Research

My research work is in the area of quantum computation and information. I am currently working on theoretical aspects and implementations of quantum walks. I am also interested in quantum optimization and quantum machine learning algorithms. My other interest is in computational finance, particularly, I am interested in fast numerical and Monte Carlo methods for pricing of financial option contracts. In the past I have worked on the applications of quantum spin systems, in quantum information theory. More details can be found in my research statement.

Current Project:

1. Quantum walks and their implementations

Summary:  Quantum walks  are quantum analogues of a classical walk. In this project we study the quantum hitting time of a particle taking a quantum walk on a hypercube. Just like classical Markov chains quantum walks can be discrete or continuous. A quantum walk describes the motion of a quantum mechanical particle on a lattice or a graph. A discrete time quantum walk  has a Hilbert space $H_C \otimes H_P$, where $H_C$ is the Hilbert space of the coin and  $H_P$ is the position space. At each time step a coin (Unitary operator) acts on a coin space and conditioned on the outcome of the coin a shift operator acts on the position space, thus moving the particle on the lattice. After several steps a quantum measurement operation is done on the position space resulting  in a probability distribution of the particle to be found on any point on the lattice or graph.  A well known result is that the hitting time of a quantum walker on a $n$ dimensional hypercube from one vertex to the antipodal vertex is linear in the dimension of the hypercube. This is exponentially faster than the classical case. In [3] we have worked on the hitting times cube-like graphs; a type of Cayley graphs that are generalization of the hypercube. 

A continuous-time quantum walk describes the motion of a quantum mechanical particle on an underlying graph which is associated with a Hilbert space of dimension equal to the number of vertices. The dynamics of the walk is governed by the unitary operator $U(t) = e^{iAt}$, where, $A$ is the adjacency matrix of the graph.  A crucial feature of a continuous-time quantum walk is the transfer of a quantum state from one vertex to another, which plays a significant role in various quantum information processing tasks. If the fidelity of the transfer is unity, we call it a perfect state transfer (PST). The PST problem in graphs uses several techniques from algebraic graph theory. In this project we showed that perfect state transfer occurs in rational weighted cube-like graphs at a time with is a multiple of $\frac{\pi}{2}$ . We are also working on PST in weighted Abelian Cayley graphs [1,4]. 

An efficient quantum circuit is one in which the number of quantum gates scale polynomial in the number of input qubits. A key feature of both these projects was the construction of efficient quantum circuits [2,3] for the discrete time and continuous time walks. Several undergraduate students are involved in writing python code to construct the quantum circuits for quantum walks.

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Past Projects:

1. Quantum spin systems in quantum information theory

Summary: Quantum spin chains are a natural test bed for study of condensed matter systems and quantum information theory. Of  particular interested are the ground states of a large class of spin systems called matrix product states (MPS). MPS are universal for measurement based quantum computation. Numerical techniques based on higher dimensional MPS (tensor networks) are  being used to study phenomenon such as high temperature superconductivity and exotic phases of matter. In [1] we have investigated the low lying spectrum of the ferromagnetic XXZ chain and shown that isolated eigen values exist in the thermodynamic limit. Further in [2] we constructed one and two qubit gates using the subspace corresponding to the isolated eigen values. We have used an algorithm (DMRG) based on MPS to numerically approximate the ground states of the XXZ model and time evolve the system to construct certain elementary high fidelity gates. More recently in [3] we have studied the channel capacities of a quantum memory channel that are correlated by matrix product states. 

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2. Entropy rates of Stochastic processes

Summary: Entropy rate of a stochastic process is the average amount of information per symbol. In the generalization of Shannon's theorems to stationary and ergodic  processes entropy rate plays an important role. While there is an exact formula for the entropy rate of a Markov chain an efficient formula for the entropy rate of a general hidden Markov model (HMM) is one of the outstanding problems of information theory.  This assumes importance as hidden Markov Models are ubiquitous in practical applications such as speech and image processing, bioinformatics and communication and information theory. In [1] we were be to obtain a formula for the entropy rate of a HMM in the special case where there is at least one unambiguous symbol. Our derivation for the entropy rate formula was based on connecting the measure of the entropy rate to the classical analogues of quantum states called matrix product states (see above). In [2] we gave an efficient algorithm for the entropy rate of a hidden Markov model with unambiguous symbols. In [3] we obtain formulas for the entropy rates of Gaussian random vector processes.

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3. Capacities of quantum memory channels

Summary: Reliable transmission of classical information over quantum channels is an important problem in quantum information theory. In recent years there has been a lot of interest in quantum channels with memory. Quantum memory channels model the many physical situations where the noise effects are correlated. An unmodulated spin chain and a micro maser have been proposed as physical models of quantum channels with memory effects. In general computing capacity formulas for memory channels is difficult since it involves an optimization task over quantum states in a high dimensions. A subclass of quantum memory channels are the forgetful channels where the effects of the initial memory configuration are forgotten over time. In [1]  we derive the classical capacity look at a forgetful quantum memory channel where the noise correlations come from an ergodic Markov chain. In [2] we extended this result to compute the entanglement assisted capacity of this channel. In [3] we compute the quantum capacity of quantum memory channels correlated by matrix product states.

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