High-Fidelity Measurement of Qubits Encoded in Multilevel Superconducting Circuits

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 relaxa...
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Autor*in:	Salvatore S. Elder [verfasserIn] Christopher S. Wang [verfasserIn] Philip Reinhold [verfasserIn] Connor T. Hann [verfasserIn] Kevin S. Chou [verfasserIn] Brian J. Lester [verfasserIn] Serge Rosenblum [verfasserIn] Luigi Frunzio [verfasserIn] Liang Jiang [verfasserIn] Robert J. Schoelkopf [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2020

Übergeordnetes Werk:	In: Physical Review X - American Physical Society, 2011, 10(2020), 1, p 011001
Übergeordnetes Werk:	volume:10 ; year:2020 ; number:1, p 011001

Links:	Link aufrufen Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.1103/PhysRevX.10.011001

Katalog-ID:	DOAJ052358240

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allfields	10.1103/PhysRevX.10.011001 doi (DE-627)DOAJ052358240 (DE-599)DOAJd8d78fc8871c4fcba325384b109124f3 DE-627 ger DE-627 rakwb eng QC1-999 Salvatore S. Elder verfasserin aut High-Fidelity Measurement of Qubits Encoded in Multilevel Superconducting Circuits 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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)PLRAAN2469-992610.1103/PhysRevA.98.022305]. 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=2, N=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. Physics Christopher S. Wang verfasserin aut Philip Reinhold verfasserin aut Connor T. Hann verfasserin aut Kevin S. Chou verfasserin aut Brian J. Lester verfasserin aut Serge Rosenblum verfasserin aut Luigi Frunzio verfasserin aut Liang Jiang verfasserin aut Robert J. Schoelkopf verfasserin aut In Physical Review X American Physical Society, 2011 10(2020), 1, p 011001 (DE-627)666214115 (DE-600)2622565-7 21603308 nnns volume:10 year:2020 number:1, p 011001 https://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/article/d8d78fc8871c4fcba325384b109124f3 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/toc/2160-3308 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2020 1, p 011001
spelling	10.1103/PhysRevX.10.011001 doi (DE-627)DOAJ052358240 (DE-599)DOAJd8d78fc8871c4fcba325384b109124f3 DE-627 ger DE-627 rakwb eng QC1-999 Salvatore S. Elder verfasserin aut High-Fidelity Measurement of Qubits Encoded in Multilevel Superconducting Circuits 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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)PLRAAN2469-992610.1103/PhysRevA.98.022305]. 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=2, N=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. Physics Christopher S. Wang verfasserin aut Philip Reinhold verfasserin aut Connor T. Hann verfasserin aut Kevin S. Chou verfasserin aut Brian J. Lester verfasserin aut Serge Rosenblum verfasserin aut Luigi Frunzio verfasserin aut Liang Jiang verfasserin aut Robert J. Schoelkopf verfasserin aut In Physical Review X American Physical Society, 2011 10(2020), 1, p 011001 (DE-627)666214115 (DE-600)2622565-7 21603308 nnns volume:10 year:2020 number:1, p 011001 https://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/article/d8d78fc8871c4fcba325384b109124f3 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/toc/2160-3308 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2020 1, p 011001
allfields_unstemmed	10.1103/PhysRevX.10.011001 doi (DE-627)DOAJ052358240 (DE-599)DOAJd8d78fc8871c4fcba325384b109124f3 DE-627 ger DE-627 rakwb eng QC1-999 Salvatore S. Elder verfasserin aut High-Fidelity Measurement of Qubits Encoded in Multilevel Superconducting Circuits 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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)PLRAAN2469-992610.1103/PhysRevA.98.022305]. 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=2, N=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. Physics Christopher S. Wang verfasserin aut Philip Reinhold verfasserin aut Connor T. Hann verfasserin aut Kevin S. Chou verfasserin aut Brian J. Lester verfasserin aut Serge Rosenblum verfasserin aut Luigi Frunzio verfasserin aut Liang Jiang verfasserin aut Robert J. Schoelkopf verfasserin aut In Physical Review X American Physical Society, 2011 10(2020), 1, p 011001 (DE-627)666214115 (DE-600)2622565-7 21603308 nnns volume:10 year:2020 number:1, p 011001 https://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/article/d8d78fc8871c4fcba325384b109124f3 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/toc/2160-3308 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2020 1, p 011001
allfieldsGer	10.1103/PhysRevX.10.011001 doi (DE-627)DOAJ052358240 (DE-599)DOAJd8d78fc8871c4fcba325384b109124f3 DE-627 ger DE-627 rakwb eng QC1-999 Salvatore S. Elder verfasserin aut High-Fidelity Measurement of Qubits Encoded in Multilevel Superconducting Circuits 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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)PLRAAN2469-992610.1103/PhysRevA.98.022305]. 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=2, N=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. Physics Christopher S. Wang verfasserin aut Philip Reinhold verfasserin aut Connor T. Hann verfasserin aut Kevin S. Chou verfasserin aut Brian J. Lester verfasserin aut Serge Rosenblum verfasserin aut Luigi Frunzio verfasserin aut Liang Jiang verfasserin aut Robert J. Schoelkopf verfasserin aut In Physical Review X American Physical Society, 2011 10(2020), 1, p 011001 (DE-627)666214115 (DE-600)2622565-7 21603308 nnns volume:10 year:2020 number:1, p 011001 https://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/article/d8d78fc8871c4fcba325384b109124f3 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/toc/2160-3308 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2020 1, p 011001
allfieldsSound	10.1103/PhysRevX.10.011001 doi (DE-627)DOAJ052358240 (DE-599)DOAJd8d78fc8871c4fcba325384b109124f3 DE-627 ger DE-627 rakwb eng QC1-999 Salvatore S. Elder verfasserin aut High-Fidelity Measurement of Qubits Encoded in Multilevel Superconducting Circuits 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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)PLRAAN2469-992610.1103/PhysRevA.98.022305]. 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=2, N=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. Physics Christopher S. Wang verfasserin aut Philip Reinhold verfasserin aut Connor T. Hann verfasserin aut Kevin S. Chou verfasserin aut Brian J. Lester verfasserin aut Serge Rosenblum verfasserin aut Luigi Frunzio verfasserin aut Liang Jiang verfasserin aut Robert J. Schoelkopf verfasserin aut In Physical Review X American Physical Society, 2011 10(2020), 1, p 011001 (DE-627)666214115 (DE-600)2622565-7 21603308 nnns volume:10 year:2020 number:1, p 011001 https://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/article/d8d78fc8871c4fcba325384b109124f3 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei http://doi.org/10.1103/PhysRevX.10.011001 kostenfrei https://doaj.org/toc/2160-3308 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2020 1, p 011001
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author	Salvatore S. Elder
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title_auth	High-Fidelity Measurement of Qubits Encoded in Multilevel Superconducting Circuits
abstract	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)PLRAAN2469-992610.1103/PhysRevA.98.022305]. 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=2, N=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.
abstractGer	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)PLRAAN2469-992610.1103/PhysRevA.98.022305]. 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=2, N=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.
abstract_unstemmed	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)PLRAAN2469-992610.1103/PhysRevA.98.022305]. 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=2, N=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.
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title_short	High-Fidelity Measurement of Qubits Encoded in Multilevel Superconducting Circuits
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