Z. M. McIntyre and colleagues at the University of Basel report a potential path toward dramatically extending qubit coherence times. The researchers estimate a coherence time of 100 seconds is achievable using a novel qubit encoding, based on the nuclear spins of antimony (^(123)Sb) embedded in silicon. This approach uses the antimony’s quadrupolar nuclei to suppress noise that leads to qubit dephasing. The work also proposes a two-qubit gate design capable of 99% fidelity with a quadrupolar splitting enhancement of approximately four times current measurements.
Quadrupolar Interactions and Z_2 Symmetry in Nuclear Spins
Measurements of antimony-123 (^(123)Sb) nuclear spins in silicon revealed times reaching tens of milliseconds, alongside exceptionally long times exceeding tens of seconds. This work introduces a nuclear-spin qubit encoding designed to extend the dephasing time, , of quantum information stored within these nuclear spins by using a clock transition in the nuclear-spin spectrum. The foundation of this approach lies in the quadrupolar interaction between spins with I ≥ 1 and the electric-field gradient at the nuclear spin’s location, a phenomenon important for manipulating nuclear spins.
Spin cat states, demonstrated experimentally with ^(123)Sb donors in silicon, offer a pathway to passively suppress errors at the hardware level through careful qubit encoding, rather than relying on post-error correction or dynamical decoupling. Z. M.
McIntyre and Daniel Loss compute 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 symmetry by charge-noise-induced fluctuations of the quadrupolar tensor. Using measured parameters for antimony (^(123)Sb) donors in silicon, they estimate that a coherence time of = 100 s could be achieved with this encoding.
Establishing a substantial electric-field gradient requires breaking the tetrahedral symmetry of silicon, achievable through the application of lattice strain, and is a key component of the encoding process. The interaction between the electric-field gradient and the nucleus’s electric quadrupole moment enables electrical control of the nuclear spin via nuclear electric resonance, offering a method for coherent manipulation using radio frequency electric fields instead of traditional nuclear magnetic resonance.
The study emphasizes the importance of orienting the applied magnetic field to ensure the required symmetry of the nuclear-spin Hamiltonian, a critical aspect of the encoding scheme. This encoding method relies on nuclei with spins I ≥ 1 possessing an electric quadrupole moment due to their non-spherical charge distribution, a property that interacts with electric-field gradients at the nucleus’s site. The layout of the article, the researchers note, begins with an overview of these quadrupolar interactions and the necessary conditions for achieving symmetry, providing a detailed theoretical framework for the proposed qubit encoding.
Spin Kerr-cat Qubit Encoding with Nuclear-Spin States
The spin Kerr-cat qubit uses a clock transition within quadrupolar nuclei to achieve a first-order suppression of noise leading to qubit dephasing. Researchers detail how the basis states of this qubit are defined by the two lowest energy levels of a -symmetric nuclear-spin Hamiltonian and are well approximated by spin cat states. These spin cat states are constructed from linear combinations of spin coherent states, maximizing their distinguishability and ensuring an equal projection along the axis.
This approach mirrors the principles behind Kerr-cat qubits used in circuit QED, which are encoded in the lowest energy levels of a squeezed Kerr-nonlinear oscillator, using cat states with fixed photon-number parity. While both qubit types employ cat states, the spin Kerr-cat qubit is uniquely positioned to address dephasing, a critical limitation in spin qubit technology, by using the inherent properties of nuclear spins.
The spin cat states have been previously considered as logical basis states for quantum error correcting codes designed to detect dephasing errors. These states are designed to be maximally distinguishable in their azimuthal coordinate, further enhancing their resilience against noise. Single-qubit states are then defined by points on the Bloch sphere, achievable through rotations of a reference state, such as |0⟩, by angles θ and φ, allowing for full control over the qubit’s quantum state.
Dephasing Time Enhancement via Clock Transitions
Researchers detail how tuning the qubit splitting to this specific transition results in a first-order insensitivity to magnetic-field fluctuations, a critical factor in extending coherence times. This approach uses the unique properties of nuclei with a spin length of I ≥ 1, distinguishing it from many current qubit designs focused on electron spins. These findings, detailed in the work, suggest a strong foundation for building more stable qubits.
The team’s analysis focuses on quantifying the enhancement in dephasing time, a measure of how long a qubit maintains its quantum state, when operating at this clock transition, and also investigates relaxation mechanisms caused by charge noise. This noise couples the nuclear spin to fluctuations in the electric-field gradient at the nuclear site.
The concept of clock transitions, where the frequency of a transition remains stable against external disturbances, originates in atomic physics and is used in precise timekeeping standards. While previously explored in optical quantum memories using rare-earth dopants and proposed for donor-bound electron-spin qubits, this work applies the principle directly to nuclear-spin qubits. The researchers found that the lowest number-parity doublet, the two lowest energy levels, can define a qubit with a significantly extended dephasing time compared to conventional approaches.
“Clock transitions are characterized by a vanishing first derivative of the transition frequency with respect to magnetic-field fluctuations,” the study explains, highlighting the mechanism behind the improved stability. The analysis also considers practical implementation details, including single-qubit gates, initialization and readout methods, and A proposed two-qubit gate relies on electron-spin mediation, compatible with techniques for shuttling qubits between locations.
The team estimates that achieving a 99% fidelity for this gate requires enhancing measured quadrupolar splittings by a factor of approximately four, a specific engineering target for future development. The dispersion of eigenenergies, or energy levels, will exhibit avoided crossings between states of the same parity, while states of opposite parity may cross, a phenomenon the researchers accounted for in their calculations.
^(123)Sb Nuclear Spins Achieve 100s Coherence Potential
Antimony-123 (^(123)Sb) nuclear spins in silicon offer a potential pathway to a coherence time of = 100 s, according to a new analysis of qubit encoding strategies. This estimate, based on measured parameters for antimony donors, significantly exceeds previously demonstrated coherence in similar systems and suggests a promising material for building more stable quantum computers. This work builds on prior demonstrations of spin-cat coherence in ^(123)Sb donors, where tens of milliseconds of coherence have already been observed.
However, the new analysis focuses on extending this lifetime through a noise-robust encoding, rather than relying on error correction techniques that address errors after they occur. The team’s calculations consider the impact of charge-noise-induced fluctuations on the quadrupolar tensor.
The selection of antimony, with a spin length of I = 7/2 and a gyromagnetic ratio of 5.55 MHz/T, is not arbitrary. While earlier efforts focused on phosphorous-31 (^(31)P), antimony offers distinct advantages for this particular encoding scheme. Bismuth-209 (^(209)Bi), with a spin length of I = 9/2, is also identified as a potential high-spin donor for silicon-based quantum computing architectures. The analysis also acknowledges that the fidelity estimate for the two-qubit gate assumes negligible errors impacting the electron during the shuttling and readout processes, an area for future investigation.
The researchers’ work demonstrates a shift toward hardware-level error suppression, complementing ongoing efforts in quantum error correction. By carefully selecting qubit encodings and materials, it may be possible to build quantum computers that are inherently more resilient to noise.
The proposed qubit design relies on tuning the qubit splitting to a clock transition, a phenomenon originating from the interaction between the nuclear spin and the electric-field gradient at the donor site. The work also details how single-qubit gates, initialization, and readout could be performed.
👉 More information
🗞 Spin Kerr-cat qubits
✍️ Z. M. McIntyre and Daniel Loss
🧠 DOI: http://link.aps.org/doi/10.1103/z3dj-95j6
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