# Quantum Computers

It was about four decades ago that Richard Feynman gave a seminal lecture [R. P. Feynman, Int. J. Theor. Phys. 21, 467-488 (1982)] about computer simulations of physics which marked the birth of quantum computing. Ever since, scientists and researchers alike have been on a quest to build practical quantum devices to efficiently simulate a quantum system. The exponentially increasing complexity in computation required to simulate a physical system using classical resources is going to make classical computers substandard in the near future. Exploiting the properties of quantum parallelism, one therefore turns to quantum devices which offers an appealing solution to the computational problems currently considered intractable on classical devices.

The basic unit of quantum information is called qubit. Unlike bits in classical computers, qubits can be in both state $|0\rangle$ and state $|1\rangle$ at the same time. Two qubits can be in states $|00\rangle$, $|01\rangle$, $|10\rangle$ and $|11\rangle$ at the same time. For each additional qubits, the total number of states doubles. Although such states cannot be measured directly, it is the natural existence of these kinds of states that give quantum computers their ability to process large amounts of data quicker than any classical computers. Quantum computers hold remarkable promises to speed up one’s computational task on day-to-day basis. It promises to aid researchers to solve complex problems like simulations of large biological molecules or factoring large numbers using Shor’s algorithm. In addition, blue print for provably secure cryptography techniques exists that is only possible within the realm of quantum computers. In the following, I will present results of my semester research conducted at ETH Zurich’s quantum device lab headed by Prof. Andreas Wallraff. Most of the text and plots in this page are adapted from my semester thesis which is available here.

Quantum gates are fundamentally important to process information in quantum devices. Manipulation of qubits deﬁnes the unitary evolution of quantum information. Clasical computers are built of electrical circuits, and it involves logic gates to manipulate the classical bits. In quantum computers, qubit states are manipulated with the help of quantum gates. One such quantum gate is a controlled phase gate (or CPHASE gate). It is a two-qubit gate, and performs a controlled operation on the second qubit based on the state of the first qubit – if the first qubit is in state $|1\rangle$ and the second qubit is in state $|1\rangle$, the application of CPHASE unitary introduces a minus sign in front of the state of the two qubits as depicted by the circuit diagram below,

### Theory:

We consider two superconducting qutrits in our simulation. The Hamiltonian has bare component and interaction component,

$\hat{H}(t)=\hat{H}_{bare}+\hat{H}_{int}$

The relevant part of the interaction component is,

$\hat{H}_{int}=J_0[ (|20\rangle \langle11|+|11\rangle\langle20|)$+c.c],

where $J_0$ is the coupling between the states $|11\rangle$ and $|20\rangle$. Energy spectrum of  the Hamiltonian can be tuned by sending a magnetic ﬂux pulse through one or both of the qutrit frequency biases. To obtain the time-dynamics of these qutrits, we numerically solve the Schrodinger equation using the so-called Dyson series. Therefore the desired CPHASE propagator $\hat{U}_{sim}(t)$ is approximated simply by consecutively multiplying several unitaries at small time slices as follows:

$\hat{U}_{sim}(t)=\prod_{n=0}^N e^{-i \hat{H}(ndt') dt'}$

where $N=t/dt'-1$. To increase the accuracy, one should use smaller time slice $dt$ and large $N$. The following are the simulation results for population and phase of the $|11\rangle$ state. I have also included my simulation result below for the theoretically obtainable CPHASE gate fidelity using this approximate propagator.

### Experiment:

Our quantum computing architecture comprises of superconducting transmon-type qubits. The image below shows a prototype superconducting quantum processor with four qubits. The qubits are made out of non-linear inductor called Josephson junction. In such a junction, the cooper pairs tunnel though a potential barrier which is well approximated by a qubic-quadratic function. In this barrier, the reactant is a metastable state, and the cooper pairs tunneling from this quantum mechanical state into a continuum state of the product, thereby altering the phase of the wavefunction across the barrier (and which also ultimately represents the state of qubits). The tunneling enhancement can be obtained by semiclassical approaches such as WKB or instanton theory or using path-integral sampling approaches such as quantum instanton or quantum transition-state theory. The latter is able to account for anharmonicity in potential energy barrier, and is seen to be more accurate than the former.