CMSC 33001: Novel Computing Architectures and Technologies

Lecturer: Pranav Gokhale Scribe: Pranav Gokhale

Lecture 05: Superconducting Qubits

October 16, 2018

1 Outlook and High Level Overview

Superconducting technology is one of the frontrunners for scalable quantum computing.There are several different qubit designs (and more still to be invented!) with differenttradeoffs and properties. First, some of the key features of superconducting circuits:

• Leverage existing semiconductor technology

• Quantum states are macroscopic

• Nanosecond single qubit gates and per gate errors in 0.1% range

The two simplest superconducting qubits are are flux qubits and Cooper Pair Boxes.In flux qubits (RF-SQUID), the quantum state is defined by the direction of current flow:clockwise or counterclockwise (or a superposition). For Cooper Pair Boxes, the quantumstate is given by the number of charges that have tunneled across a barrier.

Progress in superconducting technologies has been rapid. “Schoelkopf’s Law” has beencoined to capture the ˜doubling in qubit lifetimes every year, shown in Figure 1.

1.1 DiVincenzo Criteria

As a short digression, we consider the DiVinenzo criteria [DI00] that are needed for a quan-tum computer.

1. Scalable system with well characterized qubits

2. Ability to initialize qubits (e.g. |0〉)

3. Long decoherence times

4. Universal gate set (e.g. single qubit gates + CNOT)

5. Ability to measure

Note that these goals are in tension with each other. In particular, being able to initialize,perform gates, and measure requires interactions between the system and environment, butlong decoherence times require isolating the system from the environment. This is thefundamentally difficult part about building a quantum computer.

CMSC 33001 (Autumn 2018) Lecture 05

Figure 1: Superconducting qubit lifetimes have doubled about every year [Joh18].

2 Box on Spring Review

We begin by considering the box-on-spring problem of classical mechanics: a box of mass mattached to a spring of spring constant k. As we’ll see, this problem is equivalent to the LCcircuit in the next section.

2.1 Newton’s Laws Formulation

We can solve the box-on-spring problem by observing that

F = −kx = ma = mx.

A solution to this differential equation is

x(t) = sin

(√k

mt

).

2.2 Hamiltonian Formulation

Another formulation for this is the Hamiltonian formulation1. We express the Hamiltonian(total energy) in terms of the position x and momentum p = mv = mx. Since, kinetic energy

is mv2

2= p2

2mand potential energy is kx2

2, we have:

1Note that I am slightly abusing the Hamiltonian formulation in these notes for simplicity, in particularthe choice of coordinates x and p

2

CMSC 33001 (Autumn 2018) Lecture 05

H(x, p) = KE + PE =p2

2m+kx2

2.

The Hamiltonian equations of motion are:

∂H

∂p=dx

dt

∂H

∂x= −dp

dt

The first equation gives us pm

= x = v, which is just the definition of momentum. Thesecond equation gives us kx = −p = −mv = −mx. This is the same as the dynamicsfrom the Newton’s Laws Formulation above. So we again get the sinusoidal solution withfrequency

√k/m.

3 LC Circuit

Let’s now consider an LC circuit. The energy of a capacitor is

EC =1

2CV 2.

The energy of an inductor is

EL =1

2LI2.

So the Hamiltonian is

H =1

2CV 2 +

1

2LI2.

Before proceeding, we must re-write the Hamiltonian in terms of “conjugate coordinates”that mimic the momentum-position relationship from the box-on-spring problem. The mathbehind the choice of conjugate coordinates is beyond the scope of this lecture, but instead,I’ll just tell you that we need to express the Hamiltonian in terms of charge Q and flux Φ.For capacitors we have Q = CV and for inductors, we have Φ = LI. Plugging this in givesus:

H =Q2

2C+

Φ2

2L

which correspond to kinetic and potential energy respectively.Instead of directly applying the Hamiltonian equations, we can just compare to the box-

on-spring problem, with p corresponding to Q and x corresponding to Φ. This gives us thesolution,

Q(t) = sin(t/√LC).

This is the same as the solution to the box-on-spring with 1/√LC in place of

√k/m.

3

CMSC 33001 (Autumn 2018) Lecture 05

4 Quantum LC Circuit

To consider the quantum version of the LC circuit, i.e. charges are quantized discrete quan-tities instead of continuous quantities, we just add “hat” symbols to the quantized quantitiesin the Hamiltonian:

H =Q2

2C+

Φ2

2L.

The hat symbols mean that Q and Φ are no longer just numbers, but rather operators, whichcorrespond to matrices. The eigenvalues of these matrices are the spectrum of values thatthese operators can physically take.

If we zoom in on the potential energy term, we have a quadratic potential (also knownas a harmonic oscillator), Φ2/(2L). The quantum quadratic potential is well studied andhas infinite discrete solutions of energies:

En = (n+1

2)~ω

Figure 2: Energy levels of the quantum harmonic oscillator. Figure from HyperPhysics.

For example, the ground energy is E0 = 12~ω, the next lowest energy is E1 = 3

2~ω, the

next lowest energy is E2 = 52~ω, and so forth.

But the equal spacing between energy levels makes the quantum LC circuit a bad qubit!If we want to take a qubit from the ground state to the next lowest energy, we would shinephotons carrying ~ω of energy. But once the qubit is in the next bigher energy, if it absorbsanother photon, it will rise another energy level and so forth. We have no way of selectivelyisolating specific transitions.

This property of equal energy spacings, called harmonicity, is bad for qubits. We seekanharmonicity, or unequal energy spacings. This is where we turn to the properties ofsuperconducting circuits.

4

CMSC 33001 (Autumn 2018) Lecture 05

5 Superconducting

Here is a quick summary of the key properties of superconducting physics:

• Below a critical temperature, resistance drops to 0. For quantum applications, we needzero resistance, because otherwise, we would lose energy to the environment and oursystem would decohere.

• Below the critical temperature, electrons in superconducting metals form bonds knownas Cooper pairs. These Cooper pairs have a total spin of 0, making them bosons.Ordinarily, electrons have ±1

2spin, making them fermions which are described by

energy levels, spins, momenta, etc. But the Cooper pair bosons are indistinguishableand each share the same state. The only degree of freedom in the state space is thenumber of Cooper pairs. So in summary, in the superconducting realm, we reducean otherwise vast quantum state to a single number. This is what is meant by amacroscopic quantum state.

• Fun fact: the same phenomenon (phonon vibrations) that causes Cooper Pair formationalso obstructs conduction at normal temperatures. So superconducting materials arebad conductors are normal temperatures.

6 Josephson Junction

The solution to our harmonicity conundrum from Section 4 is to introduce a potential termthat is non-quadratic. We use the Josephson Junction, which is an insulator sandwichedbetween two superconductors. As explained above, the quantum state of the system is asingle macroscopic property: the number of Cooper pairs on each of the superconductors.

A Cooper pair can tunnel across the insulator from one superconductor to the other, andthe frequency of these tunneling events has energy EJ . Without getting into the microscopicphenomonology, we simply state that the current passing through the Josephson Junctionis:

I = I0 sin(2πΦ/Φ0)

which corresponds to a potential energy of

EJ cos

(2πΦ

Φ0

).

Unlike the quadratic potential energy of the LC circuit, the potential is now sinusoidal, whichgives rise to anharmonicity–working out the math would show that the energy gap betweenthe lowest and 2nd lowest energy level is different from the the energy gap between all otherpairs of energy levels. This is exactly what we want.

Let’s now revisit the Hamiltonian, substituting in our new potential energy term:

5

CMSC 33001 (Autumn 2018) Lecture 05

H =Q2

2C− EJ cos

(2πΦ

Φ0

)Before proceeding, let’s also replace Q = 2eN , to explicitly point out that the charge is

quantized in 2-electron (Cooper pair) units:

H =2e2

CN2 − EJ cos

(2πΦ

Φ0

)= ECN

2 − EJ cos

(2πΦ

Φ0

)

7 Cooper Pair Box

The Hamiltonian above is a good starting point, but it doesn’t have any mechanism forcontrol. To fix that, let’s add a gate voltage bias to our circuit. The effect of this is to createan “offset” to the baseline number of Cooper pairs on the capacitor. If Ng is this baseline(which we the experimentalist can control), we have a Hamiltonian of:

EC(N −Ng)2 − EJ cos

(2πΦ

Φ0

)

To analyze this Hamiltonian, let’s first start with EJ = 0 for simplicity, so that we have

H = EC(N −Ng)2

Now, if we pick a value of N , we are picking a particular energy level. And rememberthat we the experimentalist can control the value of Ng. Observe that after picking a value

for N and sweeping Ng, we obtain a set of parabolas. For example:

• N = 0, the ground state, has parabola ECN2g

• N = 1 has parabola EC(Ng − 1)2

• N = −1 has parabola EC(Ng + 1)2

At Ng = 0, these three parabolas have energies of 0, EC , and EC respectively. There is alarge gap between the lowest energy and the next two lowest energies, which means we nowhave a good procedure for initialization: pick Ng = 0 and let the system spontaneouslycool. Since the energy gap between the ground state and next excited state is large, thesystem will settle into |0〉 with high probability.

At Ng = 1/2, our three energies are EC

4, EC

4, and 9EC

4. So the gap between the two lowest

energies and the next highest energy is 9EC/4 − EC/4 = 2EC , which is quite large. Thisgives us a good qubit: once we have the system in the lowest or second lowest state and weset Ng = 1/2, it is unlikely to leak into the third lowest state. You may observe though,that the energies of the lowest and second lowest states are equal, which would indeed be

6

CMSC 33001 (Autumn 2018) Lecture 05

problematic. Fortunately, if we were to re-introduce EJ , we’d find that the two energiesbecome slightly unequal. In physics parlance, we say that EJ , the tunnel coupling, lifts thedegeneracy between the two lowest energies.

This analysis demonstrates that the Cooper Pair Box is a viable qubit. We know how toinitialize it to |0〉 and then re-adjust the gate voltage to stop leakage to states higher than|1〉.

The gate voltage Ng also gives us a way to execute gates. The details are beyond thislecture, but the TLDR is that after rapidly switching from Ng = 0 to Ng = 1/2 with a |0〉initial state, the qubit will oscillate between |0〉 and |1〉 sinusoidally. This is an Rx rotationon the Bloch sphere.

8 Transmon

Traditionally, qubits were operated in the regime of small EJ/EC ratios, but look at whathappens when we increase the ratio:

Figure 3: Energy levels plotted for different EJ/EC ’s. From [KYG+07].

On the downside, you can see that as we increase EJ/EC , the anharmonicity decreases.In the final graph, the gaps E1 − E0 and E2 − E1 is almost 0. However, there is also ahuge upside: as EJ/EC increases, the plot of each energy level flattens, which increases thequbit’s resilience to charge sensitivity. For example, suppose we wiggle the gate voltage ng

(which corresponds to charge noise). In plot (a), where the sine curves are steep, small x-axischanges could switch our qubit state. By contrast, in plot (d), the sine curves are almostflat, meaning that wiggling in the x-axis is unlikely to affect our qubit state.

7

CMSC 33001 (Autumn 2018) Lecture 05

It turns out that the decrease in anharmonicity happens at a much slower rate than theincrease in charge noise resilience. So increasing EJ/EC is worthwhile, and qubits with ahigh EJ/EC ratio, or equivalently a large capacitor, are called transmon qubits.

References

[DI00] David P. DiVincenzo and IBM. The physical implementation of quantum com-putation. 2000.

[Joh18] Blake Johnson. Rigetti qc overview. https://hpcuserforum.com/

presentations/tuscon2018/QCOverview_Rigetti_UFTucson2018.pdf, 2018.

[KYG+07] Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer,Alexandre Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. Chargeinsensitive qubit design derived from the cooper pair box. 2007.

8