leakage detection circuits simulations

  • The circuit simulation uses the Choi-Jamiolkowski isomorphism to decompose a circuit into Kraus operators, with the described calculation involving matrices of size 28 × 28.Source: Appendix D · pp. 18–19
  • For a system with n data qubits, the protocol simulates a circuit on 2n qubits; the studied circuits require 18 qubits and are simulated with QuEST.The protocol applies the circuit to halves of Bell pairs and reshapes the resulting state vector into a matrix; it extends to noise channels and measured ancillas.Source: Appendix D · p. 18

b. Twirling approximation

  • The twirling approximation replaces general noisy Kraus operators with Pauli errors for the decoder, and the resulting error weights depend on physical parameters such as exchange coupling and g-factors.Source: Appendix D · pp. 18–19

Twirling approximation

  • First-order events in the X-stabiliser error distribution are described as easily correctable, while second-order errors—including multi-data-qubit errors or unpaired leakage—are more harmful.The figure shows the distribution after twirling at p = 0.6%.Source: Figure 9 · p. 19

c. QEC simulator

  • The QEC simulator samples Pauli errors, computes leakage-aware syndromes, decodes with MWPM and leakage information, then checks whether the correction leaves a logical error.Source: Appendix D · p. 19

d. Monte-Carlo simulations

  • Logical error rates are estimated with 10,000 to 10,000,000 Monte Carlo repetitions, assuming perfect final stabiliser measurements and leakage detections.Source: Appendix D · pp. 19–20
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