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arXiv:2407.17407 · Physical Review Applied

High- E J / E C transmon qudits with up to 12 levels

Zihao Wang, Rayleigh W. Parker, Elizabeth Champion, Machiel Blok

Published Mar 20, 2025• 29 citations
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Abstract

Qudits provide a resource-efficient alternative to qubits for quantum information processing. The multilevel nature of the transmon, with its individually resolvable transition frequencies, makes it an attractive platform for superconducting circuit-based qudits. In this work, we systematically analyze the trade-offs associated with encoding high-dimensional quantum information in fixed-frequency transmons. Designing high ${E}_{J}/{E}_{C}$ ratios of up to 325, we observe up to 12 levels ($d=12$) on a single transmon. Despite the decreased anharmonicity, we demonstrate process infidelities ${e}_{f}<3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ for qubitlike operations in each adjacent-level qubit subspace in the lowest ten levels. Furthermore, we achieve a ten-state readout assignment fidelity of 93.8% with the assistance of deep neural network classification of a multitone dispersive measurement. We find that the Hahn echo time ${T}_{2E}$ for the higher levels is close to the limit of ${T}_{1}$ decay, primarily limited by bosonic enhancement. We verify the recently observed Josephson harmonic model, finding that it yields better predictions for the transition frequencies and charge dispersion. Finally, we show strong $ZZ$-like coupling between the higher energy levels in a two-transmon system. Our high-fidelity control and readout methods, in combination with our comprehensive characterization of the transmon model, suggest that the high-${E}_{J}/{E}_{C}$ transmon is a powerful tool for exploring excited states in circuit quantum electrodynamics.

DOI
10.1103/physrevapplied.23.034046

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Qudits hold great promise for efficient quantum computation and the simulation of high-dimensional quantum systems [1]. However, existing control and measurement schemes for qudit systems scale unfavorably with qudit dimension since they decompose S U ( d ) operations into series of qubitlike rotations and perform measurements on a small number of states [2–6]. Here, we address these challenges by employing simultaneous multifrequency drives to generate rotations and projections in an effective spin- 7 / 2 system mapped onto the energy eigenstates of a superconducting circuit. We implement single-shot readout of the eight states using a multitone dispersive readout ( F assignment = 88.3 % ) and exploit the strong nonlinearity in a high- E J / E C transmon to simultaneously address each transition and realize a spin displacement operator. Combining this displacement operator with a virtual SNAP gate, we realize arbitrary single-qudit unitary operations in O ( d ) physical pulses and extract spin displacement gate fidelities ranging from 0.997 to 0.989 for virtual spins of size j = 1 to j = 7 / 2 . We demonstrate the potential of our control scheme in three ways: the direct measurement of the spin qudit Wigner function, randomized benchmarking of a logical qubit encoded into the qudit state, and randomized benchmarking of the full qudit Clifford group. In the latter experiment, we implement the d -dimensional quantum Fourier transform with an average gate fidelity of 0.91(6) in d = 8 . Our multifrequency approach to qudit control and measurement can be readily extended to other physical platforms that realize a multilevel system coupled to a cavity and can become a building block for efficient qudit-based quantum computation and simulation.
The fidelity and quantum nondemolition character of the dispersive readout in circuit QED are limited by unwanted transitions to highly excited states at specific photon numbers in the readout resonator. This observation can be explained by multiphoton resonances between computational states and highly excited states in strongly driven nonlinear systems, analogous to multiphoton ionization in atoms and molecules. In this work, we utilize the multilevel nature of high- E J / E C transmons to probe the excited-state dynamics induced by strong drives during readout. With up to ten resolvable states, we quantify the critical photon number of ionization, the resulting state after ionization, and the fraction of the population transferred to highly excited states. Moreover, using pulse shaping to control the photon number in the readout resonator in the high-power regime, we tune the adiabaticity of the transition and verify that transmon ionization is a Landau-Zener-type transition. We further extend these methods to a typical transmon with E J / E C ≈ 55 and probe the offset-charge dependence of ionization dynamics in a timed-resolved manner. Our experimental results agree well with the theoretical prediction from a semiclassical driven transmon model and may guide future exploration of strongly driven nonlinear oscillators.
Qubit measurements are central to quantum information processing.In the field of superconducting qubits, standard readout techniques are limited not only by the signal-to-noise ratio, but also by state relaxation during the measurement.In this work, we demonstrate that the limitation due to relaxation can be suppressed by using the many-level Hilbert space of superconducting circuits: In a multilevel encoding, the measurement is corrupted only when multiple errors occur.Employing this technique, we show that we can directly resolve transmon gate errors at the level of one part in 10 3 .Extending this idea, we apply the same principles to the measurement of a logical qubit encoded in a bosonic mode and detected with a transmon ancilla, implementing a proposal by Hann et al. [Phys. Rev. A 98, 022305 (2018)].Qubit state assignments are made based on a sequence of repeated readouts, further reducing the overall infidelity.This approach is quite general, and several encodings are studied; the codewords are more distinguishable when the distance between them is increased with respect to photon loss.The trade-off between multiple readouts and state relaxation is explored and shown to be consistent with the photon-loss model.We report a logical assignment infidelity of 5.8 × 10 -5 for a Fock-based encoding and 4.2 × 10 -3 for a quantum error correction code (the S ¼ 2, N ¼ 1 binomial code).Our results not only improve the fidelity of quantum information applications, but also enable more precise characterization of process or gate errors.
Superconducting qubits are a promising technology for building a scalable quantum computer. An important architecture employed in the field is called Circuit Quantum Electrodynamics (circuit QED), where such qubits are combined with high quality microwave cavities to study the interaction between artificial atoms and single microwave photons. The ultra-strong coupling achieved in these systems allows for control and readout of the quantum state of qubits to perform quantum information processing. The work on circuit QED performed in this thesis consisted of realizing an experimental setup for qubit experiments in a new laboratory, investigating the coherence and decay of higher energy levels of superconducting transmon qubits and finally demonstrating a novel coaxial form of circuit QED. Designing and building a 3D circuit QED setup involved the following main accomplishments: producing high quality 3D cavities; designing and installing the cryogenic microwave setup as well as the room temperature amplification and data acquisition circuitry; successfully developing a recipe for the fabrication of Josephson junctions; controlling and measuring superconducting 3D transmon qubits at 10mK. Several qubits were fully characterised and have shown coherence times of several microseconds and relaxation times up to 25μs. Superconducting qubits in fact possess higher energy levels that can provide significant computational advantages in quantum information applications. In experiments performed at MIT, preparation and control of the five lowest states of a transmon qubit was demonstrated, followed by an investigation of the phase coherence and decay dynamics of these higher energy levels. The decay was found to proceed mainly sequentially with relaxation times in excess of 20μs for all transitions. A direct measurement of the charge dispersion of these levels was performed to explore their characteristics of dephasing. This experiment was also reproduced on a 3D transmon fabricated and measured in Oxford, where due to a higher effective qubit temperature a multi-level decay model including thermal excitations was developed to explain the observed relaxation dynamics. Finally, a coaxial transmon, which we name the coaxmon, is presented and measured with a coaxial LC readout resonator and input/output coupling ports placed inline along the third dimension. This novel coaxial circuit QED architecture holds great promise for developing a scalable planar grid of qubits to build a quantum computer.
Transmon qubits are a cornerstone of modern superconducting quantum computing platforms. Temporal fluctuations of energy relaxation in these qubits are widely attributed to microscopic two-level systems (TLSs) in device dielectrics and interfaces, yet isolating individual defects typically relies on tuning the qubit or the TLS into resonance. We demonstrate a novel spectroscopy method for fixed-frequency transmons based on multilevel relaxation: repeated preparation of the second excited state and simultaneous T1T_1T1​ extraction of the first and second excited states reveals characteristic correlations in the decay rates of adjacent transitions. From these correlations we identify one or more dominant TLSs and reconstruct their frequency drift over time. Remarkably, we find that TLSs detuned by ≳100 MHz\gtrsim 100\,\mathrm{MHz}≳100MHz from the qubit transition can still significantly influence relaxation. The proposed method provides a powerful tool for TLS spectroscopy without the need to tune the transmon frequency, either via a flux-tunable inductor or AC-Stark shifts.
Quantum processors based on superconducting circuitry have reached the scale of hundreds of qubits and demonstrated computational capabilities that challenge state-of-the-art classical methods for carefully designed problems. Despite these advances, fundamental questions persist over how to best improve and scale the computational power of this emerging technology. Most transmon-based quantum processors encode information in two-level systems, or qubits, leaving the rich multilevel structure of the transmon largely unexploited. In this thesis, we demonstrate how the capabilities of these devices can be meaningfully extended by harnessing this intrinsic multilevel structure to encode qudits — d-level quantum systems that access a larger and more connected Hilbert space than their qubit counterparts. We begin by demonstrating universal single-qudit operation before turning to microwave-activated entanglement to realize scalable two-qudit gates. We present a high-fidelity, fully entangling two-qutrit CZ gate based on a novel tunable cross-Kerr mechanism. We further explore two-qudit gates that selectively entangle subspaces of the full Hilbert space via two-photon multipartite Raman transitions, employing these operations to generate large entangled states across a transmon array. To address the noisy nature of near-term qudit computation, we introduce and characterize error suppression techniques for qudit circuits, including dynamical decoupling and quantum error mitigation. We then synthesize these directions by exploring simulations of quantum systems that have innate multilevel interactions and are therefore well tailored to qudit operation. First we investigate digital quantum simulations of collective neutrino oscillations and the spin-1 truncated Abelian Higgs model, finding error mitigation methods critical for improving circuit performance. Finally, out of equilibrium --- or Floquet --- spin-1 systems are realized using an array of 15 superconducting qutrits to probe a novel Z3 discrete time crystalline (DTC) phase of matter with intrinsically chiral interactions. This DTC result demonstrates that native qudit hardware provides powerful access to non-equilibrium phases rooted in higher discrete symmetries, opening a route to a broader landscape of dynamical orders beyond the Z2 Ising paradigm.