### What is a quantum computer?

The term quantum computer is used in the general literature to embrace a broad range of quantum technologies capable of performing or aiding in the execution of some sort of computation, either digital or analog.

In general terms, a quantum computer is a device that takes advantage of the quantum nature of some physical system to encode data and perform operations. Thanks to its nature, a quantum computer operates in a way profoundly different from that of a classical computer opening the path to new ways of performing computation in a way that is, in some cases, algorithmically more efficient than the best strategy that can be implemented on a classical computer.

In this page, we will use the term quantum computer to discuss the case of universal digital quantum computers that represent the most general and powerful platforms on which quantum algorithms can be implemented.

### How does a quantum computer work?

A classical digital computer operates by manipulating, through logic operations, data encoded in a series of bits. Each bit represents a binary value that can either be set in a $0$ or a $1$ logic level. If a register of $n$ bits is used to encode data, a single value can be stored by independently setting each one of the $n$ bits.

Similarly, a quantum computer operates by manipulating quantum bits or, more shortly, qubits. Each qubit has access to two orthogonal logic states $\lvert 0 \rangle$ and $\lvert 1 \rangle$ but, differently to a classical bit, a qubit can be set in whatever linear combination of the two states. In more formal terms, the state $\lvert \phi \rangle$ of a qubit can be written as:

$$ \lvert \phi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle $$

with $\alpha$ and $\beta$ complex numbers with $\lvert \alpha \rvert ^2 + \lvert \beta \rvert ^2 = 1$.Thanks to this fact, a register of $n$ qubits can be set in a state $\lvert \psi \rangle$ encoding simultaneously all the $2^n$ possible states of an equally big classical register. If we indicate with $\lvert i \rangle$ the state setting the register to the binary representation of the number $i$, the state of the quantum register can be written as:

$$\lvert \psi \rangle = \sum_{i=0}^{2^n-1} c_i \lvert i \rangle \qquad \text{with} \qquad \sum_{i=0}^{2^n-1} \lvert c_i \rvert^2 = 1$$

Similarly to classical computers, where bits are manipulated using logical operations setting the value stored in the register to a new bitstring, a qubit register can be manipulated by means of unitary operations $\hat{U}$, where $\hat{U}\hat{U}^\dagger=\hat{I}$, capable of changing the state $\lvert \psi \rangle$ of the whole qubit register to a new state $\lvert \psi’ \rangle = \hat{U} \lvert \psi \rangle$. Please notice how the requirement of $\hat{U}$ being unitary ensures normalization of the final state.

A computation performed on a quantum computer can be imagined as a series of unitary operations acting on the state of the computer itself or, in more practical terms, a series of $2^n \times 2^n$ unitary matrices acting on a vector of $2^n$ coefficients associated to each bitstring state $\lvert i \rangle$ of the quantum register.

In this context, it is easy to see how a quantum computer is potentially exponentially more efficient than a classical one. While the classical computer can elaborate only one register state at a time, the quantum computer can do the same on all the possible states of the register at the same time. This is the great power of harnessing the superposition and entanglement between quantum states of the register.

This however is only part of the story since the quantum results obtained from the quantum algorithm must be obtained through some sort of measurement. This represents the critical point of an algorithm design since the measurement randomly collapses the state of the computer to one of its possible states. To obtain a realistic picture of the result the quantum circuit must be run many times to obtain a good approximation of the result coefficients. This, for a generic quantum state, has the potential to be exponentially expensive in computational terms voiding the quantum advantage described before. As such a good quantum algorithm must be designed taking into account the measurement and collapsing the solution in an easily samplable distribution of a few important register states.

### What can be done at the moment?

What has been discussed so far is an ideal picture of a quantum computer in which the state of the quantum register can be perfectly controlled and is not perturbed by external interactions. The system has been assumed to be in a pure quantum state and, in principle, can be kept in that state indefinitely. This however is not what happens in the real world where a perfectly isolated quantum object can hardly be obtained. A real quantum device is a fragile object and its state can be perturbed by many external factors causing dephasing and relaxations that, in the long term, end up destroying the coherent quantum state of the system preventing it from being useful in a computation. Systems of quantum error correction can in principle be developed but they are usually complex and expensive in terms of qubit counts requiring many physical qubits to represent the state of a hypothetical logical qubit.

At the moment the literature describes the period we live in as the noisy intermediate-scale quantum era or, more shortly, the NISQ era. The devices of the NISQ era are characterized by a relatively small qubit count (usually less than 1000), not capable of fault tolerance, and still not capable of achieving a marked quantum advantage. The noise of these devices is however sufficiently low to carry out simple quantum experiments and to be useful in the development of NISQ algorithms. These algorithms can, for example, use the low qubit count to speed up parts of a bigger operation carried out on a classical computer using the quantum computer as an accelerator. An example of this is the Variational Quantum Eigensolver or, more shortly, the VQE algorithm.

### What is the VQE algorithm?

The Variational Quantum Eigensolver or, more shortly, the VQE algorithm, is a quantum-classical algorithm that can be used to find the ground state of a quantum system.

The process starts by defining a mapping. The mapping essentially establishes how the states of the system to be studied are encoded on the quantum computer in terms of the states of the quantum register. Once this correspondence has been established the expectation value of the Hamiltonian, that is to say, the energy of an arbitrary state represented on the quantum register, can be computed.

The algorithm uses a parametrized quantum circuit, called the ansatz, to construct a trial quantum state. The energy of the constructed state is measured on the quantum devices and fed to a classical device. The classical computer implements an optimization routine tasked with minimizing the energy expectation value by acting on the parameters of the ansatz. The ground state energy of the system can be obtained by iterating until convergence to the minimum is reached.

### What did I work on?

My research work aimed to adapt the VQE algorithm to the solution of stochastic diffusive problems associated with the thermally activated kinetic transition of population between potential energy minima.

The starting point of our work was that of employing the isomorphic relation existing between the quantum Hamiltonian and the Fokker-Planck-Smoluchowski operator to map the latter as a fictitious Hamiltonian in a VQE scheme. In this context, our work was different from the usual VQE approach since the interesting state in the field of stochastic analysis is usually the first excited one, associated with the kinetic transition rate between sites. To avoid the use of penalty terms to offset the ground state and, as such, converge to the first excited one, we decided to operate in symmetrical systems in which the Hilbert space is parted due to symmetry and, as such, in which the first excited state represents the ground state of the Hilbert subspace of odd functions.

As a model system, we decided to study a system of rotors with a nearest neighbors interaction scheme in which one of the rotors experienced a symmetrical bistable potential. We decided to use a very dense binary mapping to map the state of the system on the quantum register. A trial wave function was constructed using a heuristic RyRz ansatz.

Various configurations of the system and the algorithm have been tested demonstrating the feasibility of the approach but also highlighting many critical points. The full discussion of the problem has been presented in an open-access paper also available on ArXiv:

- Pierpaolo Pravatto
*et al*,*New J. Phys.*, 23, 123045, (**2021**) [Paper link] [ArXiv link]