Bosonic quantum Fourier codes
Summarize this article with:
AbstractWhile 2-level systems, aka qubits, are a natural choice to perform a logical quantum computation, the situation is less clear at the physical level. Encoding information in higher-dimensional physical systems can indeed provide a first level of redundancy and error correction that simplifies the overall fault-tolerant architecture. A challenge then is to ensure universal control over the encoded qubits. Here, we explore an approach where information is encoded in an irreducible representation of a finite subgroup of $U(2)$ through an inverse quantum Fourier transform. We illustrate this idea by applying it to the real Pauli group $\langle X, Z\rangle$ in the bosonic setting. The resulting two-mode Fourier cat code displays good error correction properties and admits an experimentally-friendly universal gate set that we discuss in detail.Featured image: Entanglement infidelity $1-F_{\mathrm{ent}}$ of cat-type bosonic qubit encodings for the pure-loss channel of parameter $\gamma$, using the Petz recovery map. Shown are the two-mode Fourier cat code (logical qubit with the multiplicity register fixed to $|0\rangle_M$), the 4-legged cat code, the 2-repetition cat code, and the pair-cat code. Left: $1-F_{\mathrm{ent}}$ versus coherent amplitude $\alpha$ at fixed $\gamma=0.01$. Right: $1-F_{\mathrm{ent}}$ versus $\gamma$ using $\alpha=\sqrt{\pi/2}$ for the two-mode Fourier cat and 2-repetition cat codes, and $\alpha$ is chosen to minimize the left-panel infidelity at $\gamma=0.01$ for the 4-legged cat and pair-cat codes. Pair-cat values are evaluated in a truncated two-mode Fock basis with cutoffs chosen to ensure numerical convergence.► BibTeX data@article{Leverrier2026bosonicquantum, doi = {10.22331/q-2026-02-09-2000}, url = {https://doi.org/10.22331/q-2026-02-09-2000}, title = {Bosonic quantum {F}ourier codes}, author = {Leverrier, Anthony}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2000}, month = feb, year = {2026} }► References [1] Bryan Eastin and Emanuel Knill. ``Restrictions on transversal encoded quantum gate sets''. Phys. Rev. Lett. 102, 110502 (2009). https://doi.org/10.1103/PhysRevLett.102.110502 [2] Mazyar Mirrahimi, Zaki Leghtas, Victor V Albert, Steven Touzard, Robert J Schoelkopf, Liang Jiang, and Michel H Devoret. ``Dynamically protected cat-qubits: a new paradigm for universal quantum computation''. New Journal of Physics 16, 045014 (2014). https://doi.org/10.1088/1367-2630/16/4/045014 [3] Jérémie Guillaud and Mazyar Mirrahimi. ``Repetition cat qubits for fault-tolerant quantum computation''. Phys. Rev. X 9, 041053 (2019). https://doi.org/10.1103/PhysRevX.9.041053 [4] Christophe Vuillot, Hamed Asasi, Yang Wang, Leonid P. Pryadko, and Barbara M. Terhal. ``Quantum error correction with the toric Gottesman-Kitaev-Preskill code''. Phys. Rev. A 99, 032344 (2019). https://doi.org/10.1103/PhysRevA.99.032344 [5] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, et al. ``Building a fault-tolerant quantum computer using concatenated cat codes''. PRX Quantum 3, 010329 (2022). https://doi.org/10.1103/PRXQuantum.3.010329 [6] Yijia Xu, Yixu Wang, Christophe Vuillot, and Victor V. Albert. ``Letting the tiger out of its cage: Bosonic coding without concatenation''. Phys. Rev. X 15, 041025 (2025). https://doi.org/10.1103/ls5r-vj7r [7] Alexander Grimm, Nicholas E Frattini, Shruti Puri, Shantanu O Mundhada, Steven Touzard, Mazyar Mirrahimi, Steven M Girvin, Shyam Shankar, and Michel H Devoret. ``Stabilization and operation of a Kerr-cat qubit''. Nature 584, 205–209 (2020). https://doi.org/10.1038/s41586-020-2587-z [8] Ulysse Réglade, Adrien Bocquet, Ronan Gautier, et al. ``Quantum control of a cat qubit with bit-flip times exceeding ten seconds''. Nature 629, 778–783 (2024). https://doi.org/10.1038/s41586-024-07294-3 [9] Harald Putterman, Kyungjoo Noh, Connor T Hann, et al. ``Hardware-efficient quantum error correction via concatenated bosonic qubits''. Nature 638, 927–934 (2025). https://doi.org/10.1038/s41586-025-08642-7 [10] Christa Flühmann, Thanh Long Nguyen, Matteo Marinelli, Vlad Negnevitsky, Karan Mehta, and Jonathan Home. ``Encoding a qubit in a trapped-ion mechanical oscillator''. Nature 566, 513–517 (2019). https://doi.org/10.1038/s41586-019-0960-6 [11] Philippe Campagne-Ibarcq, Alec Eickbusch, Steven Touzard, et al. ``Quantum error correction of a qubit encoded in grid states of an oscillator''. Nature 584, 368–372 (2020). https://doi.org/10.1038/s41586-020-2603-3 [12] Volodymyr V Sivak, Alec Eickbusch, Baptiste Royer, et al. ``Real-time quantum error correction beyond break-even''. Nature 616, 50–55 (2023). https://doi.org/10.1038/s41586-023-05782-6 [13] James D. Teoh, Patrick Winkel, Harshvardhan K. Babla, et al. ``Dual-rail encoding with superconducting cavities''. Proceedings of the National Academy of Sciences 120, e2221736120 (2023). https://doi.org/10.1073/pnas.2221736120 [14] Daniel Gottesman, Alexei Kitaev, and John Preskill. ``Encoding a qubit in an oscillator''. Phys. Rev. A 64, 012310 (2001). https://doi.org/10.1103/PhysRevA.64.012310 [15] Victor V Albert, Shantanu O Mundhada, Alexander Grimm, Steven Touzard, Michel H Devoret, and Liang Jiang. ``Pair-cat codes: autonomous error-correction with low-order nonlinearity''. Quantum Science and Technology 4, 035007 (2019). https://doi.org/10.1088/2058-9565/ab1e69 [16] Marios H. Michael, Matti Silveri, R. T. Brierley, Victor V. Albert, Juha Salmilehto, Liang Jiang, and S. M. Girvin. ``New class of quantum error-correcting codes for a bosonic mode''. Phys. Rev. X 6, 031006 (2016). https://doi.org/10.1103/PhysRevX.6.031006 [17] Arne L. Grimsmo, Joshua Combes, and Ben Q. Baragiola. ``Quantum computing with rotation-symmetric bosonic codes''. Phys. Rev. X 10, 011058 (2020). https://doi.org/10.1103/PhysRevX.10.011058 [18] Shubham P Jain, Joseph T Iosue, Alexander Barg, and Victor V Albert. ``Quantum spherical codes''. Nature Physics 20, 1300–1305 (2024). https://doi.org/10.1038/s41567-024-02496-y [19] Jonathan A. Gross. ``Designing codes around interactions: The case of a spin''. Phys. Rev. Lett. 127, 010504 (2021). https://doi.org/10.1103/PhysRevLett.127.010504 [20] Eric Kubischta and Ian Teixeira. ``Family of quantum codes with exotic transversal gates''. Phys. Rev. Lett. 131, 240601 (2023). https://doi.org/10.1103/PhysRevLett.131.240601 [21] Eric Kubischta and Ian Teixeira. ``Quantum codes from twisted unitary $t$-groups''. Phys. Rev. Lett. 133, 030602 (2024). https://doi.org/10.1103/PhysRevLett.133.030602 [22] Aurélie Denys and Anthony Leverrier. ``Quantum error-correcting codes with a covariant encoding''. Phys. Rev. Lett. 133, 240603 (2024). https://doi.org/10.1103/PhysRevLett.133.240603 [23] Andrew M. Childs and Wim van Dam. ``Quantum algorithms for algebraic problems''. Rev. Mod. Phys. 82, 1–52 (2010). https://doi.org/10.1103/RevModPhys.82.1 [24] Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian Vlastakis, Yehan Liu, Luigi Frunzio, Steven M Girvin, Liang Jiang, et al. ``Extending the lifetime of a quantum bit with error correction in superconducting circuits''. Nature 536, 441–445 (2016). https://doi.org/10.1038/nature18949 [25] Arne L. Grimsmo and Shruti Puri. ``Quantum Error Correction with the Gottesman-Kitaev-Preskill Code''. PRX Quantum 2, 020101 (2021). https://doi.org/10.1103/PRXQuantum.2.020101 [26] L.-A. Sellem, A. Sarlette, Z. Leghtas, M. Mirrahimi, P. Rouchon, and P. Campagne-Ibarcq. ``Dissipative Protection of a GKP Qubit in a High-Impedance Superconducting Circuit Driven by a Microwave Frequency Comb''. Phys. Rev. X 15, 011011 (2025). https://doi.org/10.1103/PhysRevX.15.011011 [27] Isaac L. Chuang, Debbie W. Leung, and Yoshihisa Yamamoto. ``Bosonic quantum codes for amplitude damping''. Phys. Rev. A 56, 1114–1125 (1997). https://doi.org/10.1103/PhysRevA.56.1114 [28] Reinier W Heeres, Brian Vlastakis, Eric Holland, Stefan Krastanov, Victor V Albert, Luigi Frunzio, Liang Jiang, and Robert J Schoelkopf. ``Cavity state manipulation using photon-number selective phase gates''. Phys. Rev. Lett. 115, 137002 (2015). https://doi.org/10.1103/PhysRevLett.115.137002 [29] Philip Reinhold, Serge Rosenblum, Wen-Long Ma, Luigi Frunzio, Liang Jiang, and Robert J Schoelkopf. ``Error-corrected gates on an encoded qubit''. Nature Physics 16, 822–826 (2020). https://doi.org/10.1038/s41567-020-0931-8 [30] Andrew S. Fletcher, Peter W. Shor, and Moe Z. Win. ``Optimum quantum error recovery using semidefinite programming''. Phys. Rev. A 75, 012338 (2007). https://doi.org/10.1103/PhysRevA.75.012338 [31] Dénes Petz. ``Sufficiency of channels over von Neumann algebras''.
The Quarterly Journal of Mathematics 39, 97–108 (1988). https://doi.org/10.1093/qmath/39.1.97 [32] András Gilyén, Seth Lloyd, Iman Marvian, Yihui Quek, and Mark M. Wilde. ``Quantum algorithm for petz recovery channels and pretty good measurements''. Phys. Rev. Lett. 128, 220502 (2022). https://doi.org/10.1103/PhysRevLett.128.220502 [33] Guo Zheng, Wenhao He, Gideon Lee, and Liang Jiang. ``Near-optimal performance of quantum error correction codes''. Phys. Rev. Lett. 132, 250602 (2024). https://doi.org/10.1103/PhysRevLett.132.250602 [34] Aurélie Denys and Anthony Leverrier. ``The $2T$-qutrit, a two-mode bosonic qutrit''. Quantum 7, 1032 (2023). https://doi.org/10.22331/q-2023-06-05-1032 [35] Peter Leviant, Qian Xu, Liang Jiang, and Serge Rosenblum. ``Quantum capacity and codes for the bosonic loss-dephasing channel''. Quantum 6, 821 (2022). https://doi.org/10.22331/q-2022-09-29-821 [36] Yao Lu, Aniket Maiti, John W. O. Garmon, et al. ``High-fidelity parametric beamsplitting with a parity-protected converter''. Nature Communications 14, 5767 (2023). https://doi.org/10.1038/s41467-023-41104-0 [37] Benjamin J. Chapman, Stijn J. de Graaf, Sophia H. Xue, et al. ``High-on-off-ratio beam-splitter interaction for gates on bosonically encoded qubits''. PRX Quantum 4, 020355 (2023). https://doi.org/10.1103/PRXQuantum.4.020355Cited byCould not fetch Crossref cited-by data during last attempt 2026-02-09 11:38:27: Could not fetch cited-by data for 10.22331/q-2026-02-09-2000 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-09 11:38:27: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWhile 2-level systems, aka qubits, are a natural choice to perform a logical quantum computation, the situation is less clear at the physical level. Encoding information in higher-dimensional physical systems can indeed provide a first level of redundancy and error correction that simplifies the overall fault-tolerant architecture. A challenge then is to ensure universal control over the encoded qubits. Here, we explore an approach where information is encoded in an irreducible representation of a finite subgroup of $U(2)$ through an inverse quantum Fourier transform. We illustrate this idea by applying it to the real Pauli group $\langle X, Z\rangle$ in the bosonic setting. The resulting two-mode Fourier cat code displays good error correction properties and admits an experimentally-friendly universal gate set that we discuss in detail.Featured image: Entanglement infidelity $1-F_{\mathrm{ent}}$ of cat-type bosonic qubit encodings for the pure-loss channel of parameter $\gamma$, using the Petz recovery map. Shown are the two-mode Fourier cat code (logical qubit with the multiplicity register fixed to $|0\rangle_M$), the 4-legged cat code, the 2-repetition cat code, and the pair-cat code. Left: $1-F_{\mathrm{ent}}$ versus coherent amplitude $\alpha$ at fixed $\gamma=0.01$. Right: $1-F_{\mathrm{ent}}$ versus $\gamma$ using $\alpha=\sqrt{\pi/2}$ for the two-mode Fourier cat and 2-repetition cat codes, and $\alpha$ is chosen to minimize the left-panel infidelity at $\gamma=0.01$ for the 4-legged cat and pair-cat codes. Pair-cat values are evaluated in a truncated two-mode Fock basis with cutoffs chosen to ensure numerical convergence.► BibTeX data@article{Leverrier2026bosonicquantum, doi = {10.22331/q-2026-02-09-2000}, url = {https://doi.org/10.22331/q-2026-02-09-2000}, title = {Bosonic quantum {F}ourier codes}, author = {Leverrier, Anthony}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2000}, month = feb, year = {2026} }► References [1] Bryan Eastin and Emanuel Knill. ``Restrictions on transversal encoded quantum gate sets''. Phys. Rev. Lett. 102, 110502 (2009). https://doi.org/10.1103/PhysRevLett.102.110502 [2] Mazyar Mirrahimi, Zaki Leghtas, Victor V Albert, Steven Touzard, Robert J Schoelkopf, Liang Jiang, and Michel H Devoret. ``Dynamically protected cat-qubits: a new paradigm for universal quantum computation''. New Journal of Physics 16, 045014 (2014). https://doi.org/10.1088/1367-2630/16/4/045014 [3] Jérémie Guillaud and Mazyar Mirrahimi. ``Repetition cat qubits for fault-tolerant quantum computation''. Phys. Rev. X 9, 041053 (2019). https://doi.org/10.1103/PhysRevX.9.041053 [4] Christophe Vuillot, Hamed Asasi, Yang Wang, Leonid P. Pryadko, and Barbara M. Terhal. ``Quantum error correction with the toric Gottesman-Kitaev-Preskill code''. Phys. Rev. A 99, 032344 (2019). https://doi.org/10.1103/PhysRevA.99.032344 [5] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, et al. ``Building a fault-tolerant quantum computer using concatenated cat codes''. PRX Quantum 3, 010329 (2022). https://doi.org/10.1103/PRXQuantum.3.010329 [6] Yijia Xu, Yixu Wang, Christophe Vuillot, and Victor V. Albert. ``Letting the tiger out of its cage: Bosonic coding without concatenation''. Phys. Rev. X 15, 041025 (2025). https://doi.org/10.1103/ls5r-vj7r [7] Alexander Grimm, Nicholas E Frattini, Shruti Puri, Shantanu O Mundhada, Steven Touzard, Mazyar Mirrahimi, Steven M Girvin, Shyam Shankar, and Michel H Devoret. ``Stabilization and operation of a Kerr-cat qubit''. Nature 584, 205–209 (2020). https://doi.org/10.1038/s41586-020-2587-z [8] Ulysse Réglade, Adrien Bocquet, Ronan Gautier, et al. ``Quantum control of a cat qubit with bit-flip times exceeding ten seconds''. Nature 629, 778–783 (2024). https://doi.org/10.1038/s41586-024-07294-3 [9] Harald Putterman, Kyungjoo Noh, Connor T Hann, et al. ``Hardware-efficient quantum error correction via concatenated bosonic qubits''. Nature 638, 927–934 (2025). https://doi.org/10.1038/s41586-025-08642-7 [10] Christa Flühmann, Thanh Long Nguyen, Matteo Marinelli, Vlad Negnevitsky, Karan Mehta, and Jonathan Home. ``Encoding a qubit in a trapped-ion mechanical oscillator''. Nature 566, 513–517 (2019). https://doi.org/10.1038/s41586-019-0960-6 [11] Philippe Campagne-Ibarcq, Alec Eickbusch, Steven Touzard, et al. ``Quantum error correction of a qubit encoded in grid states of an oscillator''. Nature 584, 368–372 (2020). https://doi.org/10.1038/s41586-020-2603-3 [12] Volodymyr V Sivak, Alec Eickbusch, Baptiste Royer, et al. ``Real-time quantum error correction beyond break-even''. Nature 616, 50–55 (2023). https://doi.org/10.1038/s41586-023-05782-6 [13] James D. Teoh, Patrick Winkel, Harshvardhan K. Babla, et al. ``Dual-rail encoding with superconducting cavities''. Proceedings of the National Academy of Sciences 120, e2221736120 (2023). https://doi.org/10.1073/pnas.2221736120 [14] Daniel Gottesman, Alexei Kitaev, and John Preskill. ``Encoding a qubit in an oscillator''. Phys. Rev. A 64, 012310 (2001). https://doi.org/10.1103/PhysRevA.64.012310 [15] Victor V Albert, Shantanu O Mundhada, Alexander Grimm, Steven Touzard, Michel H Devoret, and Liang Jiang. ``Pair-cat codes: autonomous error-correction with low-order nonlinearity''. Quantum Science and Technology 4, 035007 (2019). https://doi.org/10.1088/2058-9565/ab1e69 [16] Marios H. Michael, Matti Silveri, R. T. Brierley, Victor V. Albert, Juha Salmilehto, Liang Jiang, and S. M. Girvin. ``New class of quantum error-correcting codes for a bosonic mode''. Phys. Rev. X 6, 031006 (2016). https://doi.org/10.1103/PhysRevX.6.031006 [17] Arne L. Grimsmo, Joshua Combes, and Ben Q. Baragiola. ``Quantum computing with rotation-symmetric bosonic codes''. Phys. Rev. X 10, 011058 (2020). https://doi.org/10.1103/PhysRevX.10.011058 [18] Shubham P Jain, Joseph T Iosue, Alexander Barg, and Victor V Albert. ``Quantum spherical codes''. Nature Physics 20, 1300–1305 (2024). https://doi.org/10.1038/s41567-024-02496-y [19] Jonathan A. Gross. ``Designing codes around interactions: The case of a spin''. Phys. Rev. Lett. 127, 010504 (2021). https://doi.org/10.1103/PhysRevLett.127.010504 [20] Eric Kubischta and Ian Teixeira. ``Family of quantum codes with exotic transversal gates''. Phys. Rev. Lett. 131, 240601 (2023). https://doi.org/10.1103/PhysRevLett.131.240601 [21] Eric Kubischta and Ian Teixeira. ``Quantum codes from twisted unitary $t$-groups''. Phys. Rev. Lett. 133, 030602 (2024). https://doi.org/10.1103/PhysRevLett.133.030602 [22] Aurélie Denys and Anthony Leverrier. ``Quantum error-correcting codes with a covariant encoding''. Phys. Rev. Lett. 133, 240603 (2024). https://doi.org/10.1103/PhysRevLett.133.240603 [23] Andrew M. Childs and Wim van Dam. ``Quantum algorithms for algebraic problems''. Rev. Mod. Phys. 82, 1–52 (2010). https://doi.org/10.1103/RevModPhys.82.1 [24] Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian Vlastakis, Yehan Liu, Luigi Frunzio, Steven M Girvin, Liang Jiang, et al. ``Extending the lifetime of a quantum bit with error correction in superconducting circuits''. Nature 536, 441–445 (2016). https://doi.org/10.1038/nature18949 [25] Arne L. Grimsmo and Shruti Puri. ``Quantum Error Correction with the Gottesman-Kitaev-Preskill Code''. PRX Quantum 2, 020101 (2021). https://doi.org/10.1103/PRXQuantum.2.020101 [26] L.-A. Sellem, A. Sarlette, Z. Leghtas, M. Mirrahimi, P. Rouchon, and P. Campagne-Ibarcq. ``Dissipative Protection of a GKP Qubit in a High-Impedance Superconducting Circuit Driven by a Microwave Frequency Comb''. Phys. Rev. X 15, 011011 (2025). https://doi.org/10.1103/PhysRevX.15.011011 [27] Isaac L. Chuang, Debbie W. Leung, and Yoshihisa Yamamoto. ``Bosonic quantum codes for amplitude damping''. Phys. Rev. A 56, 1114–1125 (1997). https://doi.org/10.1103/PhysRevA.56.1114 [28] Reinier W Heeres, Brian Vlastakis, Eric Holland, Stefan Krastanov, Victor V Albert, Luigi Frunzio, Liang Jiang, and Robert J Schoelkopf. ``Cavity state manipulation using photon-number selective phase gates''. Phys. Rev. Lett. 115, 137002 (2015). https://doi.org/10.1103/PhysRevLett.115.137002 [29] Philip Reinhold, Serge Rosenblum, Wen-Long Ma, Luigi Frunzio, Liang Jiang, and Robert J Schoelkopf. ``Error-corrected gates on an encoded qubit''. Nature Physics 16, 822–826 (2020). https://doi.org/10.1038/s41567-020-0931-8 [30] Andrew S. Fletcher, Peter W. Shor, and Moe Z. Win. ``Optimum quantum error recovery using semidefinite programming''. Phys. Rev. A 75, 012338 (2007). https://doi.org/10.1103/PhysRevA.75.012338 [31] Dénes Petz. ``Sufficiency of channels over von Neumann algebras''.
The Quarterly Journal of Mathematics 39, 97–108 (1988). https://doi.org/10.1093/qmath/39.1.97 [32] András Gilyén, Seth Lloyd, Iman Marvian, Yihui Quek, and Mark M. Wilde. ``Quantum algorithm for petz recovery channels and pretty good measurements''. Phys. Rev. Lett. 128, 220502 (2022). https://doi.org/10.1103/PhysRevLett.128.220502 [33] Guo Zheng, Wenhao He, Gideon Lee, and Liang Jiang. ``Near-optimal performance of quantum error correction codes''. Phys. Rev. Lett. 132, 250602 (2024). https://doi.org/10.1103/PhysRevLett.132.250602 [34] Aurélie Denys and Anthony Leverrier. ``The $2T$-qutrit, a two-mode bosonic qutrit''. Quantum 7, 1032 (2023). https://doi.org/10.22331/q-2023-06-05-1032 [35] Peter Leviant, Qian Xu, Liang Jiang, and Serge Rosenblum. ``Quantum capacity and codes for the bosonic loss-dephasing channel''. Quantum 6, 821 (2022). https://doi.org/10.22331/q-2022-09-29-821 [36] Yao Lu, Aniket Maiti, John W. O. Garmon, et al. ``High-fidelity parametric beamsplitting with a parity-protected converter''. Nature Communications 14, 5767 (2023). https://doi.org/10.1038/s41467-023-41104-0 [37] Benjamin J. Chapman, Stijn J. de Graaf, Sophia H. Xue, et al. ``High-on-off-ratio beam-splitter interaction for gates on bosonically encoded qubits''. PRX Quantum 4, 020355 (2023). https://doi.org/10.1103/PRXQuantum.4.020355Cited byCould not fetch Crossref cited-by data during last attempt 2026-02-09 11:38:27: Could not fetch cited-by data for 10.22331/q-2026-02-09-2000 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-09 11:38:27: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.