Acceptor Spin Qubits in Silicon

• Acceptor spin qubits are defined by bound holes in silicon acceptors, leveraging spin–orbit coupling to create a distinct two-level system.
• They enable all-electrical manipulation via quadrupolar couplings to electric, elastic, and magnetic fields, facilitating long-range interactions.
• Integration with interface engineering, strain tuning, and advanced coupling schemes underpins experimental progress toward scalable silicon quantum architectures.

Acceptor spin qubits are spin-orbit qubits formed not by an electron bound to a donor, but by a hole bound to an acceptor dopant atom in silicon. Their defining microscopic feature is that the qubit degrees of freedom arise from the acceptor-bound hole at the valence-band edge, where spin-orbit coupling quantizes total angular momentum $J=3/2$ rather than spin, producing heavy-hole and light-hole states that can be split by magnetic, electric, elastic, and interfacial perturbations into a usable two-level system. This $J=3/2$ structure gives acceptor qubits quadrupolar couplings to electric and elastic fields, enabling electrical manipulation and several routes to long-distance coupling through phonons, capacitive links, or microwave photons; it also distinguishes them sharply from donor-electron qubits, which are mostly spin-$\tfrac12$ systems with comparatively weak spin-orbit coupling (Salfi, 2020).

1. Physical basis and qubit manifold

Acceptor spin qubits are built from substitutional acceptors, such as boron in silicon, that bind a single valence hole. In silicon acceptors, the bound state is quantized into a $J=3/2$ multiplet with projected states $\lvert J,m_J\rangle$, conventionally organized as heavy-hole-like states $\lvert m_J=\pm 3/2\rangle$ and light-hole-like states $\lvert m_J=\pm 1/2\rangle$. A magnetic field splits this manifold and can define a two-level qubit; near interfaces and under confinement or strain, the two lowest eigenstates are typically mixed heavy-hole/light-hole states rather than pure spin states (Salfi, 2020).

This state structure is the central distinction from donor platforms. In donor-electron qubits, the relevant low-energy degree of freedom is predominantly spin-$\tfrac12$. In acceptor qubits, by contrast, the qubit is intrinsically spin-orbit-active: electric fields, strain, and symmetry breaking reshape the qubit wavefunctions and their effective $g$-tensor. In the low-energy language used for acceptor devices near interfaces, the qubit is often encoded in the lowest Kramers doublet, for example

$$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$

or, in a related notation,

$J=3/2$0

where the coefficients reflect heavy-hole/light-hole admixture controlled by vertical electric field, confinement, and strain (Salfi et al., 2015, Zhang et al., 2022).

The general single-acceptor description is correspondingly multicomponent. One representative Hamiltonian for silicon acceptors near an interface is

$J=3/2$1

while a related formulation writes

$J=3/2$2

These forms emphasize that the qubit eigenstates are shaped jointly by valence-band physics, acceptor Coulomb binding, interface confinement, magnetic field, gate fields, and the acceptor-specific tetrahedral-symmetry term $J=3/2$3 (Salfi et al., 2016, Abadillo-Uriel et al., 2017).

A common misconception is to treat acceptor pairs or acceptor qubits as essentially donor-like with an effective mass rescaling. The acceptor literature explicitly rejects that simplification. Because the valence band is degenerate and spin-orbit coupling entangles orbital and spin character, acceptor-acceptor interactions are not well captured by the hydrogen-molecule singlet-triplet picture; a Heitler–London treatment based on Baldereschi–Lipari wavefunctions yields a rich ten-level spectrum rather than a two-level splitting (Durst et al., 2017).

2. Interfaces, strain, and energetic isolation of the qubit subspace

A recurring requirement in acceptor-based qubit design is the energetic isolation of a single Kramers doublet from the rest of the $J=3/2$4 manifold. In bulk silicon, a boron acceptor ground state is fourfold degenerate. Near a surface or interface, however, symmetry reduction splits that fourfold manifold into two Kramers doublets with different hole character, one predominantly heavy-hole and one predominantly light-hole. Low-temperature scanning tunneling spectroscopy on individual sub-surface boron acceptors directly observed two distinct resonances in the gap and identified them as the heavy-hole and light-hole Kramers doublets; for experimentally accessible acceptors within less than $J=3/2$5 nm of the surface, the splitting is greater than $J=3/2$6 meV and increases as the acceptor approaches the surface (Mol et al., 2015).

This interface-induced splitting is not a higher orbital excitation or a two-hole charging state. The spectroscopy data showed two resonances with the same spatial extent, the same overall symmetry, and no anti-nodes in the surface plane, which is consistent with transport through the lowest two acceptor states within the $J=3/2$7-like manifold. The splitting was extracted through thermally broadened Lorentzian fits with

$J=3/2$8

and the depth dependence agreed reasonably with fully atomistic tight-binding calculations using a 1.4-million-atom silicon cluster and a 20-orbital $J=3/2$9 basis (Mol et al., 2015).

Strain provides a second major control axis. In the strained-qubit formulations, the heavy-hole/light-hole splitting is written as

$\tfrac12$0

while the spin-hole coupling contains both electric-field and strain contributions,

$\tfrac12$1

Symmetric strain mainly changes the heavy-hole/light-hole splitting, whereas asymmetric strain directly generates additional spin-hole coupling. The resulting advantage is that strain gives independent control over the heavy-hole/light-hole splitting and the spin-hole coupling, avoiding the need to obtain both solely through large electric fields (Zhang et al., 2022).

The operating-point concept that organizes much of acceptor-qubit design is the sweet spot. A first-order sweet spot is defined by

$\tfrac12$2

and a second-order sweet spot additionally satisfies

$\tfrac12$3

These conditions identify gate biases where the qubit splitting is locally flat against electric-field noise. In silicon acceptors, theory has described both anisotropic and isotropic sweet spots, and special in-plane magnetic-field directions can merge them, producing a regime that is second-order insensitive over a finite field window of a few MV/m (Abadillo-Uriel et al., 2017). In the strain-tunable analysis, asymmetric strain can create a new low-field sweet spot, induce two sweet spots, and merge them into a second-order sweet spot; the reported quality factor can reach $\tfrac12$4 for single-qubit operation, with high tolerance for electric-field variation (Zhang et al., 2022).

3. Electrical control, EDSR, and entangling mechanisms

The principal operational advantage of acceptor spin qubits is that the same spin-orbit physics that complicates their spectrum also enables all-electrical control. Because the qubit states are not pure spin states, they acquire quadrupolar coupling to electric and elastic fields. In silicon, the acceptor-specific $\tfrac12$5-symmetry term is

$\tfrac12$6

and it has no analogue for spin-$\tfrac12$7 electrons. Together with interface-induced inversion asymmetry, this term permits electric dipole spin resonance and gate-dependent rotation of the effective spin polarization and electric dipole orientation (Abadillo-Uriel et al., 2017).

Several silicon models write the effective qubit Hamiltonian in the electrically driven form

$\tfrac12$8

or equivalent variants in which the Rabi frequency is

$\tfrac12$9

The dipole matrix element $J=3/2$0 depends on heavy-hole/light-hole mixing and on the relative orientation of the magnetic field and gate field. In the charge-insensitive single-atom proposal, the interface-enhanced inversion-asymmetry parameter is reported as $J=3/2$1, about 100 times larger than the bare $J=3/2$2 parameter $J=3/2$3, and the estimated $J=3/2$4-pulse times at the sweet spot are $J=3/2$5 ns for $J=3/2$6 nm and $J=3/2$7 ns for $J=3/2$8 nm under $J=3/2$9 V/cm (Salfi et al., 2015). In group-IV quantum wells, the same acceptor-bound hole physics also supports ESR, EDSR, and $\lvert J,m_J\rangle$0-tensor modulation resonance, with Rabi frequencies up to hundreds of MHz for heavy-hole qubits and of the order of GHz for light-hole qubits (Abadillo-Uriel et al., 2017).

Two-qubit coupling need not rely on short-range exchange. The silicon roadmap emphasizes electric dipole-dipole coupling, elastic dipole-dipole coupling, and indirect cavity-QED-like schemes mediated by microwave phonons or microwave photons (Salfi, 2020). In dipolar form,

$\lvert J,m_J\rangle$1

and the strain-engineered acceptor analysis uses this interaction to construct a $\lvert J,m_J\rangle$2 gate with

$\lvert J,m_J\rangle$3

For $\lvert J,m_J\rangle$4 and $\lvert J,m_J\rangle$5 nm, the reported two-qubit gate fidelity can reach about $\lvert J,m_J\rangle$6 (Zhang et al., 2022).

A particularly distinctive theoretical result is the electrically switchable “magic angle” mechanism. By electrically tuning the direction of the spin-dependent electric dipole through the interplay of $\lvert J,m_J\rangle$7 symmetry, heavy-hole/light-hole mixing, and magnetic-field orientation, one can make the dipole-dipole interaction vanish while retaining the ability to manipulate the individual qubits. In that framework the interaction takes the form

$\lvert J,m_J\rangle$8

with

$\lvert J,m_J\rangle$9

defining the zero-coupling condition. The proposal is explicitly intended to overcome the drawback of earlier electrical coupling mechanisms that could not support single-qubit operations while two-qubit couplings were off (Abadillo-Uriel et al., 2017).

Circuit-QED-style architectures constitute a second long-range route. The charge-insensitive silicon proposal reports $\lvert m_J=\pm 3/2\rangle$0 neV for $\lvert m_J=\pm 3/2\rangle$1 nm and $\lvert m_J=\pm 3/2\rangle$2 neV for $\lvert m_J=\pm 3/2\rangle$3 nm, corresponding to about $\lvert m_J=\pm 3/2\rangle$4 MHz resonant spin-photon coupling in the abstract, together with dispersive readout, cavity-mediated entanglement, and spin-photon entanglement (Salfi et al., 2015). The roadmap also highlights nanomechanical resonators and microwave phonons as a route to spin-phonon coupling, phonon-mediated two-qubit interactions, and even microwave-to-optical transduction for quantum networks (Salfi, 2020).

4. Experimental progress in silicon

Experimentally, acceptor spin qubits in silicon have progressed from spectroscopy proposals to single-atom devices with state-sensitive readout. The first single-atom acceptor transistor was demonstrated in an industrially fabricated device and exhibited the expected $\lvert m_J=\pm 3/2\rangle$5 Zeeman spectrum. Readout by spin-to-charge conversion was then demonstrated in an industrially fabricated two-acceptor device using gate-based reflectometry (Salfi, 2020).

A representative single-atom experiment used a commercial-like silicon tri-gate CMOS transistor operated in the single-atom regime, with channel dimensions about $\lvert m_J=\pm 3/2\rangle$6 nm high, $\lvert m_J=\pm 3/2\rangle$7 nm wide, and $\lvert m_J=\pm 3/2\rangle$8 nm long, and an rf tank circuit with resonance near $\lvert m_J=\pm 3/2\rangle$9 MHz. The device isolated a pair of coupled boron acceptors and used spin-selective tunneling at the $\lvert m_J=\pm 1/2\rangle$0 transition as the readout mechanism. This is the single-atom version of Pauli spin blockade: when the $\lvert m_J=\pm 1/2\rangle$1 triplet becomes the ground state, tunneling into the $\lvert m_J=\pm 1/2\rangle$2 singlet-like state is forbidden or strongly suppressed, and the reflectometry signal weakens accordingly (Heijden et al., 2017).

The same experiment directly exposed spin-orbit dynamics in the two-hole spectrum. A pronounced relaxation hotspot in $\lvert m_J=\pm 1/2\rangle$3 was measured near a magnetic field of about $\lvert m_J=\pm 1/2\rangle$4 T and explained by mixing between the $\lvert m_J=\pm 1/2\rangle$5 heavy-hole singlet $\lvert m_J=\pm 1/2\rangle$6 and the mixed heavy/light state $\lvert m_J=\pm 1/2\rangle$7. The fitted parameters were $\lvert m_J=\pm 1/2\rangle$8 T, $\lvert m_J=\pm 1/2\rangle$9 T in magnetic-field units, and $\tfrac12$0, showing that relaxation via the hotspot is over three orders of magnitude stronger than the ordinary heavy-hole relaxation channel (Heijden et al., 2017).

Excited-state spectroscopy in the same device yielded $\tfrac12$1 and $\tfrac12$2, and the extracted low-field effective $\tfrac12$3 from the triplet ground state was about $\tfrac12$4. The interpretation advanced in that work is that the quantization axis of angular momentum is not fixed by the magnetic field at low $\tfrac12$5, but instead by local strain, electric field, or interface effects; as the field increases, Zeeman energy dominates and the quantization axis aligns more with $\tfrac12$6 (Heijden et al., 2017).

A separate experimental milestone concerns energetic isolation by interface engineering. Scanning tunneling spectroscopy on individual sub-surface acceptors directly showed that a nearby silicon/vacuum interface lifts the fourfold ground-state degeneracy into two Kramers doublets. This established that a naturally occurring nanoscale interface can isolate a low-energy doublet with a robust meV-scale gap to the next doublet, providing a realistic route to tunable acceptor-based qubits in silicon (Mol et al., 2015).

5. Coherence, relaxation, hyperfine physics, and noise protection

The coherence record that has most strongly shaped the field is the observation that boron acceptors in isotope-purified $\tfrac12$7Si can exhibit ultra-long coherence times $\tfrac12$8 in a moderate static strain environment. The same work noted that isotope purification narrows ensemble EPR linewidths by removing random strain from the host crystal. This result is notable because one might expect strong spin-orbit mixing to shorten coherence, yet acceptors can still be exceptionally coherent when isotopic disorder is removed and strain is controlled (Salfi, 2020).

Theoretical and device studies explain this apparent tension by emphasizing that the same electric sensitivity that enables control must be balanced against operating points with suppressed charge-noise response. In the charge-insensitive silicon proposal, a sweet spot around $\tfrac12$9 MV/m for acceptor depths around $g$0 nm yields first-order insensitivity to vertical electric noise and no first-order dependence on in-plane electric fields. For a fluctuating trap $g$1 nm away, the relevant electric-field fluctuations were estimated as $g$2 and $g$3, with a large gate-voltage window in which $g$4; Johnson-limited dephasing was predicted to be $g$5 times weaker than charge-trap noise, and TLS dephasing $g$6 times weaker (Salfi et al., 2015).

Relaxation is predominantly phonon-mediated. In silicon, deformation-potential phonons dominate because Si is not piezoelectric, and several treatments show that $g$7 depends sensitively on heavy-hole/light-hole admixture and magnetic-field orientation. One magnetic-field-orientation study writes

$g$8

and argues that choosing field directions near the decoherence-free-subspace condition can dramatically suppress relaxation. In that analysis, $g$9 improves faster than the single-qubit drive slows down because $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$0 while the EDSR coupling scales only linearly, $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$1 (Abadillo-Uriel et al., 2017).

The hyperfine interaction is weak compared with donor-electron systems but remains diagnostically important. For boron acceptors in silicon, the hole-spin-echo envelope modulation analysis shows that in unstrained silicon, both the hyperfine and Zeeman Hamiltonians are approximately isotropic, leading to negligible envelope modulations, whereas in strained silicon, where light-hole spin qubits can be energetically isolated, the hyperfine Hamiltonian and $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$2-tensor are sufficiently anisotropic to produce substantial HSEEM. The predicted maximum modulation depth reaches $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$3 at a moderate laboratory magnetic field $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$4, and recent experiments reporting $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$5 ms at $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$6 mT in isotopically purified strained silicon were identified as more than sufficient to resolve the effect (Philippopoulos et al., 2019).

The broader lesson is that strain is both a resource and a liability. Strain engineering isolates the qubit subspace, creates sweet spots, and can suppress some relaxation channels, but strain inhomogeneity broadens $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$7-factors and heavy-hole/light-hole splittings. This suggests that materials purity and microscopic strain control are not ancillary device concerns; they are part of the qubit Hamiltonian.

6. Material extensions, comparative status, and architectural outlook

Although silicon remains the principal experimental host, acceptor spin qubits have been generalized to several material systems. In group-IV quantum wells, acceptor-bound hole spins in engineered SiGe heterostructures can exploit both the acceptor $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$8 symmetry term and heterostructure-induced Rashba-like interaction, with the wave function shifted between well and barrier layers to tune the effective $$|l\pm\rangle = a_L|\pm 1/2\rangle \mp i a_H|\mp 3/2\rangle,$$9-factor and enable ESR, EDSR, and $J=3/2$00-tensor modulation resonance (Abadillo-Uriel et al., 2017). In strained GaAs, optical pumping and coherent population trapping have been demonstrated for acceptor-bound holes, with $J=3/2$01 at $J=3/2$02 K and $J=3/2$03 T and $J=3/2$04, the latter attributed primarily to $J=3/2$05-factor broadening due to strain inhomogeneity rather than to hyperfine noise (Linpeng et al., 2020). In ZnSe, an optically active single acceptor-bound hole spin has been isolated, with an effective hole $J=3/2$06-factor of $J=3/2$07, a radiative lifetime of $J=3/2$08 ps, and an optical resonance linewidth of $J=3/2$09, establishing a single-emitter acceptor-spin platform with optical access (Alizadehherfati et al., 30 Jun 2026).

Germanium has recently been assessed as a particularly interesting but comparatively immature host. In comparative work on Ge-based qubit modalities, acceptors are described as the most atom-like realization of Ge valence-band spin physics and also as the least mature of the four Ge qubit modalities. Their appeal lies in the $J=3/2$10-point valence band, spin-$J=3/2$11 structure, heavy-hole/light-hole mixing, electric-field and strain sensitivity, quadrupolar response, and possible roles as hybrid spin-phonon or spin-strain nodes. Their principal weakness is heightened sensitivity to interface, strain, and central-cell details, together with the absence of a mature Ge acceptor stack demonstrating the level of benchmarked performance seen in gate-defined Ge hole qubits (Mei et al., 13 May 2026).

A more architecture-focused Ge design study proposes indium acceptor-bound hole spins in detector-grade Ge as a low-disorder intermediate platform between donor-based impurity qubits and fully gate-defined Ge hole-spin hardware. The proposed materials strategy uses a residual impurity background near $J=3/2$12 and a target In density of approximately $J=3/2$13, corresponding to an acceptor spacing of about $J=3/2$14 nm. In that picture, a $J=3/2$15m-long active channel can statistically contain about five acceptors on average, allowing a post-fabrication selected register rather than a deterministically placed chain. The design hierarchy places all-electrical single-qubit control first, dipolar coupling second, phonon-mediated coupling third, and exchange only as a close-pair or gate-enhanced channel (Mei et al., 20 Jun 2026).

The present architectural picture in silicon is therefore twofold. On one side, acceptor qubits offer mature silicon processing, a fundamentally different and highly useful spin-orbit physics, electrically addressable qubit states, and multiple routes to long-distance coupling (Salfi, 2020). On the other side, the remaining challenges are explicitly engineering and architectural: robust single-qubit measurement in scalable arrays, coherent coupling that does not sacrifice the sweet spots required for long $J=3/2$16, integration with nanomechanical cavities and superconducting resonators, practical microwave-to-optical transducers, and architectures that preserve coherence while exploiting the electric and elastic couplings that make acceptors attractive (Salfi, 2020). A plausible implication is that acceptor spin qubits are best understood not as a variant of donor exchange qubits, but as a distinct impurity-defined spin-orbit platform whose scalability depends on controlled use of quadrupolar, dipolar, phononic, and cavity-mediated interactions rather than on nearest-neighbor exchange alone.