Schrödinger cat states of a nuclear spin qudit in silicon
Xi Yu, Benjamin Wilhelm, Holmes, Danielle, Arjen Vaartjes, Daniel Schwienbacher, Martin Nurizzo, Anders Kringhøj, Mark R. van Blankenstein, Alexander Jakob, Pragati Gupta, Fay E. Hudson, Kohei M. Itoh, Riley J. Murray, Robin Blume-Kohout, Thaddeus D. Ladd, Anand, Namit, Andrew S. Dzurak, Barry C. Sanders, David N. Jamieson, Andrea Morello
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.
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.
We construct a fault-tolerant quantum error-correcting protocol based on a qubit encoded in a large spin qudit using a spin-cat code, analogous to the continuous-variable cat encoding. With this, we can correct the dominant error sources, namely processes that can be expressed as error operators that are linear or quadratic in the components of angular momentum. Such codes tailored to dominant error sources can exhibit superior thresholds and lower resource overheads when compared to those designed for unstructured noise models. A key component is the gate that preserves the rank of spherical tensor operators. Categorizing the dominant errors as phase and amplitude errors, we demonstrate how phase errors, analogous to phase-flip errors for qubits, can be effectively corrected. Furthermore, we propose a measurement-free error-correction scheme to address amplitude errors without relying on syndrome measurements. Through an in-depth analysis of logical gate errors, we establish that the fault-tolerant threshold for error correction in the spin-cat encoding surpasses that of standard qubit-based encodings. We consider a specific implementation based on neutral-atom quantum computing, with qudits encoded in the nuclear spin of 87 Sr , and show how to generate the universal gate set, including the rank-preserving gate, using quantum control and the Rydberg blockade. These findings pave the way for encoding a qubit in a large spin with the potential to achieve fault tolerance, high threshold, and reduced resource overhead in quantum information processing. Published by the American Physical Society 2024
Spin precession is a textbook example of the dynamics of a quantum system that exactly mimics its classical counterpart. Here, we challenge this view by certifying the quantumness of exotic states of a nuclear spin through its uniform precession. The key to this result is measuring the positivity, instead of the expectation value, of the x -projection of the precessing spin, and using a spin >1/2 qudit that is not restricted to semi-classical spin coherent states. The experiment is performed on a single spin-7/2 123 Sb nucleus, implanted in a silicon nanoelectronic device, amenable to high-fidelity preparation, control, and projective single-shot readout. Using Schrödinger cat states and other bespoke states of the nucleus, we violate the classical bound by 18 standard deviations, proving that no classical probability distribution can explain the statistic of this spin precession and highlighting our ability to prepare quantum resource states with high fidelity in a single-atomic-scale qudit.
The realization of effective quantum error correction protocols remains a central challenge in the development of scalable quantum computers. Employing high-dimensional quantum systems (qudits) can offer more hardware-efficient protocols than qubit-based approaches. Using electron-nuclear double resonance, we implement a logical qubit encoded on the four states of a I=3/2 nuclear spin hyperfine-coupled to an S=1/2 electron spin qubit; the encoding protects against the dominant decoherence mechanism in such systems, i.e., fluctuations of the quantizing magnetic field. We explore the dynamics of the encoded state both under a controlled application of the fluctuation and under natural decoherence processes. Our results confirm the potential of these proposals for practical, implementable, fault-tolerant quantum memories.
Encoding quantum information in d-dimensional systems (qudits) with d>2 offers appealing new directions for resource-efficient quantum computing. However, protecting the coherence of nonclassical qudit resource states remains an open challenge. Dynamical decoupling (DD) is a well-established technique for extending coherence, but its application to qudits is hindered by the quadratic scaling of the decoupling sequence overhead with the dimension d. As a result, demonstrations of qudit DD have so far been limited to low-dimensional qudits. Here, we show that noise-biased qudits enable a nontrivial reduction of this overhead, for example, from quadratic to linear. We perform qudit DD on a model d=8 qudit, the nuclear spin of an antimony donor in silicon. We benchmark DD performance on highly nonclassical states such as uniform superpositions, cat states, and Haar random states and measure coherence improvements up to a factor 5 to 10. For a spin-cat error correction code, continuous DD extends the coherence time of the logical superposition ∣+⟩L to 225~ms, compared to 19~ms under free evolution and 85~ms with a Hahn echo. This improvement shows that DD sequences tailored to a system's noise bias can significantly enhance the lifetime of encoded logical qubits.
Nonclassical states are essential resources for quantum computing that cannot be described by classical probability theory. In this thesis, we engineer nonclassical states in two separate systems: finite-dimensional high-spin nuclei, and infinite-dimensional bosonic cavity modes. In the finite-dimensional case, we study a spin−7/2 antimony (123Sb) donor in silicon, forming a D = 8 qudit. The required non-linearity for nonclassicality is provided by the quadratic quadrupole interaction. We fully characterize the 3 × 3 quadrupole with nuclear magnetic resonance (NMR) spectroscopy in a rotating magnetic field. Inspired by this model, we improve the QND readout of the 123Sb nucleus by developing an adaptive readout protocol. In the ancilla-mediated nuclear spin readout, we induce unwanted spin-flip transitions. The adaptive protocol minimizes errors by relying on negative-resultmeasurements, achieving an increase in readout fidelity from 98.93(7) % to 99.61(4) %. We proceed by improving the coherence of the qudit by applying qudit dynamical decoupling (DD). We show that implicit noise biases can simplify lengthy pulse sequences. By cyclically permuting the spin eigenstates, we decouple all dephasing noise and gain a factor 5-10 increase in the coherence time of various nonclassical states. We additionally protect logical spin cat qubit states. By applying a continuous spin-locking drive tailored to the noise bias, we achieve an order-of-magnitude improvement. We then directly address one of the main questions of this thesis by certifying the quantumness of nonclassical states in the 123Sb qudit through a uniform precession protocol. We provide a first experimental demonstration of nonclassicality in the dynamics of a spin. The quantum protocol score exceeds the classical bound by 18 standard deviations, ruling out any classical explanation for the observed set of measurements. We complement the finite-dimensional results by creating nonclassical bosonic (D = ∞) squeezed states with a frequency-tunable kinetic inductance parametric amplifier (KIPA). With a squeezed-state-receiver setup, we demonstrate 7.8(2) dBs of intercavity squeezing. These high levels of squeezing stay robust against the exposure of high magnetic fields over 1 T and elevated temperatures above 1 K. Lastly, we provide a future outlook by developing methods to scale spin qubits, industrially fabricated on 300mm wafers, aimed at improving the scalability of spin qubits
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.