What are Superconducting Qubits? A mildly mathematical excursion

Quantum Computing seems to be all the rage again, given the recent coverage including Wired's article on the race to sell true quantum computers and a full Economist Technology Quarterly segment on Quantum Leaps. This motivated me to review my basic understanding of the architectures that are underlying the efforts to build a scalable Quantum Computer. The physical implementation all these systems --in one way or another-- have in common is the Superconducting Qubit (SCQ). What kind of beast is this? Is it similar to an atom? Unfortunately it's not quite as simple as the qubit relies heavily on superconductivity -- a macroscopic quantum state that involves many particles and cannot be described using a simple atom pictures. Hence, I decided to write up a more or less rigorous mathematical excursion to motivate and clarify the origins of the SCQ.

On the way we need to introduce some complex concepts that help us understand how we can quantize a macroscopic electromechanical circuit element. Moreover, we briefly touch on two surprising quantum effects that underlie the SCQ: The quantization of the magnetic flux and the Josephson effect -- both consequences of superconductivity. After introducing these ingredients we will be ready to discuss what the SCQ is and how we can think about it.

As always, I hope you enjoy reading the post!

Brief Introduction

We have all heard about single atoms and electrons that behave in strange ways described by quantum mechanics and --to a certain extent-- have gotten used to it. However, when it comes to macroscopic objects as big as electric circuits we have not yet fully embraced the fact that they can also exhibit quantum-weirdness. One such device type is the Josephson Junction that gives rise to "macroscopic" quantum states that form the basis of modern qubit architectures to lead the quest to quantum computing.

The quantum world enters into these devices through an element called a Josephson Junction, that relies on superconductivity, a quantum state consisting of many electron-electron pairs and allowing electric currents to flow without resistance. Being a quantum state the superconductivity has an amplitude described by the number of electron pairs --corresponding to classical charge and current-- and a quantum phase which has no classical equivalent.

A simplistic picture of a Superconducting Qubit is a bound state of locked phase and charge in a phase-potential created by a Josephson Junction. The potential exhibits several such states and we choose two of them to encode the logical (quantum) states $\vert 0 \rangle$ and $\vert 1 \rangle$. This is very similar to a quantum particle in a deep well which exhibits discrete oscillatory motions similar to a (quantum) harmonic oscillator due to the well trapping the particle.

To create a quantum computer we need many of these qubits to talk to each other and also need to control their behavior. Since SCQs are emerging in electric circuit elements we can use all the tools of electrical engineering to manipulate the qubits and perform computations -- hopefully giving rise to quantum computers at some point in the future.

If you are interested in more details about the theory behind Superconducting Qubits, and in particular the Superconducting Phase Qubit, please read on, but also brace yourself for some math ;)

Quantization of the LC Circuit

To get started we introduce a few concepts of how to quantize electric circuits and start with a simple LC circuit consisting of a capacitor and an inductor. As the final equation for the LC circuit looks like that of a quantum mechanical harmonic oscillator, we will review the latter one first and remind us of some basic properties and the framework of canonical quantization.

Canonical Quantization

In order to approach the challenge of deriving the physics of a SCQ we follow the approach of canonical quantization. This framework consists in identifying the canonical, conjugate variables of the classical counterpart of the system under consideration and replacing the Poisson bracket algebra with canonical commutation relations. The standard example is a harmonic oscillator whose classical Hamiltonian is given by

where $m$ is the mass of the oscillator and $\omega$ is the resonance frequency. In classical mechanics the position $x$ and momentum $p$ are canonically conjugate and fulfill the Poisson bracket algebra

Using the rules of canonical quantization we replace the Poisson bracket with commutation relations leading to

and promote the conjugate variables to operators leading to the quantum mechanical Hamiltonian

An elegant way to solve the equations for the quantum mechanical harmonic oscillator is to introduce annihilation and creation operators defined as

These new operators fulfill the commutation relation

and replacing $\hat x$ and $\hat p$ by those new operators transforms the Hamiltonian into

We call $\hat n = \hat a^\dagger \hat a$ the number operator and its value labels the quantum state of the oscillator. It turns out that is can only take on integer values and hence each state has an energy that is an integer multiple of the oscillator frequency $\omega$.

There is plenty of information on the quantum harmonic oscillator; feel free to look around if you need more background.

Lagrangian of the LC circuit

On our journey to derive the origin of a SCQ we need to understand how to quantize electromechanical circuit elements. We start this by exploring a simple linear system -- the LC-circuit, an element consisting of a capacitor and an inductor.

To derive the equation governing the system (the equations of motion) we will need Kirchhoff voltage law which states that the sum over all voltages in a closed circuit is zero

The voltages dropping in the simple LC-circuit are

• the capacitor voltage $V_C\,=\,\frac{Q}{C}$ where $Q$ is the total charge and $C$ is the capacitance of the capacitor.
• the inductor voltage $V_L\,=\, L \frac{dI}{dt}$ with $I\,=\,\frac{dQ}{dt}$ being the current and $L$ the inductance

Putting those together and using Kirchhoff's law we find

While we could solve the LC-circuit at this point ^{fn-LC-solution} it is instrumental for us to identify the canonically conjugate variable of this problem. To do this we re-express Kirchhoff's law using the energies connected with capacitor and inductor:

• the capacitor energy is given by $E_C\,=\,Q^2/2C$.
• the inductor energy is $E_L\,=\,LI^2/2\,=\,\Phi^2/2L$, where we introduced the magnetic flux $\Phi = LI$.

At this point we need to perform some magic for people that are not familiar with Lagrangian mechanics or the Euler-Lagrange formalism. It turns out that when studying the energy balance of the LC circuit we can interpret $E_L$ as kinetic energy $T$ and $E_C$ as potential energy $U$. With this we can define the Lagrangian of the LC-circuit as

where we did the replacement $I\,=\,\frac{dQ}{dt}$. Using the Euler-Lagrange equation we find

which is exactly the law we found using Kirchhoff's voltage rule above! The Euler-Lagrange formalism also allows us to compute the conjugate variable to the charge $Q$ as

Now, we are very close: Using the canonically conjugate pair $Q$ and $\Phi$ we can write the Hamiltonian^fn-legendre as

Promoting those variables to quantum operators and employing canonical quantization we find the dynamics of the system described by the Hamiltonian

with commutation relations

Comparing this to the results of the previous section we see that this system mimics a harmonic oscillator with frequency $\omega=\sqrt{1/LC}$ and quantum mechanical variables given by charge $\hat Q$ and magnetic flux $\hat\Phi$. The implications are similarly profound as for the standard position-momentum oscillator describing a particle in a square well. We can either know the charge precisely but not know the flux at all and vice versa. However, this system is describing a macroscopic electromagnetic circuit element rather than a single atomic particle!

The Josephson Junction

With the procedure to quantize an electromagnetic circuit element we are now equipped to tackle the derivation of a SCQ. First we need to introduce the concept of a quantized magnetic flux (we now also explain what that actually is) and need to understand some basic properties of a Josephson junction.

Let's get started!

Flux Quantization

When describing the energy of an inductor we introduced the quantity

which we called magnetic flux. Using Faraday's law this quantity can also be written as

where $B$ is a constant magnetic field permeating a closed area $A$, e.g. a magnetic field permeating a wire loop. This law was one of the foundational observations in the postulation of Maxwell's equations.

This is where some magical macroscopic quantum physics happens^{fn-flux-quantization}. Consider now a superconductor whose quantum state is described by a macroscopic wave-function

Here $\rho$ describes the charge density of the conductor and $\theta$ is the phase of the wave-function. If the superconductor is forms a solid ring then the requirement of a single-valued wave-function leads to the relation that

i.e. we can fix the value of the phase function only up to a factor of $2\pi n$, $n\in \mathbb{Z}$ due to the periodicity of the complex exponential. Now it turns out that the phase of the quantum mechanical wave-function is proportional to the current flowing in the superconductor ^fn-wf-phase. This is the origin of the resistance-free perma-current in superconductors. However, as we saw the current also determines the magnetic flux in a system and putting all these things together we arrive at a surprising observations of a quantized magnetic flux

The basic flux quantum is defined as

where $e$ is the electron charge. This is remarkable if you think about it. A macroscopic object, i.e. a superconducting ring, exposes quantized behavior in the magnetic field that can penetrate the ring. All non-integer components of the flux cannot enter the area enclosed by the conductor!

Josephson Effect

Now on to part II of the quantum magic. Let us now investigate what happens if we couple two superconductors A, B via an insulating junction C. This setup is called a Josephson Junction and is depicted in the picture. We can describe this system using coupled Schrödinger-equations as

where $K$ is a small constant describing the coupling of the states $\psi_A$ and $\psi_B$ through the junction. The energy $qV$ is proportional to the voltage that is dropping over the junction if the superconductors are terminated by a battery. Making the ansatz for a macroscopic wave-function $\psi_i\,=\,\sqrt{\rho_i}\textrm{e}^{i\theta_i}$ it follows that we can describe the current through the junction by

where $\phi = \theta_A -\theta_B$ is the relative phase difference and $I_0$ is a constant depending on the junction details. This equation implies that no matter what the charge imbalance between the different superconductors, for certain phase differences there can no tunneling current happen!

The above ansatz lets us also connect the phase difference to the voltage

which will become important when deriving the SCQ in the next section.

The Current-Biased Josephson Junction

To gain insight into the emergence of a superconducting qubit we limit ourselves to a special type of qubit -- the superconducting phase qubit (SPQ). The electric circuit consists of a capacitor (C) in series with a Josephson Junction (JJ). Those elements are furthermore biased with an additional current $I_b$ (see figure).

We will in the following derive the semi-classical Lagrangian for the system by using the macroscopic wave function of the JJ and afterwards use canonical quantization to arrive at the Hamiltonian of the current-biased Josephson Junction that gives rise to the washboard potential and superconducting phase qubits.

A more detailed review article also goes over other circuits and connects those to the computing aspect.

Lagrangian and the Washboard potential

To derive the Hamiltonian governing the SCQ we use the same approach when quantizing the LC circuit. The relevant currents are

• at the capacitor: $I_C\,= C \frac{d V_C}{dt}\,=\,\frac{\hbar}{q}\ddot \phi$ where we used the relation with the phase difference of the superconductors
• at the junction: $I_J\,=\, I_0 \text{sin}(\phi)$
• the circuit bias current $I_b$

Using Kirchhoff's current conservation law we find

With this we can use the definition of the Euler-Lagrange equations and identify the different energy contributions as kinetic energy

with the Cooper pair energy $E_C = (2e)^2/2C$ and potential energy

with the Josephson energy $E_J = \hbar I_0 / 2e$. It is instructive to construct the conjugate variable to the phase $\phi$. Using the Lagrange formalism from before we compute

where we introduced the number of charge pairs $n=Q/2e$. As in the case of the LC-circuit we find that the total charge (measured in units of $2e$) is canonically conjugate to the phase difference and hence we cannot measure both phase difference and charge in the circuit with arbitrary precision.

Applying the rules of canonical quantization we can define the conjugate operator of the phase as $\hat p_\phi\,=\,-i\hbar\frac{\partial}{\partial \phi}$ which then leads to the pair number operator

resulting in the commutation relations

and the Hamiltonian of the current-biased JJ (omitting a constant)

It is worthwhile to take a look at this for a bit: The pair number term takes the role of the kinetic energy, which moves in a potential generated by the phase operator. Due to its form the potential is also called the washboard potential (see image below).

The ratio $r_I = \frac{I_b}{I_0}$ of bias to junction current controls the depth of the potential. For a ratio larger than $\vert r_I\vert>1$ all minima vanish which will have implications for the qubit we discuss below. Moreover, for a finite bias-current the potential is slanted and due to the standard tunneling effect particles can still escape the local minima of the potential.

The SPQ as Bound States of the Washboard

The origin of the superconducting phase qubit lies in the bound states of the washboard potential. Like every potential with minima it exposes long-lived states that in the case at hand lock the phase and pair number together ^{fn-tunneling-influence}. To get a feeling of how the bound-states look like and what their energy scales are we expand the washboard potential around one of the local minima $\phi_0$ to quadratic order and obtain:

Looking hard at this equation we recognize the standard quantum harmonic oscillator from before. To solve the system we perform a change of variables $x = (E_J \text{cos}(\phi_0)/2E_C)^{1/4}(\hat \phi-\phi_0)$ and replace $\hat n = -i\frac{\partial}{\partial x}$. Introducing the annihilation and creation operators $\hat a = (x +\frac{\partial}{\partial x})/\sqrt{2}$ and $\hat a^\dagger = (x -\frac{\partial}{\partial x})/\sqrt{2}$ we can rewrite the Hamiltonian as

with the energy of the superconducting phase qubit (SPQ) is given by the plasma frequency

Remember that $r=I_b/I_0$ and hence we can control the frequency and thus energy spacing of the state with the bias current.

Referring back to the introduction we can now choose two states of our liking to define the logical qubit states and be done -- almost. In practice an ideal harmonic oscillator would not be of great use. The energy spacing between the different states of the oscillator $\langle \hat a^\dagger \hat a\rangle$ have the same energy and it would be hard to resolve and address just two individual states that we need to create a qubit. Luckily the washboard potential is not harmonic and the second-order expansion is not very good. We can get slightly better using a higher expansion and calculate the states numerically. The influence of the higher-order contributions result in non-equally spaced energies of the states and we can use this fact to resolve two states of interest for us to use as qubit. For a sixth-order expansion the expansion and lowest eigenstates look like

With this we arrived at the final step to understand what SPQ actually is: It consists of two states of locked phase and charge in a biased Josephson-Junction. This allows us to use standard electromechanical tools to read and manipulate the qubits and potentially build a quantum computer out of those qubits.

As always here is a link to the notebook I used to create some of the blog's graphs.