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1 Outlook and High Level Overview 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 different tradeoffs 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 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. j0i) 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 mechanics: 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 energy) in terms of the position x and momentum p = mv = mx_. Since, kinetic energy mv2 p2 kx2 is 2 = 2m and potential energy 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 2 CMSC 33001 (Autumn 2018) Lecture 05 p2 kx2 H(x; p) = KE + PE = + : 2m 2 The Hamiltonian equations of motion 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 frequency pk=m. 3 LC Circuit Let's now consider an LC circuit. The energy of a capacitor is 1 E = CV 2: C 2 The energy of an inductor 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 capacitors we have Q = CV and for inductors, 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, p Q(t) = sin(t= LC): p This is the same as the solution to the box-on-spring with 1= LC in place of pk=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 quantities in the Hamiltonian: Q^2 Φ^ 2 H^ = + : 2C 2L The hat symbols mean that Q^ and Φ^ are no longer just numbers, but rather operators, which correspond to matrices. The eigenvalues of these matrices are the spectrum 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 harmonic oscillator), Φ^ 2=(2L). The quantum quadratic potential is well studied and has infinite discrete solutions of energies: 1 E = (n + ) ! n 2 ~ Figure 2: Energy levels of the quantum harmonic oscillator. Figure from HyperPhysics. 1 3 For example, the ground energy is E0 = 2 ~!, the next lowest energy is E1 = 2 ~!, the 5 next lowest energy is E2 = 2 ~!, 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 shine photons carrying ~! of energy. But once the qubit is in the next bigher energy, if it absorbs another photon, 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. 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 need zero resistance, because otherwise, we would lose energy to the environment and our system would decohere. • Below the critical temperature, electrons in superconducting metals form bonds known as Cooper pairs. These Cooper pairs have a total spin 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 temperatures. 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: ^ ^ I = I0 sin(2πΦ=Φ0) which corresponds to a potential energy of ! 2πΦ^ 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: 5 CMSC 33001 (Autumn 2018) Lecture 05 ^2 ^ ! ^ Q 2πΦ H = − EJ cos 2C Φ0 Before proceeding, let's also replace Q^ = 2eN^, to explicitly point out that the charge is quantized in 2-electron (Cooper pair) units: 2 ^ ! ^ ! ^ 2e ^ 2 2πΦ ^ 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: ^ ! ^ 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 ^ ^ 2 H = EC (N − Ng) Now, if we pick a value of N^, 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 ^ for N and sweeping Ng, we obtain a set of parabolas. For example: ^ 2 • N = 0, the ground state, has parabola EC Ng ^ 2 • N = 1 has parabola EC (Ng − 1) ^ 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.
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