Symplectic Geometry and Circuit Quantization

Circuit quantization is an extraordinarily successful theory that describes the behavior of quantum circuits with high precision. The most widely used approach of circuit quantization relies on introducing a classical Lagrangian whose degrees of freedom are either magnetic fluxes or electric charges in the circuit. By combining nonlinear circuit elements (such as Josephson junctions or quantum phase slips), it is possible to build circuits where a standard Lagrangian description (and thus the standard quantization method) does not exist. Inspired by the mathematics of symplectic geometry and graph theory, we address this challenge, and present a Hamiltonian formulation of nondissipative electrodynamic circuits. The resulting procedure for circuit quantization is independent of whether circuit elements are linear or nonlinear, or if the circuit is driven by external biases. We explain how to rederive known results from our formalism, and provide an efficient algorithm for quantizing circuits, including those that cannot be quantized using existing methods.

Popular Summary

One of the most promising routes to building a quantum computer, both in academia and industry, is to build extremely high-fidelity qubits out of superconducting circuits. The theoretical underpinning to this effort is circuit quantization, which tells us how to quantize a circuit and subsequently deduce its spectrum and robustness to environmental noise. As in a conventional mechanical system, one aims to identify canonically conjugate degrees of freedom along with a Hamiltonian. However, electrical circuits have many constraints, and thus far no generic prescription has been found to identify degrees of freedom and ultimately quantize a circuit. Experimentalists can build circuits that theorists do not know how to quantize, raising both a fundamental theoretical question and a practical experimental challenge of how to confirm or analyze the quality of a proposed novel qubit.

We have solved this longstanding challenge. Inspired by the 20th-century mathematics of symplectic geometry, which abstracts notions of Hamiltonian mechanics to more general settings, we present an efficient and universal procedure for identifying the degrees of freedom and ultimately quantizing general electrical circuits built out of inductors and capacitors. Existing theories for circuit quantization arise as limits of our more general framework.

Our approach leads to an efficient numerical algorithm for circuit quantization on classical computers, which will aid present-day researchers in the design of increasingly large and complex qudits and quantum circuits. Looking forward, the deep connection between superconducting circuits and symplectic geometry suggests that the physics of robustly protected qubits may have a surprising connection to the mathematics of geometric quantization.

Figure 1

Examples of quantum circuits with Lagrangians using one or two types of generalized coordinates. (a) An inductor shunted by two capacitors: the Lagrangian of the circuit contains a single type of variable, the flux across the inductor, such that $L(\varphi ,\dot{\varphi })$. The charge-flux conjugate pairs are defined at the Lagrangian level. (b) A Josephson junction and a quantum phase-slip junction forming a loop. This quantum circuit cannot be described by a Lagrangian using a single type of variable but only with a Lagrangian that contains both charge and flux variables, $L(\varphi ,\dot{\varphi },q,\dot{q})$. (c) A more complex circuit of multiple Josephson junctions and quantum phase slips, where writing down a Hamiltonian requires geometrical arguments.

Figure 2

Graph theory and quantum circuits. (a) An arbitrary circuit containing all four types of superconducting circuit elements: capacitors, inductors, Josephson junctions, and quantum phase slips. The node fluxes are ${\varphi }_{v_i}$, while the branch charges across the capacitive elements are $q_{e_i}$. (b) The graph of the full circuit. The vertices of the graph are label as $v_i$, while the edges are denoted as $e_i$. Inductive branches are colored with blue lines, while capacitive ones are highlighted with red lines. The incidence matrix of the graph of the full circuit is $A_{ev}$. (c) The capacitive subgraph of the circuit containing only capacitive edges. The capacitive incidence matrix is ${\mathrm{\Omega }}_{ev}$.

Figure 3

Circuit elements and their constitutive relations. The circuit elements that we consider in this work connect two variables (flux, charge, and their derivatives). Inductive elements relate flux $\varphi$ and the current $\dot{q}$, such as linear inductors ($L$) and Josephson junctions ($\mathrm{JJ}$). On the other hand, capacitive elements connect charge $q$ with voltage $\dot{\varphi }$, such as linear capacitors ($C$) and quantum phase-slip elements ($\mathrm{QPS}$). The memristor ($M$) [36] and resistor ($R$) connect directly charge with flux or their derivatives.

Figure 4

Symplectic geometry of a circuit. When describing a circuit with its capacitive graph, we can define two types of geometrical objects that correspond to null vectors of the capacitive incidence matrix of the circuit: inductively shunted islands (green-filled rectangular), and capacitive loops (orange-filled rectangular). The variables associated with these null vectors must be removed to be able to quantize the circuit. Additional variables can be removed based on the Noether charges of the circuit. The edges that are part of a (nonunique) spanning tree are highlighted with wide lines.

Figure 5

Inductively and capacitively shunted islands. (a) The inductively shunted island at node flux ${\varphi }_{v_2}$ corresponds to the right null vector of the incidence matrix ${\mathrm{\Omega }}_{ev}$; thus, we need to remove such variable to be able to arrive to a self-consistent Hamiltonian. (b) The capacitively shunted island at ${\varphi }_{v_2}$ corresponds to a Noether charge in the circuit. It is possible but not required to remove such a variable to be able to define conjugate pairs.

Figure 6

Examples for quantum circuits in the framework of symplectic geometry. The branches are colored based on the type of element they contain; red: capacitive elements, blue: inductive elements. The circuits are (a) dualmon circuit, (b) offset-charge-sensitive transmon [53], (c) external-flux-sensitive fluxonium.

Figure 7

Circuit in time-dependent external flux. A flux-tunable transmon with two Josephson junctions shunted by two capacitors. The external flux in the loop enclosed by the two junctions can be modeled as one or more additional batteries in a loop, as long as the total flux provided by these batteries equals the external flux. The batteries are modeled as capacitive elements.

Figure 8

Singular circuit. (a) Example for a one-mode circuit where a quantum phase element has no series inductance attached to it. The circuit has a capacitive loop, $\mathcal{C}=\{\{q_{e_1},q_{e_2}\}\}$ leading to a nonanalytical constraint. (b) When a series inductance is included in the circuit, the capacitive loop is broken, and the system has two degrees of freedom and an analytical Hamiltonian.