Pith. sign in

REVIEW 2 major objections 5 minor 85 references

Reciprocal lumped-element superconducting circuits: quantization, decomposition, and model extraction

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that in circuits with both Josephson junctions and quantum phase slip wires, a doubly-discrete conjugate pair $(\Phi_k, \Pi_k)$ drops out of the quantized Hamiltonian, so the phase space for these variables is $S^1 \times…

desk verdict A genuinely useful toolkit for circuit quantization, decomposition, and extraction, but the headline doubly-discrete drop-out claim is a well-labeled hypothesis that sits uneasily with the paper's own canonical brackets. read the letter →

arxiv 2412.06880 v1 pith:7Z65QMW3 submitted 2024-12-09 quant-ph

Basil M. Smitham , Andrew A. Houck This is my paper · ORCID
classification quant-ph
keywords superconductingcircuitscircuitquantizationnetworkmatrixJosephsonjunctionsquantumphaseslipsflux-chargesymmetryhybridadmittance-impedancelumped-elementmodelextraction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a flux-charge symmetric method, built on the network matrix $\Omega$, the integer $\{0,\pm1\}$ matrix that records which inductive loops pass through which capacitive nodes, for three tasks in superconducting circuit quantum electrodynamics: quantizing a lumped circuit, decomposing it into a simplest equivalent form, and extracting an exact transformerless lumped model from electromagnetic simulations. Its central new claim is about quantization: in circuits containing both Josephson junctions and quantum phase slip wires, the doubly-discrete conjugate pair $(\Phi_k, \Pi_k)$ — whose members are integer multiples of the flux quantum and of the Cooper-pair charge — drops out of the final Hamiltonian, so the phase space for these variables is $S^1 \times S^1$ rather than $\mathbb{R} \times \mathbb{R}$. The algorithms rest on reducing $\Omega$ to an identity submatrix by integer row and column operations, on the integrated equations of motion to classify discrete versus continuous spectra, and on pivoting the edge form of $\Omega$ to separate harmonic modes from the nonlinear degrees of freedom. If the central claim is right, the quantized spectra of circuits containing both Josephson junctions and phase slip wires differ from what prior continuous-spectrum treatments predict, and the same matrix operations connect quantization, decomposition, and model extraction into one toolkit.

What carries the argument

The central object is the network matrix $\Omega$ (and its edge form $\Omega_E$), an integer matrix with entries $0, \pm1$ that records which inductive loops pass through which capacitive nodes (or, in the edge basis, which inductive cotree edges lie in the fundamental cutsets of capacitive tree edges). The argument is carried by two auxiliary mechanisms: the integrated equations of motion, which express the conjugate momenta $(\Pi, P)$ in terms of the integer-valued tunneling numbers $N_J$ and $M_S$, thereby fixing which operators acquire discrete versus continuous spectra; and a canonical transformation of the $k$-sector that shifts the doubly-discrete variables out of the Hamiltonian. For decomposition, the load-bearing operation is pivoting on $\Omega_E$ while preserving the nonlinear degrees of freedom (junction fluxes and phase slip charges), which separates out harmonic and free modes. For extraction, the load-bearing identity is the zero-frequency hybrid matrix constraint relating port currents and voltages to $\Omega_E$, together with a matching of capacitance and inductance matrix blocks to the residues of the hybrid response's poles.

What would settle it

Perform high-resolution spectroscopy on the fluxonium-with-phase-slips circuit (capacitor and junction in parallel with inductor and phase slip in series, Section III B) while applying a drive designed to excite the doubly-discrete variable, such as a fast flux pulse that creates a superposition of fluxon states. The paper predicts no effect on the qubit spectrum and no extra dephasing, because $(\Phi_k, \Pi_k)$ drops out; the continuous-spectrum treatment predicts additional spectral transitions or dephasing from that degree of freedom. Seeing extra lines or decoherence would falsify the claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the network matrix $\Omega$, after integer basis changes, encodes the mode structure of a superconducting circuit, and that the integrated equations of motion — not the standard ones — decide which of the canonically conjugate variables become discrete, compact, or extended after quantization. For circuits containing both Josephson junctions and phase slip wires, the variables $(\Phi_k, \Pi_k)$ inherit integer spectra from the fluxon and Cooper-pair tunneling numbers ($\Phi_0 M_S$ and $-2e N_J$), making the pair doubly discrete. Because these integers appear inside the cosine terms only as multiples of $2\pi$, the Hamiltonian is independent of them; the conjugates $(\Phi_k, \Pi_k)$ drop out of the dynamics, leaving the remaining $k+j+s$ pairs with the usual three types of spectra. The same edge network matrix $\Omega_E$, obtained by a change of basis, then carries the circuit's topology into a 'fundamental form' with harmonic modes separated out, and — through the hybrid admittance/impedance matrix and a zero-frequency constraint in terms of $\Omega_E$ — into an exact transformerless lumped model synthesized from electromagnetic data.

Load-bearing premise

The load-bearing premise is that the counts of Cooper pairs and magnetic fluxons that tunnel through the circuit are integers, making their conjugate pair $(\Phi, \Pi)$ doubly discrete; if this discrete classification fails for circuits with both junctions and phase slips, the predicted drop-out fails and the continuous-spectrum treatment applies.

Editorial extensions

If this is right

  • For any reciprocal lumped circuit with both Josephson junctions and phase slips, the quantized Hamiltonian omits the doubly-discrete pair $(\Phi_k, \Pi_k)$, so the effective number of dynamical modes is smaller than the classical node/loop count.
  • The Hamiltonian treats external charge and external flux symmetrically: Cooper-pair offsets enter through the junction cosine and fluxon offsets through the phase-slip cosine, generalizing earlier external-flux prescriptions (Appendix D 8).
  • The fundamental decomposition reduces every junction-only circuit to a junction tree plus auxiliary harmonic modes, giving the enumeration of one-junction (two classes) and two-junction (four classes) circuit topologies.
  • The extraction procedure produces transformerless lumped models — capacitors and inductors only — whose auxiliary LC oscillators exactly reproduce each finite-frequency pole of the hybrid matrix, so the synthesized circuit can be quantized with the same network-matrix algorithm.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct experimental test is conceivable with current fluxonium-with-phase-slip devices: in the paper's picture the qubit spectrum is independent of the fluxon/Cooper-pair number pair, whereas the continuous-spectrum alternative adds an extra band of states; spectroscopy looking for those extra transitions would distinguish the two.
  • The decomposition procedure can be used as a preprocessing step for quantization: because the fundamental form separates harmonic from nonlinear modes, one can quantize only the reduced core, which shrinks the numerical Hilbert space for simulations.
  • The hybrid-matrix extraction is a multi-port, flux-charge symmetric generalisation of standard single-port lossless network synthesis; extending it to non-reciprocal or lossy systems, which the paper explicitly leaves open, would require relaxing the zero-frequency network-matrix constraint.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a flux-charge symmetric framework for lumped-element superconducting circuits based on a node-loop network matrix. It presents three contributions: a quantization algorithm that uses the integrated equations of motion to classify mode spectra and predicts that in circuits with both Josephson junctions and phase slip wires a doubly-discrete conjugate pair (Φ_k, Π_k) drops out of the Hamiltonian; a circuit decomposition method based on pivoting operations on the edge network matrix; and a hybrid-matrix technique for extracting transformerless lumped circuit models from electromagnetic simulations. The main text summarizes the results, while the appendices contain detailed derivations, including the integrated equations of motion, the quantization procedure, and the decomposition algorithm.

Significance. If the central hypothesis is correct, the network-matrix formulation is a useful and intuitive toolkit: the decomposition via edge-network-matrix pivoting and the transformerless extraction from hybrid admittance/impedance responses are concrete, parameter-free algorithms with detailed derivations in the appendices. The doubly-discrete drop-out prediction is also a sharp, falsifiable contrast with the continuous-spectrum treatment of Ref. [8]. However, the quantization claim is explicitly labeled a hypothesis, and the paper does not provide a consistent quantization of the putative S1×S1 phase space, leaving the paper's central novelty on an unproven and arguably internally inconsistent footing.

major comments (2)
  1. [Section III A and Appendix C 1 (Eqs. 32, 38, C23–C28)] The derivation of the doubly-discrete drop-out is internally inconsistent as presented. The paper sets the Poisson bracket {Φ_k, Π_k} = I (Eq. 32), then applies a canonical transformation that preserves Poisson brackets (Eq. 38), and finally hypothesizes that the transformed pair has discrete spectra and phase space S1×S1 (Section III B). Under standard canonical quantization, a pair of operators with a unit Poisson bracket cannot both have purely discrete spectra; the paper supplies no alternative quantization (such as a noncommutative torus) for this pair, and in Appendix C 2 it explicitly omits commutation relations for these variables. Since the removal of these variables in Eq. 39 relies on their integer-valued spectra, the paper's central prediction is not backed by a consistent quantization procedure.
  2. [Appendix B 1 and Section III A] The spectral classification of N_J and M_S as integer-valued is asserted on the basis of unweighted integrals of charge and flux densities, but it is not proven for the mixed case of circuits containing both Josephson junctions and phase slip wires. This classification is the load-bearing step that leads to the drop-out of the doubly-discrete pair, and the paper explicitly contrasts its prediction with the continuous-spectrum treatment of Ref. [8]. Because the claim is testable and central, the manuscript should either supply a rigorous derivation of the discrete spectra for the relevant variables or clearly restrict the quantization claims to a regime where the hypothesis can be independently justified.
minor comments (5)
  1. [Section III A] There is a repeated word: 'arising from from inductive loops' should read 'arising from inductive loops'.
  2. [Section V E] The word 'synthesiszed' should be 'synthesized'.
  3. [Section II C] The reference 'Fig. IV B(b)' should be 'Fig. 2(b)'.
  4. [Section V C] The abstract and Section V claim an 'exact, transformerless circuit model,' but Section V C states that a cutoff is applied to the number of high-frequency poles; the sense of 'exact' should be qualified, for instance by stating that the model is exact up to the retained poles.
  5. [Eq. 76 and Appendix D 2] The sign convention in the definition of the edge network matrix differs from the loop-based definition in Eq. 76 versus Eq. D15; the equivalence should be stated explicitly to avoid confusion.

Circularity Check

1 steps flagged · score 6.0 of 10

The headline 'doubly-discrete drop-out' prediction is the paper's own stated hypothesis, so the central claim reduces to its input assumption.

  1. self definitional [Section III A, quantization algorithm (around Eq. 39 and final paragraph); restated in Appendix C 1 after Eq. C28]
    "In particular, we hypothesize that certain circuits with both Josephson junctions and phase slips [13] possess doubly-discrete charge and flux conjugate pairs of variables that drop out of the final Hamiltonians. ... The presence of these removable, doubly-discrete conjugate variables is the main hypothesis of our quantization procedure."

    The paper advertises the absence of (Phi_k, Pi_k) from the final Hamiltonian as its novel prediction (Eqs. 39, 44, C28). The removal at Eq. 39/C28 is performed by discarding these variables from cosine arguments because they appear as integer multiples of 2pi, i.e., because they have been assigned a doubly-discrete spectrum. That assignment is not a consequence of the equations of motion; it is the assumption introduced before the derivation ('we hypothesize...'). The final 'central hypothesis' restates the predicted drop-out verbatim. If the conjugate pair instead had continuous spectra (the alternative in Ref. [8]), the 2pi multiples would not cancel and the variables would survive in the Hamiltonian.

full rationale

One concrete circular step is present. The paper's central new result for circuits with both Josephson junctions and phase slip wires is that the conjugate pair (Phi_k, Pi_k) becomes doubly discrete and drops out of the Hamiltonian. The paper explicitly introduces this as its 'main hypothesis' and then presents the drop-out as a prediction. The derivation in Eq. 39 and Appendix C 1 removes the pair only after assuming the integer/multiple-of-2pi spectrum; that assumption is precisely the content of the prediction. This is a self-definitional reduction rather than an external falsification. The decomposition and model-extraction sections are not circular: they are constructive matching algorithms whose output (an equivalent circuit, or a lumped model matching a given hybrid matrix) is defined by the stated matching conditions, and no constants are fitted to data to manufacture a 'prediction.' There are no load-bearing self-citations: the cited prior work is used for notation and contrast, not to justify the disputed spectral assignment. Because the central novel prediction reduces to its own hypothesis, the circularity score is 6 rather than 0-2.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The framework rests on the standard lumped-element, reciprocal, lossless circuit assumptions, plus two structural restrictions (no junction-only loops, no phase-slip-only cutsets) that guarantee the tree-cotree construction. The novel quantization prediction additionally depends on the discrete-spectrum classification of Appendix B 1, namely that Cooper pair and fluxon tunneling numbers are integer-valued; this is a physical assumption, not fitted. No free parameters or invented entities enter the central claim.

assumptions (8)
  • domain assumption The circuit is lossless and reciprocal, with symmetric capacitance and inductance matrices.
    Stated in Section II A and Appendix A; underlies the hybrid matrix synthesis and the conservative equations of motion.
  • domain assumption There are no Josephson-junction-only loops; each junction loop contains some linear inductance, ensuring AJ has full column rank.
    Introduced in Section II A; needed for the spanning tree/cotree construction and to make the decomposition well-defined.
  • domain assumption There are no phase-slip-only cutsets or nodes; each phase slip node touches a capacitor, ensuring BS has full column rank.
    Introduced in Section II A; dual to the junction-loop condition and required for tree-cotree structure.
  • domain assumption Single-particle charge and flux are quantized in units of 2e and Phi0, and unweighted integrals of charge and flux densities become integer-valued operators upon quantization.
    Appendix B 1; this is the load-bearing physical input for the discrete-spectrum classification that produces the doubly-discrete drop-out prediction.
  • domain assumption The lumped-element quasi-static limit applies, with path-independent node voltages and loop currents.
    Appendix A 1; standard cQED modeling assumption; also needed for the extraction synthesis.
  • domain assumption External charge and flux are treated as classical parameters, symmetric in form.
    Section II A and Appendix A 4; used throughout the Hamiltonian construction.
  • standard math Basis transformations U and W are integer-valued and preserve the edge network matrix structure of the circuit.
    Section II B and Appendix D 2; relies on total unimodularity and closure of network matrices under pivoting, a known result in linear programming [11,12].
  • domain assumption Capacitance and inductance matrices are invertible after grounding.
    Section II A and Appendix A 3; required to construct the Hamiltonian and to remove free modes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Reciprocal lumped-element superconducting circuits: quantization, decomposition, and model extraction." pith.science (2026). https://pith.science/paper/7Z65QMW3

@misc{pith2026241206880,
  author       = {Pith},
  title        = {Pith review of: Reciprocal lumped-element superconducting circuits: quantization, decomposition, and model extraction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Z65QMW3}},
  note         = {Machine review of arXiv:2412.06880}
}
read the original abstract

In this work, we introduce new methods for the quantization, decomposition, and extraction (from electromagnetic simulations) of lumped-element circuit models for superconducting quantum devices. Our flux-charge symmetric procedures center on the network matrix, which encodes the connectivity of a circuit's inductive loops and capacitive nodes. First, we use the network matrix to demonstrate a simple algorithm for circuit quantization, giving novel predictions for the Hamiltonians of circuits with both Josephson junctions and quantum phase slip wires. We then show that by performing pivoting operations on the network matrix, we can decompose a superconducting circuit model into its simplest equivalent "fundamental" form, in which the harmonic degrees of freedom are separated out from the Josephson junctions and phase slip wires. Finally, we illustrate how to extract an exact, transformerless circuit model from electromagnetic simulations of a device's hybrid admittance/impedance response matrix, by matching the lumped circuit's network matrix to the network topology of the physical layout. Overall, we provide a toolkit of intuitive methods that can be used to construct, analyze, and manipulate superconducting circuit models.

Figures

Figures reproduced from arXiv: 2412.06880 by the authors.

Figure 1
Figure 1. FIG. 1. Overview of the flux-charge symmetric model. Cir [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Superconducting circuit models represented in standard and tree-cotree notations. (a) Superconducting circuit in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Predicted types superconducting circuit modes. (a) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Fundamental decomposition of a Josephson junction [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Decomposition applied to an example circuit. (a) Initial four-node circuit with three Josephson junctions and two [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Classification of one and two-Josephson junction cir [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Overview of the model extraction procedure. A layout (with orange rectangles representing regions of Josephson [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Model extraction for a fluxonium qubit with external [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Circuit model illustrations of (a) the electromagnetic [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Graph model of electrical circuits with capacitance [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Capacitors and inductors. (a) Branch capacitor cir [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Constructing system matrices from branch circuit models. (a) Circuit diagram with capacitors and inductors. The [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Every LC network can be transformed into an equiv [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Illustrations of lumped inductor charge and capaci [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Semiclassical model of LC-oscillator system with [PITH_FULL_IMAGE:figures/full_fig_p033_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Josephson junctions and quantum phase slips. (a) [PITH_FULL_IMAGE:figures/full_fig_p034_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Construction of node flux and loop charge vari [PITH_FULL_IMAGE:figures/full_fig_p035_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Visual interpretation of the pivoting procedure in the decomposition algorithm. (a) Initial circuit model with eventual [PITH_FULL_IMAGE:figures/full_fig_p045_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Schematic of model extraction procedure in tree [PITH_FULL_IMAGE:figures/full_fig_p056_19.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

85 extracted references · 69 canonical work pages

  1. [8]

    Pivoting, structure-preserving basis transformations As discussed in Appendix D 2, the closure of edge net- work matrices under linear-algebraic pivoting allows us to manipulate and decompose circuits into equivalent forms. Recall that in this edge basis, the equations of motion for the Josephson fluxes and phase slip charges are enumer- ated by the incid...

    work page

  2. [1]

    edge network matrix

    Decomposition to the edge network matrix We recall from Appendix B 8 that the one effect of an integer-valued basis transformation U and W, with coordinate transformations # »Φn → UT −1 # »Φn (D1) # »Ql → WT −1 # »Ql (D2) is to perform the following operations on the circuit’s constituent incidence, loop, and network matrices: AJ → UAJ (D3) BS → WBS (D4) ...

  3. [2]

    Graph-theoretic edge network matrices In this basis, the circuit’s edge network matrix co- incides (up to an inconsequential minus sign) with the definition of a tree-cotree edge network matrix from lin- ear/integer programming. Edge network matrices (just called network matrices in that field) can be understood and manipulated by noting that they thus po...

  4. [3]

    Deleting a row or column from an edge network matrix (or replacing it with all 0) results in an edge network matrix

  5. [4]

    Multiplying the rows or column of an edge network matrix by −1 results in an edge network matrix

  6. [5]

    Taking the edge network matrix ΩE and augment- ing it as [ ΩE, I] gives an edge network matrix

  7. [6]

    Pivoting (through row or column operations) on a nonzero entry of an edge network matrix returns an edge network matrix

    work page

  8. [7]

    We give brief explanations of the intuition behind prop- erties 1, 2, and 4

    Edge network matrices are totally unimodular (the determinant of any square submatrix is −1, +1, or 0). We give brief explanations of the intuition behind prop- erties 1, 2, and 4. In Property 1, deleting a row corre- sponds to contracting the two sides of a capacitive edge into a single node, and deleting a column corresponds to removing an inductive edg...

    work page

Show all 85 references
  1. [9]

    fundamental decompo- sition

    F undamental decomposition We now describe the algorithm to reduce a circuit to an equivalent form that we call the “fundamental decompo- sition.” We start with the edge network matrix in block form (Eq. D21) and carry out a series of pivoting op- erations. If the submatrix ΩC...

  2. [10]

    Recall that a basis transformation takes: ΩE → UΩEWT , where U transforms the capacitive flux vari- ables and W the inductive charge

    Classifying circuits by equivalent fundamental network matrices The fundamental decomposition generates a reduced network matrix ΩE, whose signature makes it possi- ble to classify equivalent circuits by performing a set of structure-preserving transformations. Recall that a b...

  3. [11]

    D23) as: ΩE = ΩJf 0Jr 0rf Irr (D25) Here, the fundamental form of the circuit is easier to interpret

    F undamental circuit form of Junction-only circuits After the elimination of free modes, for circuits with no phase slip inductors, we can write down the fundamental network matrix (Eq. D23) as: ΩE = ΩJf 0Jr 0rf Irr (D25) Here, the fundamental form of the circuit is easier to ...

  4. [12]

    In addition, it allows us to more specifically classify the system’s ex- tended degrees of freedom, by the types of cosine terms in which they appear in the Hamiltonian

    Quantization from decomposition The fundamental form of the network matrix also pro- vides a straightforward way to carry our quantization procedure (shown in Appendix C 1 and C 2). In addition, it allows us to more specifically classify the system’s ex- tended degrees of free...

  5. [13]

    irro- tational gauge,

    Limits of zero inductance, capacitance In this work, we have considered every Josephson junc- tion loop to have some finite self-inductance and every phase slip island some self-capacitance. However, super- conducting circuits are often analyzed in limits where these quantitie...

  6. [14]

    From these simulations, circuit models of a device can be extracted and the tunneling effects of the small, nonlinear “quantum” elements added in afterwards

    Modeling superconductivity in electromagnetic simulation Here, we present background on the electromagnetic simulation of linear superconducting metal in a low-loss dielectric environment. From these simulations, circuit models of a device can be extracted and the tunneling ef...

  7. [15]

    black box

    Electromagnetic ports In order to perform a simulation and extract a circuit model, applied electromagnetic fields are needed, which mimic the excitation profile of the actual device. These excitations are provided by electromagnetic ports, po- sitioned in place of certain dev...

  8. [16]

    This matrix relates some com- bination input and output currents and voltages

    Admittance, impedance, and hybrid matrices To construct a circuit model for linear electromagnetic response of an N-port system, we generally utilize an N-by-N response matrix. This matrix relates some com- bination input and output currents and voltages. Com- monly simulated ...

  9. [17]

    For superconductors far below their critical temperature em- bedded in low-loss dielectrics, we make the common ap- proximation that the system is lossless

    Lossless positive real (reciprocal) matrices Construction of a circuit model is enabled by not- ing key properties obeyed by the Laplace-domain elec- tromagnetic response matrices, which correspond to the physical attributes of the system being modeled. For superconductors far...

  10. [18]

    The matrix H(s) can be written as a rational func- tion of real coefficients

  11. [19]

    Each pole of H(s) (including the pole at ∞) is sim- ple and has positive semi-definite Hermitian residue matrix K

  12. [20]

    Poles of H(s) (except the pole at ∞) occur at s = ±iω, where ω represents a real-valued frequency

  13. [21]

    For Property 1, the simulation response of a dis- tributed device will not exactly satisfy the condition

    H(s) + H†(s) = 0 for all s = iω, where ω is real and does not correspond to a pole of H(s). For Property 1, the simulation response of a dis- tributed device will not exactly satisfy the condition. However, the true response function is usually approx- imated to a high degree ...

  14. [22]

    Port placement, zero-frequency response Before detailing how to expand hybrid matrices in terms of their poles, we examine the question of port placement on an electromagnetic structure, to further constrain the form of this hybrid matrix. In so doing so, we force the zero-fre...

  15. [23]

    This will allow us to simplify the Laplace-domain response into a form that is easily compared to that of a lumped-element cir- cuit

    Hybrid matrix pole expansion Now we have a set of conditions and constraints in place that define the simulated hybrid matrix of our loss- less, reciprocal superconducting systems. This will allow us to simplify the Laplace-domain response into a form that is easily compared t...

  16. [24]

    We proceed in a fashion most similar to the approach presented in [43]

    Pole decomposition In order to facilitate comparisons to a lumped circuit, we perform one more expansion of each finite-frequency residue term—which relies on the positive semi-definite nature of the residue matrices. We proceed in a fashion most similar to the approach presen...

  17. [25]

    sCCC − s3 X r 1 Crr # »C Cr # »C T Cr s2 + ω2r # # »V C(s) −

    Comparison to equations of motion of lumped circuit Now we address the question of synthesizing the re- sponse function with a lumped circuit model. For a FIG. 19. Schematic of model extraction procedure in tree- cotree notation. Capacitive (parallel) and inductive (series) po...

  18. [26]

    Josephson tunneling can be placed in parallel with capacitive ports, and fluxoid tunneling can be inserted in series with in- ductive ports

    Port replacement Once a model has been constructed for the lossless lin- ear portion of the circuit, nonlinear and drive elements can inserted across the model’s terminals. Josephson tunneling can be placed in parallel with capacitive ports, and fluxoid tunneling can be insert...

  19. [27]

    B. D. Josephson, Possible new effects in superconductive tunnelling, Physics Letters 1, 251 (1962)

    1962

  20. [28]

    J. E. Mooij and Y. V. Nazarov, Superconducting nanowires as quantum phase-slip junctions, Nature Physics 2, 169 (2006), publisher: Nature Publishing Group

    2006

  21. [29]

    Devoret, Quantum fluctuations in electrical circuits, in Les Houches, Session LXIII, edited by S

    Michel H. Devoret, Quantum fluctuations in electrical circuits, in Les Houches, Session LXIII, edited by S. Rey- naud, E. Giacobino, and J. Zinn-Justin (Elsevier Science B.V., 1995) pp. 351–386

    1995

  22. [30]

    Burkard, R

    G. Burkard, R. H. Koch, and D. P. DiVincenzo, Mul- tilevel quantum description of decoherence in supercon- ducting qubits, Physical Review B 69, 064503 (2004)

    2004

  23. [31]

    Burkard, Circuit theory for decoherence in supercon- ducting charge qubits, Physical Review B 71, 144511 (2005)

    G. Burkard, Circuit theory for decoherence in supercon- ducting charge qubits, Physical Review B 71, 144511 (2005)

    2005

  24. [32]

    Ulrich and F

    J. Ulrich and F. Hassler, Dual approach to circuit quanti- zation using loop charges, Physical Review B 94, 094505 (2016)

    2016

  25. [33]

    I. L. Egusquiza and A. Parra-Rodriguez, Algebraic canonical quantization of lumped superconducting net- works, Physical Review B 106, 024510 (2022)

    2022

  26. [34]

    Parra-Rodriguez and I

    A. Parra-Rodriguez and I. L. Egusquiza, Geometrical de- scription and Faddeev-Jackiw quantization of electrical networks (2024), arXiv:2304.12252

    2024 arXiv

  27. [35]

    Osborne, T

    A. Osborne, T. Larson, S. G. Jones, R. W. Simmonds, A. Gyenis, and A. Lucas, Symplectic Geometry and Cir- cuit Quantization, PRX Quantum 5, 020309 (2024), pub- lisher: American Physical Society

    2024

  28. [36]

    Osborne and A

    A. Osborne and A. Lucas, Flux-charge symmetric the- ory of superconducting circuits, Physical Review B 109, 174524 (2024)

    2024

  29. [37]

    Schrijver, Theory of linear and integer programming (John Wiley & Sons, 1998)

    A. Schrijver, Theory of linear and integer programming (John Wiley & Sons, 1998)

    1998

  30. [38]

    L. A. Wolsey and G. L. Nemhauser, Integer and combi- natorial optimization (John Wiley & Sons, 2014)

    2014

  31. [39]

    D. T. Le, A. Grimsmo, C. M¨ uller, and T. M. Stace, Dou- bly nonlinear superconducting qubit, Physical Review A 100, 062321 (2019)

    2019

  32. [40]

    X. You, J. A. Sauls, and J. Koch, Circuit quantization in the presence of time-dependent external flux, Physical Review B 99, 174512 (2019)

    2019

  33. [41]

    E. J. Weissler, M. Bhat, Z. Liu, and J. Combes, Enu- meration of all superconducting circuits up to 5 nodes (2024), arXiv:2410.18497

    2024 arXiv

  34. [42]

    S. E. Nigg, H. Paik, B. Vlastakis, G. Kirchmair, S. Shankar, L. Frunzio, M. H. Devoret, R. J. Schoelkopf, and S. M. Girvin, Black-Box Superconducting Cir- cuit Quantization, Physical Review Letters 108, 240502 (2012)

    2012

  35. [43]

    Solgun, D

    F. Solgun, D. W. Abraham, and D. P. DiVincenzo, Black- box quantization of superconducting circuits using ex- act impedance synthesis, Physical Review B 90, 134504 (2014)

    2014

  36. [44]

    Labarca, O

    L. Labarca, O. Benhayoune-Khadraoui, A. Blais, and A. Parra-Rodriguez, Toolbox for nonreciprocal dispersive models in circuit quantum electrodynamics, Physical Re- view Applied 22, 034038 (2024)

    2024

  37. [45]

    R. W. Newcomb, Linear Multiport Synthesis (McGraw- Hill, 1966)

    1966

  38. [46]

    B. D. O. Anderson and S. Vongpanitlerd, Network Anal- ysis and Synthesis: A Modern Systems Theory Approach (Dover, 2013)

    2013

  39. [47]

    Vool and M

    U. Vool and M. Devoret, Introduction to quantum elec- tromagnetic circuits, International Journal of Circuit Theory and Applications 45, 897 (2017)

    2017

  40. [48]

    A. J. Kerman, Efficient numerical simulation of complex Josephson quantum circuits (2020), arXiv:2010.14929

    2020 arXiv

  41. [49]

    Parra-Rodriguez, I

    A. Parra-Rodriguez, I. L. Egusquiza, D. P. DiVincenzo, and E. Solano, Canonical circuit quantization with lin- ear nonreciprocal devices, Physical Review B 99, 014514 (2019)

    2019

  42. [50]

    Tinkham, Introduction to Superconductivity (Dover Publications, 1996)

    M. Tinkham, Introduction to Superconductivity (Dover Publications, 1996)

    1996

  43. [51]

    Strang, Introduction to Linear Algebra (Wellesley- Cambridge Press, 2016)

    G. Strang, Introduction to Linear Algebra (Wellesley- Cambridge Press, 2016)

    2016

  44. [52]

    R. B. Bapat, Graphs and Matrices (Springer London, London, 2014)

    2014

  45. [53]

    S. P. Chitta, T. Zhao, Z. Huang, I. Mondragon-Shem, and J. Koch, Computer-aided quantization and numeri- cal analysis of superconducting circuits, New Journal of Physics 24, 103020 (2022), publisher: IOP Publishing. 59

    2022

  46. [54]

    Rasmussen, Superconducting Circuit Companion—an Introduction with Worked Examples, PRX Quantum 2, 10.1103/PRXQuantum.2.040204 (2021)

    S. Rasmussen, Superconducting Circuit Companion—an Introduction with Worked Examples, PRX Quantum 2, 10.1103/PRXQuantum.2.040204 (2021)

    2021 doi

  47. [55]

    Ciani, D

    A. Ciani, D. P. DiVincenzo, and B. M. Terhal, Lecture Notes on Quantum Electrical Circuits (TU Delft OPEN Textbooks, 2024) publication Title: TU Delft OPEN Textbooks

    2024

  48. [56]

    M. H. Devoret, Does Brian Josephson’s Gauge-Invariant Phase Difference Live on a Line or a Circle?, Journal of Superconductivity and Novel Magnetism 34, 1633 (2021)

    2021

  49. [57]

    V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets, Science 326, 113 (2009), publisher: American Association for the Advancement of Science

    2009

  50. [58]

    V. E. Manucharyan, N. A. Masluk, A. Kamal, J. Koch, L. I. Glazman, and M. H. Devoret, Evidence for coherent quantum phase slips across a Josephson junction array, Physical Review B 85, 024521 (2012)

    2012

  51. [59]

    M. T. Randeria, T. M. Hazard, A. Di Paolo, K. Azar, Max Hays, L. Ding, J. An, M. Gingras, B. M. Niedzielski, H. Stickler, J. A. Grover, J. L. Yoder, M. E. Schwartz, W. D. Oliver, and K. Serniak, Dephasing in Fluxonium Qubits from Coherent Quantum Phase Slips, PRX Quan- tum 5, ...

    2024

  52. [60]

    J. G. Oxley, Matroid theory , Vol. 3 (Oxford University Press, USA, 2006)

    2006

  53. [61]

    Ding, H.-S

    D. Ding, H.-S. Ku, Y. Shi, and H.-H. Zhao, Free-mode removal and mode decoupling for simulating general su- perconducting quantum circuits, Physical Review B 103, 174501 (2021)

    2021

  54. [62]

    Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Mutus, P. O’Malley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, M. R. Geller, A. Cleland, and J. M. Martinis, Qubit ...

    2014

  55. [63]

    Pozar, Microwave Engineering, 4th ed

    David M. Pozar, Microwave Engineering, 4th ed. (John Wiley and Sons, 2012)

    2012

  56. [64]

    Solgun and D

    F. Solgun and D. P. DiVincenzo, Multiport impedance quantization, Annals of Physics 361, 605 (2015)

    2015

  57. [65]

    R. M. Foster, A reactance theorem, Bell System technical journal 3, 259 (1924), publisher: Wiley Online Library

    1924

  58. [66]

    Z. K. Minev, T. G. McConkey, M. Takita, A. D. Corcoles, and J. M. Gambetta, Circuit quantum electrodynam- ics (cQED) with modular quasi-lumped models (2021), arXiv:2103.10344

    2021 arXiv

  59. [67]

    Z. K. Minev, Z. Leghtas, S. O. Mundhada, L. Chris- takis, I. M. Pop, and M. H. Devoret, Energy-participation quantization of Josephson circuits, npj Quantum Infor- mation 7, 1 (2021), publisher: Nature Publishing Group

    2021

  60. [68]

    Yarlagadda and Y

    R. Yarlagadda and Y. Tokad, Synthesis of LC net- works—a state-model approach, Proceedings of the Insti- tution of Electrical Engineers 113, 975 (1966), publisher: The Institution of Engineering and Technology

    1966

  61. [69]

    R. Yarlagadda, Reciprocal Lossless Synthesis of Hybrid Parameters of LCT Networks by State-Space Approach, IEEE Transactions on Circuit Theory 19, 69 (1972), con- ference Name: IEEE Transactions on Circuit Theory

    1972

  62. [70]

    Gustavsen and A

    B. Gustavsen and A. Semlyen, Rational approximation of frequency domain responses by vector fitting, IEEE Transactions on Power Delivery 14, 1052 (1999), confer- ence Name: IEEE Transactions on Power Delivery

    1999

  63. [71]

    Wassaf, Efficiently Building and Characterizing Elec- tromagnetic Models of Multi-Qubit Superconducting Cir- cuits (2024), arXiv:2406.04351

    F. Wassaf, Efficiently Building and Characterizing Elec- tromagnetic Models of Multi-Qubit Superconducting Cir- cuits (2024), arXiv:2406.04351

    2024 arXiv

  64. [72]

    A. J. Kerman, Flux–charge duality and topological quan- tum phase fluctuations in quasi-one-dimensional super- conductors, New Journal of Physics 15, 105017 (2013)

    2013

  65. [73]

    D. J. Griffiths, Introduction to Electrodynamics (Pearson, 2013)

    2013

  66. [74]

    Chen, The Electrical Engineering Handbook (El- sevier Academic Press, 2004)

    W.-K. Chen, The Electrical Engineering Handbook (El- sevier Academic Press, 2004)

    2004

  67. [75]

    R. A. Diaz and W. J. Herrera, The positivity and other properties of the matrix of capacitance: Physical and mathematical implications, Journal of Electrostatics 69, 587 (2011)

    2011

  68. [76]

    B. D. Tellegen, The gyrator, a new electric network ele- ment, Philips Res. Rep 3, 81 (1948)

    1948

  69. [77]

    C. R. Paul, Inductance: loop and partial (John Wiley & Sons, 2011)

    2011

  70. [78]

    Eremenko, Simultaneous diagonalization of two quadratic forms and a generalized eigenvalue problem (2019)

    A. Eremenko, Simultaneous diagonalization of two quadratic forms and a generalized eigenvalue problem (2019)

    2019

  71. [79]

    Riwar, Charge quantization and detector resolu- tion, SciPost Physics 10, 093 (2021)

    R.-P. Riwar, Charge quantization and detector resolu- tion, SciPost Physics 10, 093 (2021)

    2021

  72. [80]

    Gross, A

    R. Gross, A. Marx, and F. Deppe, Applied superconduc- tivity: Josephson effect and superconducting electronics (De Gruyter, 2016)

    2016

  73. [81]

    Rymarz and D

    M. Rymarz and D. P. DiVincenzo, Consistent Quantiza- tion of Nearly Singular Superconducting Circuits, Phys- ical Review X 13, 021017 (2023)

    2023

  74. [82]

    I. L. Egusquiza and A. Parra-Rodriguez, On ”Consistent Quantization of Nearly Singular Superconducting Cir- cuits” (2024), arXiv:2408.05174

    2024

  75. [83]

    Kerr, Surface impedance of superconductors and nor- mal conductors in EM simulators, MMA Memo 21, 1 (1999)

    A. Kerr, Surface impedance of superconductors and nor- mal conductors in EM simulators, MMA Memo 21, 1 (1999)

    1999

  76. [84]

    S. V. Yuferev and N. Ida, Surface Impedance Boundary Conditions: A Comprehensive Approach (CRC Press, 2010)

    2010

  77. [85]

    T. E. Roth, Introduction to Computational Electromag- netics (2024), https://nanohub.org/resources/38968

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.