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CMSC 33001: Novel Computing Architectures and Technologies Lecturer: Pranav Gokhale Scribe: Pranav Gokhale

Lecture 05: Superconducting October 16, 2018

1 Outlook and High Level Overview

Superconducting technology is one of the frontrunners for scalable computing. There are several different designs (and more still to be invented!) with different tradeoffs and properties. First, some of the key features of superconducting circuits: • Leverage existing 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 is defined by the direction of current flow: clockwise or counterclockwise (or a superposition). For Cooper Pair Boxes, the quantum state is given by the number of charges that have tunneled across a barrier. Progress in superconducting technologies has been rapid. “Schoelkopf’s Law” has been coined 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. |0i) 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, but long decoherence times require isolating the system from the environment. This is the fundamentally 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 : a box of mass m attached to a spring of spring constant k. As we’ll see, this problem is equivalent to the LC circuit 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 r ! k x(t) = sin t . m

2.2 Hamiltonian Formulation Another formulation for this is the Hamiltonian formulation1. We express the Hamiltonian (total ) in terms of the position x and momentum p = mv = mx˙. Since, mv2 p2 kx2 is 2 = 2m and is 2 , we have: 1Note that I am slightly abusing the Hamiltonian formulation in these notes for simplicity, in particular the choice of coordinates x and p

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p2 kx2 H(x, p) = KE + PE = + . 2m 2 The Hamiltonian equations of are: ∂H dx = ∂p dt ∂H dp = − ∂x dt p The first equation gives us m =x ˙ = v, which is just the definition of momentum. The second equation gives us kx = −p˙ = −mv˙ = −mx¨. This is the same as the dynamics from the Newton’s Laws Formulation above. So we again get the sinusoidal solution with pk/m.

3 LC Circuit

Let’s now consider an LC circuit. The energy of a is 1 E = CV 2. C 2 The energy of an is 1 E = LI2. L 2 So the Hamiltonian is 1 1 H = CV 2 + LI2. 2 2 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 math behind 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 we have Q = CV and for , we have Φ = LI. Plugging this in gives us:

Q2 Φ2 H = + 2C 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 the solution, √ Q(t) = sin(t/ LC). √ This is the same as the solution to the box-on-spring with 1/ LC in place of pk/m.

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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 quantities in the Hamiltonian: Qˆ2 Φˆ 2 Hˆ = + . 2C 2L The hat symbols mean that Qˆ and Φˆ are no longer just numbers, but rather operators, which correspond to matrices. The eigenvalues of these matrices are the of values that these operators can physically take. If we zoom in on the potential energy term, we have a quadratic potential (also known as a ), Φˆ 2/(2L). The quantum quadratic potential is well studied and has infinite discrete solutions of : 1 E = (n + ) ω n 2 ~

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

1 3 For example, the ground energy is E0 = 2 ~ω, the next lowest energy is E1 = 2 ~ω, the 5 next lowest energy is E2 = 2 ~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 to the next lowest energy, we would shine carrying ~ω of energy. But once the qubit is in the next bigher energy, if it absorbs another , it will rise another energy level and so forth. We have no way of selectively isolating specific transitions. This property of equal energy spacings, called harmonicity, is bad for qubits. We seek anharmonicity, or unequal energy spacings. This is where we turn to the properties of superconducting circuits.

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5 Superconducting

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

• Below a critical , resistance drops to 0. For quantum applications, we need zero resistance, because otherwise, we would lose energy to the environment and our system would decohere.

• Below the critical temperature, in superconducting metals form bonds known as Cooper pairs. These Cooper pairs have a total of 0, making them bosons. 1 Ordinarily, electrons have ± 2 spin, making them fermions which are described by energy levels, spins, momenta, etc. But the Cooper pair bosons are indistinguishable and each share the same state. The only degree of freedom in the state space is the number of Cooper pairs. So in summary, in the superconducting realm, we reduce an otherwise vast quantum state to a single number. This is what is meant by a macroscopic quantum state.

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

6 Josephson Junction

The solution to our harmonicity conundrum from Section 4 is to introduce a potential term that is non-quadratic. We use the Josephson Junction, which is an insulator sandwiched between two superconductors. As explained above, the quantum state of the system is a single 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, and the frequency of these tunneling events has energy EJ . Without getting into the microscopic phenomonology, we simply state that the current passing through the Josephson Junction is:

ˆ ˆ I = I0 sin(2πΦ/Φ0) which corresponds to a potential energy of ! 2πΦˆ EJ cos . Φ0

Unlike the quadratic potential energy of the LC circuit, the potential is now sinusoidal, which gives rise to anharmonicity–working out the math would show that the energy gap between the lowest and 2nd lowest energy level is different from the the energy gap between all other pairs of energy levels. This is exactly what we want. Let’s now revisit the Hamiltonian, substituting in our new potential energy term:

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ˆ2 ˆ ! ˆ Q 2πΦ H = − EJ cos 2C Φ0

Before proceeding, let’s also replace Qˆ = 2eNˆ, to explicitly point out that the charge is quantized in 2- (Cooper pair) units:

2 ˆ ! ˆ ! ˆ 2e ˆ 2 2πΦ ˆ 2 2πΦ H = N − EJ cos = EC N − EJ cos C Φ0 Φ0

7 Cooper Pair Box

The Hamiltonian above is a good starting point, but it doesn’t have any mechanism for control. To fix that, let’s add a gate voltage bias to our circuit. The effect of this is to create an “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:

ˆ ! ˆ 2 2πΦ EC (N − Ng) − EJ cos Φ0

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

ˆ ˆ 2 H = EC (N − Ng) Now, if we pick a value of Nˆ, we are picking a particular energy level. And remember that we the experimentalist can control the value of Ng. Observe that after picking a value ˆ for N and sweeping Ng, we obtain a set of parabolas. For example:

ˆ 2 • N = 0, the ground state, has parabola EC Ng

ˆ 2 • N = 1 has parabola EC (Ng − 1)

ˆ 2 • N = −1 has parabola EC (Ng + 1)

At Ng = 0, these three parabolas have energies of 0, EC , and EC respectively. There is a large gap between the lowest energy and the next two lowest energies, which means we now have a good procedure for initialization: pick Ng = 0 and let the system spontaneously cool. Since the energy gap between the ground state and next is large, the system will settle into |0i with high probability. EC EC 9EC At Ng = 1/2, our three energies are 4 , 4 , and 4 . So the gap between the two lowest energies and the next highest energy is 9EC /4 − EC /4 = 2EC , which is quite large. This gives us a good qubit: once we have the system in the lowest or second lowest state and we set 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

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problematic. Fortunately, if we were to re-introduce EJ , we’d find that the two energies become slightly unequal. In physics parlance, we say that EJ , the tunnel coupling, lifts the degeneracy between the two lowest energies. This analysis demonstrates that the Cooper Pair Box is a viable qubit. We know how to initialize it to |0i and then re-adjust the gate voltage to stop leakage to states higher than |1i. The gate voltage Ng also gives us a way to execute gates. The details are beyond this lecture, but the TLDR is that after rapidly switching from Ng = 0 to Ng = 1/2 with a |0i initial state, the qubit will oscillate between |0i and |1i sinusoidally. This is an Rx rotation on the Bloch sphere.

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Traditionally, qubits were operated in the regime of small EJ /EC ratios, but look at what happens 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 a huge upside: as EJ /EC increases, the plot of each energy level flattens, which increases the qubit’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-axis changes could switch our qubit state. By contrast, in plot (d), the sine curves are almost flat, meaning that wiggling in the x-axis is unlikely to affect our qubit state.

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It turns out that the decrease in anharmonicity happens at a much slower rate than the increase in charge noise resilience. So increasing EJ /EC is worthwhile, and qubits with a high 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. Charge insensitive qubit design derived from the cooper pair box. 2007.

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