Singlet-Triplet Qubit

Description

The singlet-triplet ($S$–$T_0$) qubit encodes a logical qubit in the $m_s=0$ subspace of two exchange-coupled spin-$1/2$ carriers in a double quantum dot (DQD). The canonical implementation uses electrons, while recent silicon and germanium devices also use holes. The singlet $|S⟩$ and unpolarized triplet $|T_0⟩$ form the computational basis; the polarized triplets $|T_{+}⟩=|\uparrow \uparrow ⟩$ and $|T_{-}⟩=|\downarrow \downarrow ⟩$ are leakage states split from the logical manifold by the mean Zeeman field.

Universal single-qubit control uses voltage pulses together with a static magnetic-field or Zeeman-energy gradient:

• $Z$ rotations (around the logical $\hat{z}$ axis): tuning the exchange coupling $J$ via the gate voltage on the barrier or detuning between dots
• $X$ rotations (around the logical $\hat{x}$ axis): evolution under a Zeeman-energy gradient $\Delta E_Z=g{\mu }_B\Delta B_z$ between the dots, produced by a micromagnet, nuclear polarization, or a $g$-factor difference

No oscillating magnetic drive is required for the basic baseband protocol. Resonant modulation of $J$ is an alternative that permits operation near a symmetric point. Two-qubit coupling uses either a capacitive interaction, exploiting the state-dependent charge admixture of $|S⟩$, or direct exchange between adjacent physical spins of neighboring qubits.

After the single-spin Loss-DiVincenzo qubit, this is the next-simplest spin qubit — requiring only 2 dots per logical qubit — and was the first encoded spin qubit to be experimentally demonstrated (Petta et al. 2005).

Figure

Hamiltonian

In the $\{|S⟩,|T_0⟩\}$ basis, define ${\sigma }_z=|T_0⟩⟨T_0|-|S⟩⟨S|$. After removing a common energy offset,

$H=\frac{J(\epsilon ,t_c)}{2}{\sigma }_z+\frac{\Delta E_Z}{2}{\sigma }_x,\Delta E_Z=g{\mu }_B\Delta B_z.$

Here $J=E_{T_0}-E_S>0$ is the exchange splitting. In a symmetric two-site Hubbard model with on-site charging energy $U$, detuning $\epsilon$, and spin-conserving tunnel amplitude $t_c$, second-order perturbation theory in the $(1,1)$ charge sector gives

$J(\epsilon ,t_c)\simeq 2t_c^2\left(\frac{1}{U+\epsilon }+\frac{1}{U-\epsilon }\right)=\frac{4t_c^{2}U}{U^2-{\epsilon }^2},$

valid away from the $(1,1)$–$(2,0)/(0,2)$ charge anticrossings. At the symmetric operating point $\epsilon =0$, $\partial J/\partial \epsilon =0$. Barrier control changes $t_c$ while remaining at this first-order detuning-noise sweet spot.

Logical encoding

$|0_L⟩=|S⟩=\frac{1}{\sqrt{2}}(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩),|1_L⟩=|T_0⟩=\frac{1}{\sqrt{2}}(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)$

Both states have $m_s=0$, giving first-order insensitivity to uniform magnetic field fluctuations.

Two-qubit coupling

Capacitive coupling between DQDs produces state-dependent energy shifts. After absorbing single-qubit terms, its entangling component can be written

$H_{\text{cap}}=\alpha {\sigma }_z^{(1)}{\sigma }_z^{(2)}$

where $\alpha$ depends on inter-DQD capacitance and on the charge admixture of the logical states.

Exchange coupling acts first on the physical boundary spins, for example $H_c=J_c{\mathbf{s}}_2\cdot {\mathbf{s}}_3$. Its projection generates logical two-qubit interactions, but an abrupt pulse can also populate non-computational four-spin states; exchange-gate sequences therefore have to control leakage rather than treating $H_c$ as a simple logical Heisenberg term.

Motivation

• All-electrical control — no microwave drive needed (unlike Loss-DiVincenzo)
• Only 2 dots per logical qubit — simpler than exchange-only (3 dots)
• Fast gates — exchange pulses at ns timescales
• Well-established platform — demonstrated with electron and hole spins in GaAs, Si/SiGe, silicon MOS, and Ge/SiGe
• Foundation for more complex encodings (exchange-only, AEON, RX)

Experimental Status

First demonstration: Petta et al. (2005) in a GaAs/AlGaAs DQD — coherent singlet-triplet oscillations via exchange control, with $T_2^{\ast }\sim 10$ ns limited by hyperfine fluctuations.

Key experimental milestones:

• Bluhm et al. (2011): Dynamical decoupling extended $T_2$ to ~200 μs in GaAs
• Maune et al. (2012): First Si/SiGe singlet-triplet qubit — isotopic purification dramatically improved coherence
• Shulman et al. (2012): Two-qubit entangling gate via capacitive coupling, Bell state fidelity ~72% (GaAs)
• Nichol et al. (2017): ~99% single-qubit fidelity and 90% entangling-gate fidelity in capacitively coupled GaAs $S$–$T_0$ qubits
• Bøttcher et al. (2022): Parametric longitudinal coupling to high-impedance SC resonator
• Ungerer et al. (2024): Strong single-photon coupling in an InAs nanowire $S$–$T$ device, $g/2\pi =139\pm 4$ MHz
• Song et al. (2024): ~100 MHz field-gradient-driven oscillations with quality factor $Q>580$ in ${}^{28}$Si/SiGe
• Zhang et al. (2025): Universal control of four neighboring $S$–$T_{-}$ hole-spin qubits in a $2\times 4$ germanium dot array; this is a related polarized-triplet variant, not the $S$–$T_0$ encoding defined above
• Tsoukalas et al. (2026): Resonantly driven germanium hole $S$–$T_0$ qubit with 99.68(2)% average gate fidelity; continuous dressing extended rotating-frame coherence to $T_{2\rho }^{\ast }=20.3\mu \text{s}$ while retaining 99.63(7)% gates

References

Original proposal

• J. Levy, “Universal quantum computation with spin-1/2 pairs and Heisenberg exchange,” PRL 89, 147902 (2002)

Landmark experiment

• J. R. Petta et al., “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180 (2005)

Coherence advances

• H. Bluhm et al., “Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 μs,” Nature Phys. 7, 109 (2011)
• B. M. Maune et al., “Coherent singlet-triplet oscillations in a silicon-based double quantum dot,” Nature 481, 344 (2012)

Two-qubit gates

• M. D. Shulman et al., “Demonstration of entanglement of electrostatically coupled singlet-triplet qubits,” Science 336, 202 (2012)
• J. M. Nichol et al., “High-fidelity entangling gate for double-quantum-dot spin qubits,” npj Quantum Information 3, 3 (2017), arXiv:1608.04258

Readout

• C. Barthel et al., “Rapid single-shot measurement of a singlet-triplet qubit,” PRL 103, 160503 (2009)

Resonator coupling

• C. G. L. Bøttcher et al., “Parametric longitudinal coupling between a high-impedance superconducting resonator and a semiconductor quantum dot singlet-triplet spin qubit,” Nature Commun. 13, 4773 (2022)
• J. H. Ungerer et al., “Strong coupling between a microwave photon and a singlet-triplet qubit,” Nature Communications 15, 1068 (2024), arXiv:2303.16825

Recent control and scaling

• Y. Song et al., “Coherence of a field-gradient-driven singlet-triplet qubit coupled to many-electron spin states in ${}^{28}$Si/SiGe,” npj Quantum Information 10, 77 (2024), arXiv:2310.12603
• X. Zhang et al., “Universal control of four singlet-triplet qubits,” Nature Nanotechnology 20, 209 (2025), arXiv:2312.16101
• K. Tsoukalas et al., “A dressed singlet-triplet qubit in germanium,” Nature Communications 17, 699 (2026), arXiv:2501.14627

Linked Papers

• petta-2005-singlet-triplet
• barthel-2009-rapid-single-shot
• bluhm-2011-dephasing-time-gaas
• bttcher-2022-parametric-longitudinal-coupling
• levy-2002-universal-computation-spin
• loss-divincenzo-1998-quantum-dots
• martins-2016-symmetric-exchange-gates
• maune-2012-coherent-singlet-triplet
• nichol-2017-high-fidelity-entangling-gate
• reed-2016-reduced-sensitivity-charge-noise
• shulman-2012-demonstration-entanglement-electrostatically
• song-2024-field-gradient-driven-singlet-triplet
• tsoukalas-2026-dressed-singlet-triplet
• ungerer-2024-strong-coupling-microwave-photon
• zhang-2025-universal-control-four-singlet-triplet

Evergreen context

• exchange-interaction-in-quantum-dots — the physical origin of electrically tuned J(\epsilon) control
• heisenberg-exchange-in-quantum-dots — the minimal exchange Hamiltonian behind the logical $Z$ rotation generated by $J$
• decoherence-free-subspace — why the m_S = 0 manifold partially protects against uniform field noise
• charge-noise-sweet-spot — where symmetric detuning helps, and where the tradeoffs remain
• vacuum-rms-field-scaling — the resonator-side lever behind the high-impedance longitudinal-coupling route, distinct from the qubit-side charge admixture that supplies the electric response

Related Entries

• loss-divincenzo-qubit
• exchange-only-qubit
• rx-qubit
• aeon-qubit
• spin-qubit
• qubit-readout
• circuit-qed

Key Metrics

Metric	Value	Notes	Fidelity reference
Inhomogeneous coherence $T_2^{\ast }$	~10 ns (GaAs); $<1\mu$s in the ergodic ${}^{28}$Si/SiGe measurement; $1.9\mu$s (Ge holes)	Platform- and protocol-dependent; hyperfine-limited in GaAs, with distinct noise regimes in enriched Si and Ge	Petta et al. 2005; Song et al. 2024; Tsoukalas et al. 2026
Protected/dressed coherence	$>200\mu$s (GaAs CPMG); $T_{2\rho }^{\ast }=20.3\mu$s (dressed Ge holes)	Different coherence definitions and platforms; values should not be read as a direct ranking	Bluhm et al. 2011; Tsoukalas et al. 2026
Gate fidelity (1Q)	99.68(2)% bare resonant; 99.63(7)% dressed	Average physical-gate fidelity for a Ge hole $S$–$T_0$ qubit	Tsoukalas et al. 2026
Entangling-gate fidelity (2Q)	90%	Capacitively coupled GaAs $S$–$T_0$ qubits; process estimate from self-consistent tomography	Nichol et al. 2017
Representative 1Q gate time	327 ns bare; 500 ns dressed	$X_{\pi }$ gates in the 2026 Ge hole device	Tsoukalas et al. 2026
Readout fidelity	$>90\%$ in ~7 μs	Pauli spin blockade plus charge sensing in GaAs	Barthel et al. 2009
Spin-photon coupling	$g/2\pi =139\pm 4$ MHz	Strong-coupling InAs nanowire experiment; $\gamma /2\pi =116\pm 7$ MHz and $\kappa /2\pi =19.8\pm 0.2$ MHz	Ungerer et al. 2024
Scaled control milestone	Four neighboring $S$–$T_{-}$ qubits; 1Q fidelities 99.49(8)–99.84(1)%	Related polarized-triplet encoding in a $2\times 4$ Ge hole array; Bell-state fidelities 73(1)–90(1)%	Zhang et al. 2025