#### Introduction

Owing to the seemingly counter-intuitive effects of quantum superposition and entanglement, a quantum computer has the potential to perform certain calculations that would otherwise be impossible on a regular device. The possibilities, ranging from prime factorisation reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() to modelling of quantum systems reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(), seem endless. Building one, however, is a rather tricky endeavour. The DiVincenzo reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() criteria outline the required conditions to construct a working and scalable quantum computer. Arguably the most critical of which is the need for a system to effectively be isolated from the rest of the universe so as not to be a victim of decoherence, which destroys quantum information.

Experimentalists are working on creating coherent systems consisting of components such as cold neutral atoms reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(), photons reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(), and ions reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). In the developmental meantime, we can theoretically model such systems of qubits to explore how we can limit decoherence.

A classically ergodic system is considered thermalised if it has been allowed to explore its entire phase space for a sufficiently long time. Once thermalised, it will have an equal chance of being in any microstate. This system is described by the famous microcanonical ensemble reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2().

Unlike classical systems, where ergodicity and thermalisation are well understood, quantum systems are much more elusive. The eigenstate thermalisation hypothesis was proposed in an attempt to explain when and why a quantum system thermalises to a classical microcanonical ensemble reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). Consider a finite isolated system which has eigenstates and eigenenergies which can be measured using observables. The operator will have expectation values but we can also write them in terms of their microcanonical average and variance reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). The hypothesis states that the fluctuations of the expectation values between different eigenstates will be so small that we can evaluate the thermal ensemble from one eigenstate reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2().

It was previously understood that the only way to break this hypothesis was if there was a set of independent conserved quantities reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() in the system, where is the number of degrees of freedom. However, a new Hamiltonian has been discovered reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() which breaks this hypothesis. So far, no set of so many quantities have been found. The experiment involved a one-dimensional array of atoms which could each be individually prepared in either their ground or highly excited Rydberg state reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). In most preparations of the initial state, the system thermalised and became chaotic. However, certain prepared states appeared to remain coherent for longer than expected.

In an attempt to explain how this occurs we must look to quantum scars which are phenomena that arise in quantum chaos theory reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). An example reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() of this effect consists of a system comprising of a single particle in a stadium. Classically, reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() the path the particle will take is chaotic with a set of unstable periodic orbits. At the quantum level, the eigenstate amplitudes surrounding these orbits are enhanced. A recent article reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() demonstrates potential evidence for the discovery of the many-body analogue of this effect in the experiment described above. The authors also proclaim that the system demonstrates weak ergodicity breaking meaning that whether the system explores all possible states depends strongly on its initial configuration.

The following sections will explore whether quantum scars are responsible for the observed weak ergodicity breaking, investigate other potential explanations reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() and introduce the interaction distance reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() as a potential tool to further examine the origins of this notable phenomenon.

#### Rydberg Chain

The time evolution of any quantum system can be reduced to defining an initial configuration and a Hamiltonian by simply integrating the SchrÃ¶dinger equation reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() to evaluate a time evolution operator. For the case that the Hamiltonian does not depend on time its time evolution is described by

(1)

In practice, this is generally not so easy to evaluate since to find the exponential of a matrix it is first necessary to find its eigenvalues and eigenvectors. If we have a system of size N composed of individual components of size M, then the total dimension of the system is . Thus, the computational complexity to exponentiate the matrix is of order reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). As suggested by Feynman in 1981 reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(), rather than evaluating such complicated calculations, we could instead build a simulator which naturally evolves in time and exploits the quantum effect of superposition.

It is now experimentally possible to realise previously theoretical reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() Hamiltonians consisting of single atoms reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). By directing a laser onto a cold neutral atom, experimentalists reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() have been able to induce sinusoidal Rabi oscillations reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() in the amplitude of their ground and excited state. By adjusting the Rabi frequency, , and detuning, , it is possible to manipulate and control these oscillations.

The experiment reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() that inspired the investigation of many-body scars consisted of a linear array of \ce{^{87}Rb} atoms with the Hamiltonian

(2)

where the first two terms simply correspond to the summation of the atom-light interaction Hamiltonian. The Pauli matrix flips a state between its excited and ground state. In the experiment a secondary beam, contained within the existing and , was introduced to reach the highly excited Rydberg states reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). These states are highly repulsive when next to each other and thus contribute to the final interaction term. By first using optical pumping reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() to initialise the ground state, the group then varied the detuning to prepare a product state consisting of alternating excitations.

When the system was allowed to evolve in time, robust oscillations of the arrangement of excitations were observed. This is surprising as the system is breaking the eigenstate thermalisation hypothesis reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(), yet it is not many-body localised reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() and has not so far been found to be integrable reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2().

#### Theoretical Model

To investigate the origin of these oscillations and to explain this apparent weak ergodicity breaking, the Hamiltonian is first simplified. By introducing the projection operator, , we can break the Hilbert space into separate components, effectively restricting the accessible states to those which do not contain neighbouring excitations. In the limit where and the simplified Hamiltonian becomes reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2()

(3)

For further simplicity we consider periodic boundary conditions, meaning that is equivalent to .

The reduced Hilbert space for this Hamiltonian consists only of the states which do not contain adjacent excited atoms. The dimension of which is equal to reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() where the term is the Fibonacci number. This is a great reduction from the previous basis states, as can be seen in the following figure.

Despite this reduction, the evaluation of the eigenstates (and therefore the time evolution) is still not possible even for a relatively small . In order to increase the limit of the system size symmetries are employed. In this example, translational symmetry is identified but this can be further extended using particle-hole and spatial inversion symmetry reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2().

By its definition the translation operator acts on a state as

(4)

and thus we can identify that .

Utilising this fact and following the method of reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() we identify a set of momentum states with eigenvalues which are valid for the following momenta

(5)

Any momentum state can be assembled via a basis state and its translations

(6)

where is the periodicity of the basis state.

Since the Hamiltonian will be block diagonal in the basis of the eigenstates of . To see this we evaluate the action of the Hamiltonian on an arbitrary momentum state as

(7)

The output of the Hamiltonian parts on a reference state will also be related to a reference state within some number of translations. We are able to extract matrix elements of the form

(8)

which gives a set of matrices corresponding to each allowed momentum, further increasing the size of systems we can feasibly model on a classical computer. The article reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() demonstrates a simulation of size using the infinite time evolving block decimation method reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). This only gives results valid to time but this is enough to demonstrate the oscillations observed in the experiment. The following figure shows the fidelity of the time evolution of the initial state . In the momentum basis this is . An example of the entanglement entropy for starting in the state is plotted in the figure below.

In an attempt to explain these long-lasting oscillations, the article reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() demonstrates an unusually high overlap between eigenstates and so-called special states. The energy separation between these special states matches the frequency of oscillations of the entanglement entropy. A plot of these states can be seen in the above figure.

The article then goes further by demonstrating that it is possible to approximate these special states using a forward scattering approximation by splitting the Hamiltonian into operators

(9)

where and . These operators have the effect of increasing or decreasing the Hamming distance reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(), which gives the number of spins required to flip to reach some initial state.

We obtain a non-interacting Hamiltonian of the form

(10)

with the tridiagonal terms

(11)

This free model is reported to have errors less than for a chain of .

The authors propose that this demonstrates a new universality class of quantum dynamics and suggest that this motivates further research into this class of dynamics. To enhance this argument, the paper considers a potential alternative. They make the claim that no proximate integrable system could be responsible for the non-ergodic dynamics because the level statistics reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() approaches the Wigner-Dyson distribution as the system size is increased. This is in opposition to a trend towards a Poisson distribution which would be expected for integrable systems. It is noted that the “Golden chain” system reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(), which is integrable, is qualitatively similar to the Rydberg chain. The authors argue that, despite this, the two Hamiltonians differ too much to be considered a weak deformation of each other.

Since the publication of this article there have been further theoretical explorations of the experiment. Reference reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() demonstrates that a deformation of the Hamiltonian can enhance both coherence of the oscillations and proximity to integrability based on its level statistics. They do not demonstrate what the parent Hamiltonian is nor do they explain the importance of the distinction between a subclass of integrability and an entirely new class.

Reference reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() introduces a manifold of only locally entangled spin states and derives their equations of motion using matrix product states and the time-dependent variational principle reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). The equations of motion of the product state only explores a special low-entanglement manifold of the Hilbert space. This is like the single particle case of quantum scars in which the classical unstable orbits are enhanced.

Further exploration reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() by the authors of the original theory paper counters the proximate integrability argument by demonstrating that the quantum scars appear to be stable under various perturbations. Specifically, it is shown that there is a strong correlation between perturbations near the forward scattering approximation and the robustness of quantum scars.

#### Interaction Distance

Free fermion systems are surprisingly sprawling across physics given their simplicity to solve. It is possible to directly diagonalise their system Hamiltonian by constructing a new set of eigenmodes. This allows them to be solved as a set of independent and uncorrelated single particle modes. The energy spectrum of such a system is given by reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2()

(12)

where represents the energies of the eigenmodes and the value from the number operator, , can be or . When considering more complicated interacting systems it would clearly be useful to instead express it as a free fermion system or alternatively find one that is very close.

Reference reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() demonstrates a new tool, the interaction distance, for efficiently calculating the proximity of any given interacting fermion system to its closest free system by calculating its entanglement spectrum reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). In addition, it outlines a method for extracting the composition of the stated nearby free system.

A system in thermal equilibrium can be represented by the density matrix reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2()

(13)

where is the inverse thermodynamic temperature and is the partition function normalised over all temperatures. A free system can be accurately described by Equation (13) and its eigenvalues are given with respect to the eigenvalues of the energy of the system.

If we instead have an interacting system, we can construct a reduced density matrix from an arbitrary eigenstate of its Hamiltonian. This reduced density matrix is given by the partial trace reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() over the entire density matrix of the state

(14)

The goal is to identify the distance between this arbitrary density matrix and some free density matrix. To find this distance, we require a manifold of all the free fermion states which are of the form

(15)

The interaction distance is thus defined as a minimisation

(16)

where the quantity is the trace distance reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2().

The authors demonstrate its utility with the one-dimensional Ising model

(17)

with and representing the ferromagnetic and antiferromagnetic systems respectively.

The quantum critical point in the ferromagnetic Ising model occurs at . This is visible in the interaction distance phase diagram which shows a relatively high interaction distance surrounding the critical point. More interestingly a critical line emerges in the antiferromagnetic system spanning through the classical critical point , where the spins can be treated as classical objects reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). As the system size is increased the interaction distance of the regions surrounding criticality decrease whilst those on the critical line remain constant.

The Rydberg blockaded system has interacting terms and thus it would be interesting to explore the phase diagram surrounding the Hamiltonian with some perturbations. It may demonstrate that it has closest proximity to the free Hamiltonian of Equation (10), further supporting the quantum scar case, or else in unexpected regions.

#### Conclusion

This review has introduced both the concept and importance of quantum many-body scars as realised in the experiment reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() with the Rydberg blockade chain Hamiltonian and explored in reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). A demonstration of the analytical realisation of the reduced Hilbert space has been presented. A computation of the eigenstates using diagonal symmetries has also been used to plot the time evolution for the product state.

A critical analysis of the validity of the suggestion and counterargument of a new universality class was also discussed. The concept of the many-body scarred states is strongly supported by other work reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() although whether there exists a parent proximate Hamiltonian is still disputed.

A novel tool to analyse the phase space of the Hamiltonian has also been considered, along with an example of an exploration of a similar interacting fermion system as described in reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2(). This is a relevant source of research to be further explored for this project. The code to calculate the interaction distance and the closest free system to an arbitrary system is available online. The input required is simply the entanglement spectrum. The evaluation of the entanglement spectrum for the special states will be the next step in this project. By evaluating the interaction distance of these states using the code it may be possible to identify behaviours of the Rydberg system in the thermodynamic limit.

Finally, in this project, I hope to be able to develop and define some form of \textit{integrability distance}. This would allow an exploration of whether there exists some proximate Hamiltonian and a measure of its closeness to the Rydberg chain Hamiltonian (2). A stronger understanding of the theory of integrability, the interaction distance and the Bethe ansatz reference: 50c6aa33-ae00-4675-ad5b-0109b60b0fd2() will be required in order to develop this idea further. Such a metric could also give means to explore alternative Hamiltonians with scarred states that could lead to realisable demonstrations of weak ergodicity breaking. New such Hamiltonians could lead to a greater understanding of the eigenstate thermalisation hypothesis and potentially lead to new methods to develop quantum devices with longer lasting coherence.