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To obtain a quantum mechanical description of an electrical circuit, a few steps are required. Firstly, all electrical elements must be described by the condensate wave function amplitude and phase rather than by closely related macroscopic [[Electric current|current]] and [[voltage]] descriptions used for classical circuits. For instance, the square of the wave function amplitude at any arbitrary point in space corresponds to the probability of finding a charge carrier there. Therefore, the squared amplitude corresponds to a classical charge distribution. The second requirement to obtain a quantum mechanical description of an electrical circuit is that generalized [[Kirchhoff's circuit laws]] are applied at every node of the circuit network to obtain the system's [[equations of motion]]. Finally, these equations of motion must be reformulated to [[Lagrangian mechanics]] such that a [[Hamiltonian (quantum mechanics)|quantum Hamiltonian]] is derived describing the total energy of the system.
==Technology==
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==== Fluxonium ====
Fluxonium qubits are a specific type of flux qubit whose Josephson junction is shunted by a linear inductor of <math> E_{J} \gg E_{L} </math> where <math>E_L = (\hbar/2e)^2 / L </math>.<ref name="Nguyen-2019" />
<math>\hat{H} = 4 E_C \hat{n}^2 + \frac{1}{2} E_L (\hat{\phi}- \phi_\mathrm{ext})^2 - E_J \cos \hat{\phi} </math>.
One important property of the fluxonium qubit is the longer [[Coherence (physics)#Quantum coherence|qubit lifetime]] at the half flux sweet spot, which can exceed 1 millisecond
=== Charge qubit ===
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[[File:PhaseQBcirc.svg|thumb|Phase qubit circuit. A Josephson junction with energy parameter <math>E_J</math> is biased by current <math>I_0</math>.
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! [[Potential energy|Potential]]
|[[File:Charge qubit potential.svg|thumb|<math>U = -E_J\cos\phi</math> . Bias voltage is set such that <math>N_g=\frac{1}{2}</math>, minimizing the energy gap between <math>|0\rangle</math> and <math>|1\rangle</math>, consequently distinguishing the gap from other energy gaps (e.g. gap between <math>|1\rangle</math> and <math>|2\rangle</math>). The difference in gaps allows addressing transitions from <math>|0\rangle</math> to <math>|1\rangle</math> and vice versa only, without populating other states.
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===Geometric phase gate===
Higher levels (outside of the computational subspace) of a pair of coupled superconducting circuits can be used to induce a geometric phase on one of the computational states of the qubits. This leads to an entangling conditional phase shift of the relevant qubit states. This effect has been implemented by flux-tuning the qubit spectra <ref name="DiCarlo Chow Gambetta Bishop 2009 pp. 240–244">{{cite journal | last=DiCarlo | first=L. | last2=Chow | first2=J. M. | last3=Gambetta | first3=J. M. | last4=Bishop | first4=Lev S. | last5=Johnson | first5=B. R. | last6=Schuster | first6=D. I. | last7=Majer | first7=J. | last8=Blais | first8=A. | last9=Frunzio | first9=L. | last10=Girvin | first10=S. M. | last11=Schoelkopf | first11=R. J. | title=Demonstration of two-qubit algorithms with a superconducting quantum processor | journal=Nature | publisher=Springer Science and Business Media LLC | volume=460 | issue=7252 | date=2009-06-28 | issn=0028-0836 | doi=10.1038/nature08121 | pages=240–244}}</ref> and by using selective microwave driving
===Heisenberg interactions===
The Heisenberg model of interactions, written as
<math>\hat{\mathcal{H}}_\mathrm{XXZ}/\hbar =\sum_{ i,j} J_\mathrm{XY}(\hat{\sigma}_\text{x}^{i}\hat{\sigma}_\text{x}^{j} + \hat{\sigma}_\text{y}^{i}\hat{\sigma}_\text{y}^{j}) + J_\mathrm{ZZ}\hat{\sigma}_\text{z}^{i}\hat{\sigma}_\text{z}^{j}</math>,
serves as the basis for analog quantum simulation of spin systems and the primitive for an expressive set of quantum gates, sometimes referred to as ''fermionic simulation'' (or ''fSim'') gates. In superconducting circuits, this interaction model has been implemented using flux-tunable qubits with flux-tunable coupling
==Qubit readout==
Architecture-specific readout, or [[Quantum measurement|measurement]], mechanisms exist. Readout of a phase qubit is explained in the [[#Qubit archetypes|qubit archetypes table]] above. A flux qubit state is often read using an adjustable DC-[[SQUID]] [[magnetometer]]. States may also be measured using an [[electrometer]].<ref name="docs.pennylane.ai">{{Cite web |title=PennyLane Documentation — PennyLane |url=https://docs.pennylane.ai/en/stable/index.html |access-date=2022-12-11 |website=docs.pennylane.ai |language=en}}</ref> A more general readout scheme includes a coupling to a microwave [[resonator]], where resonance frequency of the resonator is dispersively shifted by the qubit state.<ref name=NatRev2017>{{cite journal |last1=Gambetta |first1=Jay M. |last2=Chow |first2=Jerry M. |last3=Steffen |first3=Matthias |title=Building logical qubits in a superconducting quantum computing system |journal=[[npj Quantum Information]] |date=13 January 2017 |volume=3 |issue=1 |pages=2 |doi=10.1038/s41534-016-0004-0 |doi-access=free |bibcode=2017npjQI...3....2G |arxiv=1510.04375 }}
</ref><ref name="Dispersive Readout">{{cite journal |last1=Blais |first1=Alexandre |last2=Huang |first2=Ren-Shou |last3=Wallraff |first3=Andreas |last4=Girvin |first4=Steven |last5=Schoelkopf |first5=Robert |title=Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation |journal=Phys. Rev. A |date=2004 |volume=69 |issue=6 |pages=062320 |doi=10.1103/PhysRevA.69.062320 |url=https://link.aps.org/doi/10.1103/PhysRevA.69.062320|arxiv=cond-mat/0402216 |bibcode=2004PhRvA..69f2320B |s2cid=20427333 }}</ref>
==DiVincenzo's criteria==
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# '''A scalable physical system with well characterized qubits.''' "Well characterized implies that that [[Hamiltonian mechanics|Hamiltonian function]] must be well-defined i.e the energy eigenstates of the qubit should be able to be quantified.. A scalable system is self-explanatory, it indicates that this ability to regulate a qubit should be augmentable for multiple more qubits. Herein lies the major issue Quantum Computers face, as more qubits are implemented it leads to a exponential increase in cost and other physical implementations which pale in comparison to the enhanced speed it may offer.<ref name="qc-at-davis.github.io"/> As superconducting qubits are fabricated on a chip, the many-qubit system is readily scalable. Qubits are allocated on the 2D surface of the chip. The demand for well characterized qubits is fulfilled with (a) qubit non-linearity (accessing only two of the available energy levels) and (b) accessing a single qubit at a time (rather than the entire many-qubit system) by way of per-qubit dedicated control lines and/or frequency separation, or tuning out, of different qubits.
# '''Ability to initialize the state of qubits to a simple fiducial state.'''<ref name="DiVincenzo-2008">{{Cite journal |last=DiVincenzo |first=David |date=February 1, 2008 |title=The Physical Implementation of Quantum Computation |journal=IBM T.J. Watson Research Center|volume=48 |issue=9–11 |pages=771–783 |doi=10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E |arxiv=quant-ph/0002077 |bibcode=2000ForPh..48..771D |s2cid=15439711 }}</ref> A fiducial state is one that is easily and consistently replicable and is useful in quantum computing as it may be used to guarantee the initial state of qubits. One simple way to initialize a superconducting qubit is to wait long enough for the qubits to relax to the ground state. Controlling qubit potential with tuning knobs allows faster initialization mechanisms.
# '''Long relevant decoherence times'''<ref name="DiVincenzo-2008" />'''.''' Decoherence of superconducting qubits is affected by multiple factors. Most decoherence is attributed to the quality of the Josephson junction and imperfections in the chip substrate. Due to their mesoscopic scale, superconducting qubits are relatively short lived. Nevertheless, thousands of gate operations have been demonstrated in these many-qubit systems.<ref>{{cite journal |last1=Devoret |first1=M. H. |last2=Schoelkopf |first2=R. J. |title=Superconducting Circuits for Quantum Information: An Outlook |journal=Science |date=7 March 2013 |volume=339 |issue=6124 |pages=1169–1174 |doi=10.1126/science.1231930|pmid=23471399 |bibcode=2013Sci...339.1169D |s2cid=10123022 }}</ref>
# '''A “universal” set of quantum gates.'''<ref name="DiVincenzo-2008" /> Superconducting qubits allow arbitrary rotations in the Bloch sphere with pulsed microwave signals, implementing single qubit gates. <math>\sigma_z \sigma_z</math> and <math>\sigma_x \sigma_x</math> couplings are shown for most implementations and for complementing the universal gate set.<ref>{{cite journal |last1=Chow |first1=Jerry M. |last2=Gambetta |first2=Jay M. |last3=Córcoles |first3=A. D. |last4=Merkel |first4=Seth T. |last5=Smolin |first5=John A. |last6=Rigetti |first6=Chad |last7=Poletto |first7=S. |last8=Keefe |first8=George A. |last9=Rothwell |first9=Mary B. |last10=Rozen |first10=J. R. |last11=Ketchen |first11=Mark B. |last12=Steffen |first12=M. |title=Universal Quantum Gate Set Approaching Fault-Tolerant Thresholds with Superconducting Qubits |arxiv=1202.5344 |journal=Physical Review Letters |date=9 August 2012 |volume=109 |issue=6 |pages=060501 |doi=10.1103/PhysRevLett.109.060501|pmid=23006254 |bibcode=2012PhRvL.109f0501C |s2cid=39874288 }}</ref><ref>{{cite journal |last1=Niskanen |first1=A. O. |last2=Harrabi |first2=K. |last3=Yoshihara |first3=F. |last4=Nakamura |first4=Y. |last5=Lloyd |first5=S. |last6=Tsai |first6=J. S. |title=Quantum Coherent Tunable Coupling of Superconducting Qubits |journal=Science |date=4 May 2007 |volume=316 |issue=5825 |pages=723–726 |doi=10.1126/science.1141324|pmid=17478714 |bibcode=2007Sci...316..723N |s2cid=43175104 }}</ref><ref name = "Nguyen-2024">{{cite journal |last1=Nguyen |first1=L.B. |last2=Kim |first2=Y. |last3=Hashim |first3=A. |last4=Goss |first4=N.|last5=Marinelli |first5=B.|last6=Bhandari |first6=B.|last7=Das |first7=D.|last8=Naik |first8=R.K.|last9=Kreikebaum |first9=J.M.|last10=Jordan |first10=A.|last11=Santiago |first11=D.I.|last12=Siddiqi |first12=I. |title=Programmable Heisenberg interactions between Floquet qubits
|journal=Nature Physics |date=16 January 2024 |volume=20 |issue=1 |pages=
# '''Qubit-specific measurement ability.'''<ref name="DiVincenzo-2008" /> In general, single superconducting qubits are used for control or for measurement.
# '''Interconvertibility of stationary and flying qubits.'''<ref name="DiVincenzo-2008" /> While stationary qubits are used to store information or perform calculations, flying qubits transmit information macroscopically. Qubits should be capable of converting from being a stationary qubit to being a flying qubit and vice versa.
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==References==
{{Reflist}}
== Further reading ==
* {{Cite book |title=Principles of Superconducting Quantum Computers |last=Stancil |first=Daniel D. |publisher=John Wiley & Sons|year=2022 |___location=Hoboken, New Jersey |isbn=978-1-119-75072-7 |edition=1st |last2=Byrd |first2=Gregory T. |oclc=1302334194 |id=978-1-119-75074-1 (ebook)}}
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