Experimental Qubit Construction Margot Young March 2021 1 Introduction Quantum bits, also called qubits, are the quantum counterparts of classical bits. Classical bits are pieces of information that can take one of two states, either 0 or 1, and form the basis of classical computing. Like classical bits, qubits also have two states, j0i and j1i, which are orthonormal and are referred to as computational basis states. The choice of states j0i and j1i as opposed to another choice such as j+i and |−i is convention for quantum computing. Unlike classical bits, qubits can be in a superposition of computational basis states j i = a j0i + b j1i with complex coefficients a and b where jaj2 + jbj2 = 1. The modulus squared of these complex coefficients gives us the probability of measuring the qubit in either of the computational basis states, hence our requirement that the sum of all the probabilities must be equal to one. When we move past single-qubit systems into multiple-qubit systems, we acquire more computational basis states. For example, a two-qubit system has computational basis states j00i, j01i, j10i, and j11i. The state of the system can again be a superposition of the computational basis states with complex P 2 coefficients ai such that i jaij = 1. In general, a system with N qubits has 2N computational basis states. Entangled states such as the Bell states have measurement outcomes that are correlated between the individual qubits. This correlation is far stronger than measurement correlation in classical computing and allows for phenomena such as quantum teleportation and quantum error- correction. Figure 1: Circuit and matrix representation of a controlled NOT (CNOT) gate.[1] Quantum computation entails the implementation and manipulation of qubits. 1 These manipulations are performed using quantum logic gates, which can alter given qubits to produce desired outcomes. These quantum gates can also be represented as matrices acting upon qubits represented as vectors. For exam- ple, the controlled NOT (CNOT) gate shown in Figure 1 uses one input qubit as a control and flips the second input qubit if the first qubit is in the state j i = j1i. This gate has eigenvalues ±1. Quantum logic gates can be used to form quantum circuits and perform computations. A simple circuit to swap two qubits using CNOT gates is shown in Figure 2. We can mathematically formu- late quantum circuits of increasing complexity and number of qubits to perform increasingly complex computations. This is the essential goal of quantum com- putation. Figure 2: Circuit representation of an exchange gate constructed from CNOT gates.[2] So far, our discussion of qubits and quantum computing has been entirely theoretical. In the study of theoretical quantum physics, qubits are largely treated as mathematical objects, and quantum circuits are logical tools used to mathematically manipulate qubits. While the theory of quantum computing is vitally important to understand, it is of equal importance to understand how these theories can be realized experimentally. In the remainder of this paper we will explore the general challenges in moving from theoretical to experimental quantum circuitry as well as two of the most advanced experimental forms of qubits currently in use: neutral atom qubits and superconducting qubits. 2 Requirements for Realizing Physical Qubits We will begin with the characteristics we require or hope to obtain in physical qubits. We first require that our physical qubits have distinctly leveled sub- systems, which form our basis states, and that the system is scalable. We also require that our qubits can maintain their given states and quantum properties, such as superposition, and are able to evolve through the operation of gates. The length of time that a qubit can maintain its given state and quantum prop- erties is called the coherence time of the qubit, which is a major limiting factor in quantum computing. The length of time it takes to perform operations on 2 qubits compared with the coherence time of the qubits imposes an upper limit on the possible length and complexity of a functional quantum circuit. Deco- herence is typically a result of quantum noise, or undesired interactions in a system, which is present in all quantum computational systems as achieving an entirely closed quantum system is impossible. Noise can take many forms, and it encompasses all processes that cause a loss of information in quantum systems.[3] Qubits that cannot be altered as desired by quantum gates have little use for quantum computation. In an environment with no external influence, quantum systems will evolve according to their Hamiltonians. In order to perform opera- tions on our qubits, we need a system with a Hamiltonian containing adjustable parameters such that we can enact unitary transformations as quantum gate operations. We also require that qubits can be prepared to be in certain known states, such as the computational basis states, and that we can measure the final output after qubits pass through a circuit. This collection of require- ments is commonly known as the DiVincenzo criteria for the implementation of quantum computing[4] and is summarized below. However, it can be difficult to achieve all these ideals at once in practice, and often compromises must be made. 2.1 DiVincenzo Criteria 1. Well-defined qubit with the ability to be scaled to the many-qubit case. 2. Ability to initialize the qubits in certain known states. 3. Long coherence time. 4. Universal set of quantum gates. 5. Ability to measure the output state of the qubit. 3 Qubit System - Quantum Harmonic Oscillator To illustrate how these conditions work in physical systems, we will explore the familiar example of the quantum harmonic oscillator as a potential physical qubit. 3.1 System Preparation The quantum harmonic oscillator is known to have discrete energy eigenstates generally described by the quantum number n and the Hamiltonian p2 1 H = + (m!2x2) (1) 2m 2 3 for the one-dimensional case. This system evolves in time according to the Schrodinger equation @ i j (t)i = H j (t)i (2) ~@t which gives the evolution of states as −iHt j (t)i = e ~ j (0)i y (3) = e−ia a(~!−1=2)t=~ j (0)i for the usual creation and annihilation operators ay and a. This system can be physically realized as, for example, a superconducting LC oscillator, which consists of an inductor and a capacitor.[5] We can use the energy eigenstates of the harmonic oscillator, which we will denote here with a subscript N, to form the computational basis states, which we will denote with a subscript CB. For example, we could encode a two qubit system using the states: j00iCB = j0iN j01iCB = j2iN 1 j10i = p (j4i + j1i ) CB 2 N N 1 j11i = p (j4i − j1i ) CB 2 N N at t = 0. Using this form of encoding, we can represent an n-qubit system with 2n energy eigenstates. In principle, as the quantum harmonic oscillator has an infinite number of energy eigenstates, a many-qubit system could be represented by a single harmonic oscillator.[3] 3.2 Implementing Gates With the system outlined above, we can perform the CNOT operation by al- lowing the system to evolve to a time t = π=!~ so that each eigenstate jniN evolves as y jn(t)i = e−iπa a jn(0)i N N (4) n(0) = (−1) jn(0)iN Under this transformation, the eigenstates j0iN , j2iN , and j4iN are invariant, but j1iN goes to − j1iN , so j10iCB goes to j11iCB and j11iCB goes to j10iCB as desired.[3] 4 3.3 Alignment with DiVincenzo Criteria Now let us examine how this system performs with regard to the DiVincenzo criteria. While our system does have discrete substates and is scalable to the many-qubit case, meeting the first criterion, its states are evenly spaced as 1 the eigenvalues are given by ~(n + 2 ). This means that, as the name harmonic oscillator would suggest, this system is not anharmonic, so unwanted excitations to other eigenstates are not suppressed and the coherence time of the qubits would be low, which does not meet the third criterion. Though we were able to easily encode this system for the CNOT gate, our encoding was dependent on the fact that the CNOT operator has eigenvalues ±1. For the case of a general unitary operator, we will not always know its eigenvalues, in which case we will not be able to encode our system to transform as desired under the operation. Therefore, this system does not meet the fourth criterion. Because this system can be modeled with many physical systems, it is likely that one with good preparation and measurement capabilities could be con- structed, so we will not consider the second and fifth criteria here. As a note, the harmonic oscillator must be confined to the one-dimensional case. In higher dimensions, a degeneracy is present and accurate readout becomes impossible. In practice, the harmonicity of the quantum harmonic oscillator makes it a poor physical implementation of a qubit. Thus, while it is useful for understand- ing the transition between theoretical and experimental quantum computing, we must move on to other systems to examine the current state of experimental qubit construction. 4 Qubit System - Neutral Atoms 4.1 System Preparation Neutral atom qubit systems are generally comprised of an array of trapped neutral atoms. There are several methods for trapping single neutral atoms, but one of the most promising methods is a light trap, which includes optical tweezers and optical lattices as its most common implementations. To optically trap atoms, they must be cooled to temperatures on the order of microkelvin using lasers before being loaded into the trap sites.