# Research

## Interests

Quantum Simulation and Computing, Quantum Field Theory, Condensed Matter Physics, Mathematical Physics

## Quantum simulation

The investigation of strongly interacting complex quantum systems remains one of the outstanding challenges of modern physics. Despite the remarkable progress on both numerical as well as analytical fronts, systematic and well-controlled non-perturbative analysis of many quantities of interest remain intractable. Quantum simulation provides a promising alternative to the aforementioned conventional techniques towards tackling these problems. A particularly promising near-term approach is analog quantum simulation, where a given quantum system is tailored to simulate a specific another.

The central theme of this research direction is the investigation of analog simulators for strongly interacting bosonic quantum field theories (QFTs). Among the many available platforms for analog simulation, we consider superconducting quantum electronic circuits. We start with integrable QFTs in 1+1 space-time dimensions with the goal of performing quantum simulation systematically around integrable points.

Key publications:

- The quantum sine-Gordon model: In this work, we considered one of the paradigmatic integrable QFTs, the quantum sine-Gordon model, which can be realized straightforwardly with quantum circuits. Using a combination of analytical (form-factor) and numerical (DMRG) techniques, we computed one and two-point correlation functions for the continuum and quantum circuit models. We compared the quantum circuit model with the XYZ spin chain realization of the quantum sine-Gordon model model and showed that the quantum circuit allows exploration of a wider range of parameters for the QFT. Finally, we computed the entanglement spectrum of the QFT for both the quantum circuit and the XYZ chain and compared with analytical predictions.
- The quantum double sine-Gordon model: In this work, we considered a two-field generalization of the quantum sine-Gordon model, the so-called quantum double sine-Gordon model. Remarkably, unlike the ordinary sine-gordon model, this model can be purely quantum-mechanically integrable. What this means is that while a classical soliton wave-packet gets scrambled as it propagates through this nonlinear wave-medium, a true quantum soliton, due to the presence of quantum fluctuations, propagates undistorted and scatters with only phase-shifts. This model is one of the simplest examples of models where classical and quantum integrability part ways. We computed the thermodynamic Bethe ansatz equations for this model and predicted logarithmic singularities in the specific heat. We also showed that this pure-quantum integrable QFT model can be realized in a controlled manner with superconducting quantum circuits.
- Soliton confinement in a quantum circuit: Confinement of topological excitations into particle-like states – typically associated with theories of elementary particles – are known to occur in condensed matter systems, arising as domain-wall confinement in quantum spin chains. However, investigation of confinement in the condensed matter setting has rarely ventured beyond lattice spin systems. Here, we analyze the confinement of sine-Gordon solitons into mesonic bound states in a one-dimensional, quantum electronic circuit (QEC) array, constructed using experimentally-demonstrated circuit elements: Josephson junctions, capacitors and 0−π qubits. The interactions occurring naturally in the QEC array, due to tunneling of Cooper-pairs and pairs of Cooper-pairs, give rise to a non-integrable, interacting, lattice model of quantum rotors. In the scaling limit, the latter is described by the quantum sine-Gordon model, perturbed by a cosine potential with a different periodicity. We compute the string tension of confinement of sine-Gordon solitons and the changes in the low-lying spectrum in the perturbed model. The scaling limit is reached faster for the QEC array compared to conventional spin chain regularizations, allowing high-precision numerical investigation of the strong-coupling regime of this non-integrable quantum field theory.
- Quantum Electronic Circuits for multicritical Ising models: Multicritical Ising models and their perturbations are paradigmatic models of statistical mechanics. In two space-time dimensions, these models provide a fertile testbed for investigation of numerous non-perturbative problems in strongly-interacting quantum field theories. In this work, analog superconducting quantum electronic circuit simulators are described for the realization of these multicritical Ising models. The latter arise as perturbations of the quantum sine-Gordon model with p-fold degenerate minima, p=2,3,4,…. The corresponding quantum circuits are constructed with Josephson junctions with cos(nϕ) potential with 1≤n≤p.The simplest case, p=2, corresponds to the quantum Ising model and can be realized using conventional Josephson junctions and the so-called 0−π qubits. The lattice models for the Ising and tricritical Ising models are analyzed numerically using the density matrix renormalization group technique.

## Entanglement in many-body systems:

Entanglement, one of the quintessential properties of quantum mechanics, plays a central role in the development of long-range correlations in quantum critical phenomena. Thus, quantification of entanglement in a quantum-critical system provides a way to characterize the universal properties of the critical point. On one hand, analysis of entanglement sheds light on fundamental questions of QFTs — a notable example being an alternate proof of the c-theorem in 1+1 dimensions using strong-subaddivity and scaling properties of entanglement entropy together with Lorentz invariance. At the same time, these entanglement properties lie at the heart of the success of classical simulation 1+1D quantum systems and provide insights into the basic differences in the power of quantum vs classical computing.

The central theme of this research direction is the investigation of entanglement measures in 1+1D QFTs. In particular, we consider conformal field theories (CFTs) and their massive (but not necessarily integrable) deformations.

Key publications:

- Entanglement spectrum of the free, compact boson CFT: In this work, we investigated the spectrum of the entanglement or modular Hamiltonian (~ logarithm of the reduced density matrix) of a subsystem upon partitioning a free, boson CFT into two halves. We computed analytically the entanglement spectrum using partition functions of the free-boson on a cylinder and compared with numerical predictions obtained using DMRG.
- Entanglement spectrum of the quantum sine-gordon model: In this work, we compute the entanglement spectrum of the quantum sine-Gordon model. We related the entanglement spectrum to the spectrum of the corner transfer matrix of the classical eight vertex model, which allowed us to make analytical predictions of the model.
- Entanglement entropy in the Ising model with topological defects: In this work, we investigated the behavior of the entanglement entropy in the Ising model in the presence of topological defects. These defects are deeply related to the symmetries of the CFT. Here, we presented, for the first time, an ab-initio calculation of the entanglement entropy starting from the lattice model. In particular, we showed that these defects come in conjunction with zero-energy modes whose effects have to be carefully taken into account to make correct predictions about the universal behavior of the entanglement entropy. A related invited review is available here.

## Quantum computation with topological codes

Quantum computation holds the promise of potential exponential speedup over classical algorithms for certain problems. At the same time, it provides a viable tool for simulation of quantum many-body systems. A scalable, universal quantum computing scheme needs to be resilient towards errors. Topological quantum computing is an approach, where the protection from errors occurs at a ‘hardware level’ and is automatically implemented by the physical system. The advantage of this approach is that it is robust to local imperfections. This scheme makes use of non-abelian exchange statistics of elementary excitations of a two-dimensional quantum many-body system (see here for a review). A closely-related, but not identical, approach uses ground space of topologically-ordered systems to realize quantum codes to encode logical qubits. In this case, the protection against errors arises due to the topological degeneracy of the ground space. However, computation can be performed without using any exotic braiding statistics, following canonical methods borrowed from non-topological codes.

The central theme of this research direction is the analysis of topologically-ordered many-body quantum systems from the perspective of quantum computation.

Key publications:

- Quantum Phase Transitions of the Majorana Toric Code in the Presence of Finite Cooper-Pair Tunneling: In this work, we analyzed the different phases and the phase-transitions of a two-dimensional array of Majorana zero modes on mesoscopic superconducting islands. The latter array exhibits the toric code phase in the limit of vanishing Cooper-pair tunneling and small single-electron tunneling. We showed, using field theoretic techniques, that the toric code phase is, in-fact, robust to finite Cooper-pair tunneling — a regime which is relevant for experimental realizations. We analyzed the rich phase-diagram of the model and showed that the phase-transition out of the toric code phase transforms from a 3D XY to a 3D Ising type through a couple of tricritical points.
- Quantum phases of a one-dimensional Majorana-Bose-Hubbard model: In this work, we analyzed a one-dimensional array of Majorana-zero modes on mesoscopic superconducting islands, which forms a numerically-tractable lattice-regularization of the Kitaev wire in the presence of phase-fluctuations. We show that the critical properties of the model is equivalent to a classical 2D XY model with two types of potentials. Using DMRG, we analyzed the phase-diagram of the model. Very importantly, we showed that phase-fluctuations destroy the topological edge-modes which are the characteristic features of the model without phase-fluctuations.
- Topological ordering in the Majorana toric code: In this work, we analyzed signatures of topological order in a lattice-gauge theory model for the toric code built out of Majorana zero modes. The lattice gauge model in 2+1 space-time dimensions consists of U(1) matter fields, whose parity is coupled to an emergent Z2 gauge field. Using generalized Wilson loops or Fredenhagen-Marcu operators, we analyzed the different phases of the model. Another related work is here.