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 electron spins in a double quantum dot (DQD). The singlet $|S⟩$ and unpolarized triplet $|T_0⟩$ form the computational basis, while the polarized triplets $|T_{+}⟩=|\uparrow \uparrow ⟩$ and $|T_{-}⟩=|\downarrow \downarrow ⟩$ are split off by a uniform magnetic field and lie outside the computational subspace.

Both single- and two-qubit gates are fully electrical:

• $Z$ 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): a magnetic field gradient $\Delta B_z$ between the two dots (from a micromagnet, nuclear polarization, or $g$-factor difference)

Two-qubit coupling uses either capacitive (dipole-dipole) interaction, exploiting the charge-dipole difference between $|S⟩$ and $|T_0⟩$ at finite detuning, or direct exchange between adjacent 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:

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

where $J(\epsilon )$ is the exchange splitting controlled by the detuning $\epsilon$ between dots (or by the tunnel barrier), and $\Delta B_z$ is the magnetic field gradient. The exchange splitting $J$ depends on detuning as:

$J(\epsilon )\approx \frac{2t_c^2}{\epsilon +\sqrt{{\epsilon }^2+4t_c^2}}+\frac{2t_c^2}{U-\epsilon +\sqrt{(U-\epsilon {)}^2+4t_c^2}}$

where $t_c$ is the tunnel coupling and $U$ the on-site Coulomb energy. At the symmetric operating point ($\epsilon =0$), $\partial J/\partial \epsilon =0$, providing a charge noise sweet spot for $J$.

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 an effective $ZZ$ interaction in the logical basis:

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

where $\alpha$ depends on the inter-dot capacitance and the charge-dipole difference between $|S⟩$ and $|T_0⟩$.

Exchange coupling between adjacent spins of neighboring qubits gives an effective Heisenberg-type interaction in the logical basis.

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 in GaAs and Si/SiGe with high fidelity
• Foundation for more complex encodings (exchange-only, AEON, RX)

Experimental Status

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

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): Two-qubit gate fidelity ~90% via capacitive coupling (GaAs)
• Jock et al. (2018): Si/SiGe with $T_2^{\ast }\sim 1\mu \text{s}$, single-qubit fidelity >99%
• Weinstein et al. (2023): Symmetric operation sweet spot, high-fidelity gates in Si/SiGe
• Bøttcher et al. (2022): Parametric longitudinal coupling to high-impedance SC resonator

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 Info. 3, 3 (2017)

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)

Linked Papers

• petta-2005-singlet-triplet

Related Entries

• loss-divincenzo-qubit
• exchange-only-qubit
• rx-qubit
• aeon-qubit

Key Metrics

Metric	Value	Notes	Fidelity reference
Qubit coherence $T_2^{\ast }$	10 ns (GaAs), ~1 μs (Si)	Nuclear-limited (GaAs), charge-limited (Si)	Petta et al. 2005
Qubit coherence $T_2^{\text{echo}}$	~200 μs (GaAs), >1 ms (Si)	With dynamical decoupling	Bluhm et al. 2011
Gate fidelity (1Q)	>99%	Exchange + gradient control (Si)	Jock et al. 2018
Gate fidelity (2Q)	~90%	Capacitive coupling (GaAs)	Nichol et al. 2017
Gate time (1Q)	1–100 ns	Exchange pulse ($Z$) or gradient ($X$)	—
Gate time (2Q)	10–200 ns	Capacitive or exchange-mediated	—
Readout fidelity	95–99%	Pauli spin blockade + charge sensor	Barthel et al. 2009
Qubit footprint	~100–200 nm pitch	2 dots per logical qubit	—
Operating temperature	20–100 mK	GaAs or Si/SiGe	—
Connectivity	Nearest-neighbor	Between adjacent double dots	—