Spin Kerr-cat qubits

The spin Kerr-cat encoding introduces a noise-robust qubit design leveraging clock transitions in quadrupolar nuclei to achieve first-order suppression of noise, potentially extending coherence times significantly.

How it works

1. The basis states of the spin Kerr-cat qubit are defined by the two lowest levels of a Z2-symmetric nuclear-spin Hamiltonian and are well approximated by spin cat states.

2. This encoding leverages a clock transition in the spectrum of quadrupolar nuclei (having spin length I ≥ 1) to achieve a first-order suppression of noise leading to qubit dephasing.

3. The definite ˆIz parity of these qubit basis states is ensured by an overall Z2 symmetry of the nuclear-spin Hamiltonian, realized through proper alignment of the applied magnetic field in relation to the EFG tensor.

4. The spin cat states Θ, ϕ⟩± are defined as symmetric and anti-symmetric linear combinations of two spin coherent states that are maximally distinguishable in their azimuthal coordinate.

5. The qubit splitting ∆ is tuned to a local maximum by tuning the magnetic-field strength so that b0 = γnB0 assumes a particular value in units of Q.

6. The basis states of the spin Kerr-cat qubit are then given by 1⟩ = Θ0, π/2⟩+,0⟩ = Θ0, π/2⟩−.

Dephasing and Noise Robustness

: The spin Kerr-cat encoding is designed to suppress the dominant error source affecting spin qubits. This contrasts with Kerr-cat encoding which suppresses photon loss errors in superconducting resonators. The paper shows that this strategy on its own could enhance the decoherence time of encoded quantum information by several orders of magnitude without active quantum error correction or dynamical decoupling. 1/f noise, which couples the nuclear spin to non-uniform electric fields, is addressed because the clock transition is determined by the strength of the applied magnetic field relative to Q, causing its first derivative with respect to small variations in Q to vanish. This vanishing first derivative ensures robustness against charge noise as well. The enhancement in dephasing time T∗2,ct relative to T∗2 is quadratic, described by T∗2,ct > A(I, η)√2πQ(T

Relaxation and Error Channels

: Relaxation due to charge noise results from coupling between the nuclear spin and electric-field gradients. The paper estimates that for a two-level charge fluctuator (TLF) located at a distance of 10 nm, bitflip errors 0⟩ ↔ 1⟩ due to random movements of the charge will only occur on a timescale exceeding 1000 s for Q = 100 kHz. The generalization to an ensemble of TLFs shows that the transition rate Γ10 scales like β2max, which falls off rapidly with increasing rmin and will ultimately be set predominantly by this TLF. Phonon-driven relaxation is estimated to be negligibly small relative to charge noise. The transition rate from state 0⟩ to m ≠ 0⟩ is bounded by Γm0 = I 2J⊥(−ϵm)t, and for a realistic model of charge noise, this rate is estimated to be less than 10-31(x′−4m4 Hz. The relaxation channel specific to this encoding results from misalignment or drift of the magnetic field in the principal axis system of the electric-field gradient at the site of the nuclear spin. The bitflip time due to charge noise is estimated to be > 10 cubed s for x′ = 10 nm and Q = 100 kHz. A reasonable estimate for T∗2,ct given Q = 50 kHz, I = 7/2, η = 1, and T

Operation and Control

**: Single-qubit control can be implemented in the absence of the electron using techniques well established in NMR. Rotations about Zˆ can be realized through free evolution under the nuclear-spin Hamiltonian H, where the spin Kerr-cat computational basis states 1⟩ = Θ0, π/2⟩+ and 0⟩ = Θ0, π/2⟩− are eigenstates separated in energy by the qubit splitting ∆ ∝ Q. Rotations about a second, non-collinear axis could be realized through standard NMR techniques by applying an oscillating (AC) magnetic field at a frequency resonant with the qubit splitting. The strength of the Rabi drive depends on a dimensionless factor f(θ) determined by the orientation of the rotation axis nˆ⊥ in the xy plane of the quadrupole tensor. Two-qubit gates can be realized by entangling spin Kerr cats with mobile electron spins that can hop on and off donor potentials. This gate is achieved by sequentially entangling an electron with each qubit and then measuring the spin of the electron (Fig. 4). The fidelity FCZ of this two-qubit gate can be estimated as FCZ = (2Fg − 1) − ϵshuttle − ϵm, where ϵshuttle and ϵm are errors impacting the electron during shuttling and readout. Readout is performed by allowing a spin-down electron to tunnel onto the donor potential, flipping its state conditioned on the state of the nuclear spin, and then using an energy-selective tunneling process (Elzerman readout [76]) to a nearby electron reservoir. Initialization can be achieved by preparing the fully polarized spin state I, I⟩, rotating about the ˆIy axis to prepare the spin coherent state Θ0, 0⟩, and measuring the nuclear spin’s ˆIz parity. A single-shot readout of the spin Kerr-cat qubit can be realized using an entangling operation UCR. The fidelity Fg is plotted as a function of Q and η in Fig. 5. The paper concludes that two-qubit gates mediated by hopping electrons could be performed with 99% fidelity given only modest improvements in the quadrupole coupling relative to the values measured experimentally to date. Given the availability of such an entangling gate between nuclear and electron spins, an interesting future direction would be to consider the use of spin Kerrcat qubits as quantum memories for electron-spin qubits.

1. Improved Qubit Encoding for Noise Robustness

The spin Kerr-cat encoding achieves a first-order suppression of noise leading to qubit dephasing by tuning the qubit splitting to a clock transition in the quadrupolar interaction, which is insensitive to magnetic field fluctuations. This allows the system to suppress dominant error source affecting spin qubits and potentially enhance coherence times by several orders of magnitude without active quantum error correction.

2. Enhanced Two-Qubit Gate Fidelity

The paper estimates that with an enhancement of measured quadrupolar splittings by a factor of approximately 4, a gate fidelity of 99% could be achieved for spin Kerr-cat qubits encoded in antimony-123 nuclear spins when mediated by hopping electrons. This capability enables the realization of high-fidelity two-qubit gates between nuclear spins using electron shuttling.

3. Robust Initialization and Readout

Initialization can be performed by preparation of the fully polarized spin state I, I⟩, followed by a rotation to prepare the spin coherent state Θ0, 0⟩. Readout is achieved through an entangling operation UCR where a measurement of the electron in the plus or minus basis maps onto a Z-basis measurement of the nuclear spin.

4. Noise Model Robustness for Decoherence Time

The encoding provides a significant enhancement in qubit dephasing time T∗2 compared to linear noise models, particularly 1/f noise, by tuning the magnetic field such that the first derivative of the qubit splitting ∆ with respect to b vanishes. This robustness against charge noise suggests that the vanishing first derivative of the spin Kerr-cat splitting should therefore provide some robustness against charge noise as well.

5. Modeling and Mitigating Charge Noise Errors

The analysis shows that errors due to charge noise can be quantified by estimating the transition rate Gamma10, which is dependent on the spectral density J⊥ evaluated at-Delta. For a single two-level fluctuator (TLF) at 10 nm, this results in a bitflip time of Gamma-10 greater than 103 seconds. This quantification allows for predicting error rates based on the distance between noise sources and the characteristic Q value.

6. Phonon Noise Suppression

The relaxation rate due to coupling to phonons is estimated to be effectively equal to Gamma10, suggesting that phonons are unlikely to dominate over charge noise as a source of qubit relaxation. This implies that the encoding is effective against both dominant dephasing and relaxation channels under the considered parameters.

The use of noise-robust qubit encodings provides a way of extending the lifetime of quantum information at the hardware level. In this work, we introduce the spin Kerr-cat encoding, which leverages a sweet spot near a clock transition in the spectrum of quadrupolar nuclei (having spin I at least 1) to achieve a suppression of charge and magnetic noise leading to qubit dephasing. The basis states of the spin Kerr-cat qubit are given by the two lowest levels of a Z 2-symmetric nuclear-spin Hamiltonian and are well approximated by spin cat states for sufficiently high asymmetry parameters. For nuclei in highly asymmetric electric-field gradients, the nuclear spin could therefore be initialized in the spin-Kerr-cat subspace using SU(2) rotations and measurement rather than full SU(2I+1) control. We calculate the dephasing time of the spin Kerr-cat qubit under a model of 1/f noise, as well as relaxation of the qubit due to breaking of the Z 2 symmetry by charge-noise-induced fluctuations of the quadrupolar tensor. Using measured parameters for antimony (123 Sb) donors in silicon, we estimate that second-scale T 2* times could be achieved with this encoding, with bitflips and leakage occurring on significantly longer timescales. We propose a two-qubit gate acting on spin Kerr-cat qubits and electron-spin qubits, enabled by adiabatically moving an electron on and off the donor potential. Based on numerical simulations, we estimate that with an enhancement of measured quadrupolar couplings by a factor of about 2, a gate fidelity exceeding 99% could be achieved for spin Kerr cats encoded in 123 Sb, with a gate time under 100 ns. By entangling the nuclear spin with an electron spin in this way, the spin Kerr-cat qubit could then be measured using a Pauli spin blockade readout of the electron.