Cat codes are one of the promising encoding schemes for quantum information that enables hardware-efficient error correction. So far, only photonic cat states produced by superconducting circuits have been under consideration. Their realization within coherent spin states (CSS) can pave the way toward nuclear magnetic resonance-based implementations of this new class of codes. Here, we show how the crucial superpositions of CSS can be generated for the nuclear spins. This is through the intrinsic electric quadrupole interaction involving a critical role by the biaxiality term that is readily available, as in strained heterostructures of semiconductors, or defect centers having nearby quadrupolar spins. The persistence of the cat states is achieved using a rotation pulse so as to harness the underlying fixed points of the classical Hamiltonian. We classify the two distinct types as polar- and equator-bound over the Bloch sphere with respect to principal axes, both of which are, so-called mod~2 cat states. Their optimal performance as well as sensitivity under numerous parameter deviations are analyzed. Finally, we show that a higher level, mod 4 cat state, with one of the legs rotating can also be produced along the same spirit using three pulses.
Cat codes are one of the promising encoding schemes for quantum information that enables hardware-efficient error correction. So far, only photonic cat states produced by superconducting circuits have been under consideration. Their realization within coherent spin states (CSS) can pave the way toward nuclear magnetic resonance-based implementations of this new class of codes. Here, we show how the crucial superpositions of CSS can be generated for the nuclear spins. This is through the intrinsic electric quadrupole interaction involving a critical role by the biaxiality term that is readily available, as in strained heterostructures of semiconductors, or defect centers having nearby quadrupolar spins. The persistence of the cat states is achieved using a rotation pulse so as to harness the underlying fixed points of the classical Hamiltonian. We classify the two distinct types as polar- and equator-bound over the Bloch sphere with respect to principal axes, both of which are, so-called mod~2 cat states. Their optimal performance as well as sensitivity under numerous parameter deviations are analyzed. Finally, we show that a higher level, mod 4 cat state, with one of the legs rotating can also be produced along the same spirit using three pulses.
Spin cat states are superpositions of two or more coherent spin states (CSSs) that are distinctly separated over the Bloch sphere. Additionally, the nuclei with angular momenta greater than 1/2 possess a quadrupolar charge distribution. At the intersection of these two phenomena, we devise a simple scheme for generating various types of nuclear-spin cat states. The native biaxial electric quadrupole interaction that is readily available in strained solid-state systems plays a key role here. However, the fact that built-in strain cannot be switched off poses a challenge for the stabilization of target cat states once they are prepared. We remedy this by abruptly diverting via a single rotation pulse the state evolution to the neighborhood of the fixed points of the underlying classical Hamiltonian flow. Optimal process parameters are obtained as a function of electric field gradient biaxiality and nuclear-spin angular momentum. The overall procedure is seen to be robust under 5% deviations from optimal values. We show that higher-level cat states with four superposed CSS can also be formed using three rotation pulses. Finally, for open systems subject to decoherence we extract the scaling of cat-state fidelity damping with respect to the spin quantum number. This reveals rates greater than the dephasing of individual CSSs. Yet, our results affirm that these cat states can preserve their fidelities for practically useful durations under the currently attainable decoherence levels.
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≥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 Z2-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 Z2 symmetry by charge-noise-induced fluctuations of the quadrupolar tensor. Using measured parameters for antimony (123Sb) donors in silicon, we estimate that second-scale T2∗ 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 ≈2, a gate fidelity exceeding 99% could be achieved for spin Kerr cats encoded in 123Sb, 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.
Bias-tailored quantum error correcting codes (QECCs) offer a higher error threshold than standard QECCs and have the potential to achieve lower logical errors with less space overhead. The spin-cat qubit, encoded in a large nuclear spin-F system, is a promising candidate for bias-tailored QECCs. Yet its feasibility is hindered by the difficulty of performing fast covariant SU(2) rotation with arbitrary rotation angles for nuclear spins and by a lack of noise characterization for gate operations in neutral atom platforms. Here we demonstrate single-qubit controls of 173Yb spin-cat qubits with nuclear spin I=5/2 in an optical tweezer array. We implement a covariant SU(2) rotation and non-linear rotations by optical beams and achieve an averaged single-Clifford gate fidelity of 0.961−5+5. The measurement of the coherence time and spin relaxation time shows that the idling error becomes increasingly biased toward dephasing errors as the magnitude of the encoded sublevel ∣mF∣ increases. Furthermore, we benchmark the noise bias of rank-preserving gates on spin-cat qubits, demonstrating a finite bias of 18−11+132, in contrast to the case of the two-level system in 171Yb, which shows no bias within the experimental uncertainty. Our work demonstrates the feasibility of spin-cat qubits for realizing bias-tailored QECCs, paving the way for achieving hardware-efficient quantum error correction.
Quantum error correction (QEC) is indispensable for scalable quantum computing, but implementing it with minimal hardware overhead remains a central challenge. Large spin systems with collective degrees of freedom offer a promising route to reducing the control complexity of qubit architectures while retaining a large Hilbert space for fault-tolerant encoding. However, existing proposals for logical gates and QEC in spin ensembles generally rely on inefficient higher-order interactions. Here we introduce spin-N-Cat codes, which encode logical qubits in superpositions of spin-coherent states and generalize bosonic Cat codes to the modular subspaces of permutationally symmetric spin ensembles. The code corrects collective and individual dephasing, excitation, and decay errors. We also present an efficient physical realization in central-spin systems, such as a quantum dot, where encoding, decoding, and a universal, fault-tolerant, and bias-preserving gate set are implemented using only first-order interactions. Numerical simulations demonstrate high logical fidelity under dephasing and excitation-decay noise, independent of noise bias, and that full QEC cycles are feasible with realistic microscopic parameters. For the large collective spins available in quantum dots, this translates into a substantial extension of coherence time. Our results establish spin-N-Cat codes as a scalable, hardware-efficient approach to QEC in spin-based quantum architectures.
High-dimensional quantum systems are a valuable resource for quantum information processing. They can be used to encode error-correctable logical qubits, which has been demonstrated using continuous-variable states in microwave cavities or the motional modes of trapped ions. For example, high-dimensional systems can be used to realise `Schrödinger cat' states, superpositions of widely displaced coherent states that can also be used to illustrate quantum effects at large scales. Recent proposals have suggested encoding qubits in high-spin atomic nuclei, finite-dimensional systems that can host hardware-efficient versions of continuous-variable codes. Here we demonstrate the creation and manipulation of Schrodinger cat states using the spin-7/2 nucleus of an antimony atom embedded in a silicon nanoelectronic device. We use a multi-frequency control scheme to produce spin rotations that preserve the symmetry of the qudit, and constitute logical Pauli operations for qubits encoded in the Schrodinger cat states. Our work demonstrates the ability to prepare and control nonclassical resource states, a prerequisite for applications in quantum information processing and quantum error correction using our scalable, manufacturable semiconductor platform.
Control over nuclear-spin fluctuations is essential for processes that rely on preserving the quantum state of an embedded system. For this purpose, squeezing is a viable alternative, so far that has not been properly exploited for the nuclear spins. Of particular relevance in solids is the electric quadrupole interaction (QI), which operates on nuclei having spin higher than 1/2. In its general form, QI involves an electric-field gradient (EFG) biaxiality term. Here, we show that as this EFG biaxiality increases, it enables continuous tuning of single-particle squeezing from the one-axis twisting to the two-axis countertwisting limits. A detailed analysis of QI squeezing is provided, exhibiting the intricate consequences of EFG biaxiality. The initial states over the Bloch sphere are mapped out to identify those favorable for fast initial squeezing, or for prolonged squeezings. Furthermore, the evolution of squeezing in the presence of a phase-damping channel and an external magnetic field are investigated. We observe that dephasing drives toward an antisqueezed terminal state, the degree of which increases with the spin angular momentum. Finally, QI squeezing in the limiting case of a two-dimensional EFG with a perpendicular magnetic field is discussed, which is of importance for two-dimensional materials, and the associated beat patterns in squeezing are revealed.