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, |0i and |1i, which are orthonormal and are referred to as computational basis states. The choice of states |0i and |1i as opposed to another choice such as |+i and |−i is convention for quantum computing. Unlike classical bits, qubits can be in a superposition of computational basis states |ψi = a |0i + b |1i with complex coefficients a and b where |a|2 + |b|2 = 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 |00i, |01i, |10i, and |11i. The state of the system can again be a superposition of the computational basis states with complex P 2 coefficients ai such that i |ai| = 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 example, 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 |ψi = |1i. 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 computation.

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 properties 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 operations 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 requirements 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 = + (mω2x2) (1) 2m 2

3 for the one-dimensional case. This system evolves in time according to the Schrodinger equation ∂ i |ψ(t)i = H |ψ(t)i (2) ~∂t which gives the evolution of states as

−iHt |ψ(t)i = e ~ |ψ(0)i

† (3) = e−ia a(~ω−1/2)t/~ |ψ(0)i for the usual creation and annihilation operators a† 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:

|00iCB = |0iN

|01iCB = |2iN 1 |10i = √ (|4i + |1i ) CB 2 N N 1 |11i = √ (|4i − |1i ) 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/ω~ so that each eigenstate |niN evolves as

† |n(t)i = e−iπa a |n(0)i N N (4) n(0) = (−1) |n(0)iN

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 constructed, 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. However, the laser used for cooling causes any two atoms occupying the same site to collide and be ejected from the array. This means that many of the traps will be left empty, but these empty traps will be in a random configuration, which will introduce issues in the computations. Identifying a method to consistently fill an entire array with atoms is an ongoing problem for this mode of qubit construction. Finally, the atoms must be cooled even further, often to their vibrational ground state, in order to perform computations. Additionally, gas atoms in the system can introduce noise by colliding with the neutral atoms, so operating the system in a vacuum can greatly increase coherence time, even to the order of tens of minutes. The atoms can be initialized to well-defined fiducial states using

5 optical pumping to reach the ground hyperﬁne states of the atoms. Because the atoms in the array are neutral, they only interact weakly with one another, which minimizes unwanted interactions between qubits.[6]

4.2 Implementing Gates Single-qubit gates can be implemented on these arrays of trapped atoms using precisely focused lasers to drive atomic transitions between hyperfine levels for individual atoms. Alternatively, the same atomic transitions can be driven using microwaves that only act upon atoms whose resonant frequencies have been tuned with lasers or magnetic fields. Both methods allow for transformations to be enacted on individual qubits despite the close spacing of the atoms in the array. These operations occur very quickly, with gate times on the order of one to a few hundred microseconds, so a large number of gate operations can be enacted in the coherence time of the qubit. Unlike the case of single-qubit gates, two-qubit gates require the qubits to be strongly interacting. Because our neutral atoms only weakly interact with one another, something about the system must be altered in order to enact two-qubit gates. One proposition is to briefly move two atoms together in a controlled collision, but this process is slow and difficult to do accurately, so the fidelity of the gate often suffers. Another more popular solution is to place the atoms in Rydberg states tem- porarily so that an electron is in an excited state far from the nucleus. In their Rydberg states, atoms have a significant dipole moment and can interact strongly with one another. This strong interaction suppresses the excitation of more than one atom into a Rydberg state at a time in a small volume. By applying a series of pulses to the individual atoms, we can cause a Ry- dberg and a non-Rydberg atom to become entangled and thus obtain a universal quantum computing gate set combining this entanglement process and the single-qubit gates described above. Because the range of interactions for the Rydberg atoms is relatively large, it is possible to entangle qubits that are separated by several sites. Additionally, the suppression of multiple Rydberg excitations at a time means that even in this process of entanglement, the atoms do not exert notable unwanted forces on each other.[6] However, these two-qubit gates have been shown to have fidelity that is significantly below the high-fidelity limit of 0.99,[7] which has been exceeded in other qubit systems such as superconducting qubits. This lower fidelity is likely in part due to the Rydberg atoms’ high sensitivity to background electric fields and finite radiative lifetime, both of which can lead to decoherence during the gate operations.

4.3 Conducting Measurements It has been demonstrated that neutral atom qubits can be nondestructively measured with ﬂuorescence detection. The ﬂuorescence measurement is performed by inducing transitions to excited states with light and measuring the

6 energy of resulting emitted photons, which indicate the state of the qubit. These measurements are lossless and have been shown to be highly accurate.[8]

4.4 Alignment with DiVincenzo Criteria How well do neutral atom qubits perform regarding the DiVincenzo criteria? These qubits have well-defined hyperfine levels that can be used as basis states and are easily scalable due to their minimal interaction with other qubits in the array. The qubits can be easily initialized using optical pumping and have long coherence times. They can also be measured easily using fluorescence. Therefore, neutral atom qubits meet requirements 1-3 and 5 of the DiVincenzo criteria. A universal set of quantum gates can be enacted on a system of neutral atom qubits, but implementations of fast two-qubit gates have yet to reach high fidelities. This means that the fourth DiVincenzo criterion is somewhat weakly met.

4.5 Future Work Neutral atom qubits currently perform quite well by the DiVincenzo criteria. Future problems for the community to tackle include improving gate fidelity for entanglement processes and scaling the system up to a size that could perform computations that are not possible with classical computers. Achieving high entanglement fidelity will improve the fidelity of computations and allow for error correction, which requires very high fidelity, to be introduced into the system. Additionally, more efficient methods of filling every site in the array are likely to be examined.

5 Qubit System - Quantum Integrated Circuits

Quantum integrated circuits, better known as superconducting qubits, utilize the properties of superconductivity. Superconductivity is a phenomenon in which electrical current encounters zero resistance in a material, so it can circu- late without loss of energy. Superconductivity occurs in certain materials, such as aluminum, when they are reduced to a very low temperature.[9]

5.1 System Preparation Superconducting qubits take the form of collective excitations in superconducting circuits made from macroscopic electrical elements. This macroscopic construction allows them to be easily and strongly coupled to each other with simple electrical elements, but it also has issues with decoherence and noise. These superconducting circuits display macroscopic quantum behaviors in opposition to classical circuitry. For example, the charge on a capacitor could be both positive and negative by a superposition of states or charge could be ﬂowing both clockwise and counterclockwise around a loop.[5]

7 There are two types of superconducting qubits that are commonly used today: charge qubits, which are based on electrical charge, and flux qubits, which are based on magnetic flux. Here we will study the transmon qubit, which is a type of charge qubit and is the most widely used qubit for gate-based quantum computation.[10] A transmon qubit consists of a large shunt capacitor and a small superconducting island joined to a larger superconducting reservoir by a Josephson tunnel junction, which behaves as non-linear inductor. Josephson junctions are constructed from two superconducting electrodes separated by a thin insula- tor and admit the coherent tunneling of Cooper pairs. Cooper pairs consist of pairs of electrons that are bound together at low temperatures, such as the temperatures required for superconductivity.[11] The charge offset on the superconducting island is controlled by a capacitively coupled gate voltage, which can be used to adjust the qubit frequency. Due to the non-linear nature of the Josephson junctions, the potential of the circuit is anharmonic, so the ground state and first excited state of the circuit can be used to define a qubit with a reduced probability of unwanted excitations to higher states. The shunt capacitor serves to make the circuit largely charge-insensitive so that environmental charge noise no longer inhibits the coherence time of the qubit. However, the shunt capacitor lowers the circuit’s anharmonicity. The Hamiltonian of the circuit is given by

2 H = 4EC (N − ng) − EJ cos φ (5) where EC is the capacitive charging energy, EJ is the Josephson energy, N is the number of excess Cooper pairs on the island, ng is the charge offset on the island, and φ describes the phase difference across the Josephson junction.[10] Superconductivity ensures that electrical signals can be transmitted without energy loss, which is one requirement for quantum coherence in this system. Transmon qubits have been demonstrated to have coherence times in the range of 50-100 µs.[10] In order to achieve superconductivity, the circuits must be cooled to very low temperatures, in the regime where the thermal fluctuations are much smaller than the energy associated with the transition between the 2 basis states of the superconducting qubit.

5.2 Implementing Gates The predominant technique for enacting single-qubit operations is irradiating the superconducting circuit with microwaves at the qubit transition frequency, which drives oscillations and can be used to implement rotations. While the use of these single-qubit gates has become relatively standard for superconducting qubits, the same consensus has not been reached for two-qubit gates. There are many kinds of two-qubit gates, with techniques including microwave irradiation, application of magnetic ﬂux, and driving coupling elements. Several gates have achieved ﬁdelities of over 0.99.[10]

8 5.3 Conducting Measurements The most common form of readout for superconducting qubits is dispersive readout, which is performed by observing the shifts in frequencies of a linear resonator that contains the given qubit. This method gives a high-ﬁdelity fast readout of the superconducting qubits, but the qubit’s coherence time is limited due to spontaneous energy decay, so many further modiﬁcations have been made to increase coherence time for measurements.[10]

5.4 Alignment with DiVincenzo Criteria Let us summarize how well the transmon superconducting qubit meets the Di- Vincenzo criteria. The circuit potential has well-defined sublevels, of which the lowest two levels are generally used to characterize the qubit. Additionally, systems of approximately 50 transmon qubits have been realized, with systems on the order of 50-100 transmon qubits in development.[10][12] Therefore, the transmon qubit system is well-defined and scalable, so it meets the first DiVin- cenzo criterion. Because the superconducting qubits are macroscopic, initialization is fairly straightforward. There are many techniques employed to initialize transmons, many of which involve performing strong projective measurements,[13] so this system also meets the second DiVincenzo criterion. The transmon qubit is charge-insensitive so its coherence time is not susceptible to environmental charge noise. However, the transmon qubit only has weak anharmonicity and is susceptible to decoherence by spontaneous energy decay during measurement, so it weakly meets the third DiVincenzo criterion. The transmon qubit system admits transformation by quantum gates, so it meets the fourth criterion. Finally, transmon qubits can be measured quickly with high fidelity, so they additionally meet the fifth DiVincenzo criterion.

5.5 Future Work Future work on transmon qubit systems is likely to involve methods to extend the coherence time of the qubits, which is currently the system’s primary weak- ness. Longer coherence times will allow for more complex computations to be possible. Additionally, the transmon qubit system is not independent of scale, so as system sizes increase to include more qubits, additional methods for main- taining coherence and ﬁdelity will have to be implemented.

6 Acknowledgments

I would like to thank Hannes Bernien and David Schuster for their valuable insight into the world of experimental quantum compuation. I would also like to thank Savdeep Sethi and Wen Han Chiu for their wonderful teaching and assistance throughout the quarter.

9 References

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