Transmon

<h2 id="comparison-to-cooper-pair-box">Comparison to Cooper-pair box</h2>

The transmon design is similar to the first design of the <a href="/facts/Charge_qubit/Dq7mtX4k">charge qubit</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> known as a "Cooper-pair box"; both are described by the same Hamiltonian, with the only difference being the 
 
 
 
 
 E
 
 
 J
 
 
 
 
 /
 
 
 E
 
 
 C
 
 
 
 
 
 {\displaystyle E_{\rm {J}}/E_{\rm {C}}}
 
 ratio. Here 
 
 
 
 
 E
 
 
 J
 
 
 
 
 
 {\displaystyle E_{\rm {J}}}
 
 is the <a href="/facts/Josephson_effect/2fvcdS8N">Josephson energy</a> of the junction, and 
 
 
 
 
 E
 
 
 C
 
 
 
 
 
 {\displaystyle E_{\rm {C}}}
 
 is the charging energy inversely proportional to the total capacitance of the qubit circuit. Transmons typically have 
 
 
 
 
 E
 
 
 J
 
 
 
 
 /
 
 
 E
 
 
 C
 
 
 
 ≫
 1
 
 
 {\displaystyle E_{\mathrm {J} }/E_{\mathrm {C} }\gg 1}
 
 (while 
 
 
 
 
 E
 
 
 J
 
 
 
 
 /
 
 
 E
 
 
 C
 
 
 
 ≲
 1
 
 
 {\displaystyle E_{\mathrm {J} }/E_{\mathrm {C} }\lesssim 1}
 
 for typical Cooper-pair-box qubits), which is achieved by shunting the <a href="/facts/Josephson_junction/2fvcdS8N">Josephson junction</a> with an additional large <a href="/facts/Capacitor/hyowqv7n">capacitor</a>. 
The benefit of increasing the 
 
 
 
 
 E
 
 
 J
 
 
 
 
 /
 
 
 E
 
 
 C
 
 
 
 
 
 {\displaystyle E_{\rm {J}}/E_{\rm {C}}}
 
 ratio is the insensitivity to charge noise—the energy levels become independent of the offset charge 
 
 
 
 
 n
 
 g
 
 
 
 
 {\displaystyle n_{g}}
 
 across the junction; thus the <a href="/facts/Dephasing/ogYdaSDL">dephasing</a> time of the qubit is prolonged. The disadvantage is the reduced anharmonicity 
 
 
 
 α
 =
 (
 
 E
 
 21
 
 
 −
 
 E
 
 10
 
 
 )
 
 /
 
 
 E
 
 10
 
 
 
 
 {\displaystyle \alpha =(E_{21}-E_{10})/E_{10}}
 
, where 
 
 
 
 
 E
 
 i
 j
 
 
 
 
 {\displaystyle E_{ij}}
 
 is the energy difference between eigenstates 
 
 
 
 
 |
 
 i
 ⟩
 
 
 {\displaystyle |i\rangle }
 
 and 
 
 
 
 
 |
 
 j
 ⟩
 
 
 {\displaystyle |j\rangle }
 
. Reduced anharmonicity complicates the device operation as a two level system, e.g. exciting the device from the ground state to the first excited state by a resonant pulse also populates the higher excited state. This complication is overcome by complex microwave pulse design, that takes into account the higher energy levels, and prohibits their excitation by destructive interference. Also, while the variation of 
 
 
 
 
 E
 
 10
 
 
 
 
 {\displaystyle E_{10}}
 
with respect to 
 
 
 
 
 n
 
 g
 
 
 
 
 {\displaystyle n_{g}}
 
 tend to decrease exponentially with 
 
 
 
 
 E
 
 
 J
 
 
 
 
 /
 
 
 E
 
 
 C
 
 
 
 
 
 {\displaystyle E_{\mathrm {J} }/E_{\mathrm {C} }}
 
, the anharmonicity only has a weaker, algebraic dependence on 
 
 
 
 
 E
 
 
 J
 
 
 
 
 /
 
 
 E
 
 
 C
 
 
 
 
 
 {\displaystyle E_{\mathrm {J} }/E_{\mathrm {C} }}
 
 as 
 
 
 
 ∼
 (
 
 E
 
 
 J
 
 
 
 
 /
 
 
 E
 
 
 C
 
 
 
 
 )
 
 −
 1
 
 /
 
 2
 
 
 
 
 {\displaystyle \sim (E_{\mathrm {J} }/E_{\mathrm {C} })^{-1/2}}
 
. The significant gain in the coherence time outweigh the decrease in the anharmonicity for controlling the states with high fidelity.
Measurement, control and coupling of transmons is performed by means of microwave resonators with techniques from <a href="/facts/Circuit_quantum_electrodynamics/dvxbMejH">circuit quantum electrodynamics</a> also applicable to <a href="/facts/Superconducting_quantum_computing/3Dj2BN2v">other superconducting qubits</a>. Coupling to the resonators is done by placing a capacitor between the qubit and the resonator, at a point where the resonator <a href="/facts/Electromagnetic_field/MacraS8y">electromagnetic field</a> is greatest. For example, in <a href="/facts/IBM_Quantum_Experience/Ixu3gUIF">IBM Quantum Experience</a> devices, the resonators are implemented with "quarter wave" <a href="/facts/Coplanar_waveguide/3G3P5uGk">coplanar waveguides</a> with maximal field at the signal-ground short at the waveguide end; thus every IBM transmon qubit has a long resonator "tail". The initial proposal included similar <a href="/facts/Transmission_line/cYlqaPo3">transmission line</a> resonators coupled to every transmon, becoming a part of the name. However, charge qubits operated at a similar 
 
 
 
 
 E
 
 
 J
 
 
 
 
 /
 
 
 E
 
 
 C
 
 
 
 
 
 {\displaystyle E_{\rm {J}}/E_{\rm {C}}}
 
 regime, coupled to different kinds of microwave cavities are referred to as transmons as well.

<h2 id="transmons-as-qubits">Transmons as qubits</h2>
Transmons have been explored for use as d-dimensional <a href="/facts/Qubit/WveeEubw">qubits</a> via the additional energy levels that naturally occur above the qubit subspace (the lowest two states). For example, the lowest three levels can be used to make a transmon <a href="/facts/Qutrit/rPYAYaw6">qutrit</a>; in the early 2020s, researchers have reported realizations of single-qutrit <a href="/facts/Quantum_logic_gate/EFV5sPCU">quantum gates</a> on transmons<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> as well as two-qutrit <a href="/facts/Quantum_entanglement/afzslwN0">entangling</a> gates.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> Entangling gates on transmons have also been explored theoretically and in simulations for the general case of qudits of arbitrary d.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Charge_qubit/Dq7mtX4k">Charge qubit</a></li>
<li><a href="/facts/Anharmonicity/GxK2WMWB">Anharmonicity</a></li>
<li><a href="/facts/Circuit_quantum_electrodynamics/dvxbMejH">Circuit quantum electrodynamics</a> (cQED)</li>
<li><a href="/facts/Dilution_refrigerator/s0RYbDAY">Dilution refrigerator</a></li>
<li><a href="/facts/List_of_quantum_processors/GWcmkMue">List of quantum processors</a></li>
<li><a href="/facts/Quantum_harmonic_oscillator/JYA5J0h5">Quantum harmonic oscillator</a></li>
<li><a href="/facts/Superconducting_quantum_computing/3Dj2BN2v">Superconducting quantum computing</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Koch, Jens; Yu, Terri M.; Gambetta, Jay; Houck, A. A.; Schuster, D. I.; Majer, J.; Blais, Alexandre; Devoret, M. H.; Girvin, S. M.; Schoelkopf, R. J. (2007-10-12). "Charge-insensitive qubit design derived from the Cooper pair box". Physical Review A. 76 (4): 042319. arXiv:cond-mat/0703002. Bibcode:2007PhRvA..76d2319K. doi:10.1103/physreva.76.042319. ISSN 1050-2947. S2CID 53983107. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Schreier, J. A.; Houck, A. A.; Koch, Jens; Schuster, D. I.; Johnson, B. R.; et al. (2008-05-12). "Suppressing charge noise decoherence in superconducting charge qubits". Physical Review B. 77 (18). American Physical Society (APS): 180402. arXiv:0712.3581. Bibcode:2008PhRvB..77r0502S. doi:10.1103/physrevb.77.180502. ISSN 1098-0121. S2CID 119181860. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Fink, Johannes M. (2010). Quantum Nonlinearities in Strong Coupling Circuit QED (Ph.D.). ETH Zurich. <a href="/wiki/ETH_Zurich" target="_blank">/wiki/ETH_Zurich</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Barends, R.; Kelly, J.; Megrant, A.; Sank, D.; Jeffrey, E.; et al. (2013-08-22). "Coherent Josephson Qubit Suitable for Scalable Quantum Integrated Circuits". Physical Review Letters. 111 (8): 080502. arXiv:1304.2322. Bibcode:2013PhRvL.111h0502B. doi:10.1103/physrevlett.111.080502. ISSN 0031-9007. PMID 24010421. S2CID 27081288. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Paik, Hanhee; Schuster, D. I.; Bishop, Lev S.; Kirchmair, G.; Catelani, G.; et al. (2011-12-05). "Observation of High Coherence in Josephson Junction Qubits Measured in a Three-Dimensional Circuit QED Architecture". Physical Review Letters. 107 (24): 240501. arXiv:1105.4652. Bibcode:2011PhRvL.107x0501P. doi:10.1103/physrevlett.107.240501. ISSN 0031-9007. PMID 22242979. S2CID 19296685. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Rigetti, Chad; Gambetta, Jay M.; Poletto, Stefano; Plourde, B. L. T.; Chow, Jerry M.; et al. (2012-09-24). "Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms". Physical Review B. 86 (10). American Physical Society (APS): 100506. arXiv:1202.5533. Bibcode:2012PhRvB..86j0506R. doi:10.1103/physrevb.86.100506. ISSN 1098-0121. S2CID 118702797. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Place, Alexander P. M.; Rodgers, Lila V. H.; Mundada, Pranav; Smitham, Basil M.; Fitzpatrick, Mattias; Leng, Zhaoqi; Premkumar, Anjali; Bryon, Jacob; Vrajitoarea, Andrei; Sussman, Sara; Cheng, Guangming; Madhavan, Trisha; Cava, Robert J.; de Leon, Nathalie; Houck, Andrew A. (2021-03-19). "New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds". Nature Communications. 12 (1): 1779. arXiv:2003.00024. Bibcode:2021NatCo..12.1779P. doi:10.1038/s41467-021-22030-5. ISSN 2041-1723. PMC 7979772. PMID 33741989. <a href="/wiki/Nathalie_de_Leon" target="_blank">/wiki/Nathalie_de_Leon</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Bouchiat, V.; Vion, D.; Joyez, P.; Esteve, D.; Devoret, M. H. (1998). "Quantum coherence with a single Cooper pair". Physica Scripta. 1998 (T76): 165. Bibcode:1998PhST...76..165B. doi:10.1238/Physica.Topical.076a00165. ISSN 1402-4896. S2CID 250887469. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Yurtalan, M. A.; Shi, J.; Kononenko, M.; Lupascu, A.; Ashhab, S. (2020-10-27). "Implementation of a Walsh-Hadamard Gate in a Superconducting Qutrit". Physical Review Letters. 125 (18): 180504. arXiv:2003.04879. Bibcode:2020PhRvL.125r0504Y. doi:10.1103/PhysRevLett.125.180504. PMID 33196217. S2CID 128064435. <a href="https://link.aps.org/doi/10.1103/PhysRevLett.125.180504" target="_blank">https://link.aps.org/doi/10.1103/PhysRevLett.125.180504</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Morvan, A.; Ramasesh, V. V.; Blok, M. S.; Kreikebaum, J. M.; O’Brien, K.; Chen, L.; Mitchell, B. K.; Naik, R. K.; Santiago, D. I.; Siddiqi, I. (2021-05-27). "Qutrit Randomized Benchmarking". Physical Review Letters. 126 (21): 210504. arXiv:2008.09134. Bibcode:2021PhRvL.126u0504M. doi:10.1103/PhysRevLett.126.210504. hdl:1721.1/143809. OSTI 1818119. PMID 34114846. S2CID 221246177. <a href="https://link.aps.org/doi/10.1103/PhysRevLett.126.210504" target="_blank">https://link.aps.org/doi/10.1103/PhysRevLett.126.210504</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Goss, Noah; Morvan, Alexis; Marinelli, Brian; Mitchell, Bradley K.; Nguyen, Long B.; Naik, Ravi K.; Chen, Larry; Jünger, Christian; Kreikebaum, John Mark; Santiago, David I.; Wallman, Joel J.; Siddiqi, Irfan (2022-12-05). "High-fidelity qutrit entangling gates for superconducting circuits". Nature Communications. 13 (1): 7481. arXiv:2206.07216. Bibcode:2022NatCo..13.7481G. doi:10.1038/s41467-022-34851-z. ISSN 2041-1723. PMC 9722686. PMID 36470858. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9722686" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9722686</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Fischer, Laurin E.; Chiesa, Alessandro; Tacchino, Francesco; Egger, Daniel J.; Carretta, Stefano; Tavernelli, Ivano (2023-08-28). "Universal Qudit Gate Synthesis for Transmons". PRX Quantum. 4 (3): 030327. arXiv:2212.04496. Bibcode:2023PRXQ....4c0327F. doi:10.1103/PRXQuantum.4.030327. S2CID 254408561. <a href="https://link.aps.org/doi/10.1103/PRXQuantum.4.030327" target="_blank">https://link.aps.org/doi/10.1103/PRXQuantum.4.030327</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
</ol>

Transmon open-in-new

Transmon