@inproceedings{bib_Impl_2024, AUTHOR = {Shantanav Chakraborty}, TITLE = {Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers}, BOOKTITLE = {Quantum Journal}. YEAR = {2024}}

We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method (Single-Ancilla LCU) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach (Analog LCU) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method (Ancilla-free LCU) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.

@inproceedings{bib_Quan_2023, AUTHOR = {Shantanav Chakraborty, Aditya Morolia, Sai Anurudh Reddy Peduri}, TITLE = {Quantum Regularized Least Squares}, BOOKTITLE = {Quantum Journal}. YEAR = {2023}}

Linear regression is a widely used technique to fit linear models and finds widespread applications across different areas such as machine learning and statistics. In most real-world scenarios, however, linear regression problems are often ill-posed or the underlying model suffers from overfitting, leading to erroneous or trivial solutions. This is often dealt with by adding extra constraints, known as regularization. In this paper, we use the frameworks of block-encoding and quantum singular value transformation (QSVT) to design the first quantum algorithms for quantum least squares with general ℓ2-regularization. These include regularized versions of quantum ordinary least squares, quantum weighted least squares, and quantum generalized least squares. Our quantum algorithms substantially improve upon prior results on quantum ridge regression (polynomial improvement in the condition number and an exponential improvement in accuracy), which is a particular case of our result. To this end, we assume approximate block-encodings of the underlying matrices as input and use robust QSVT algorithms for various linear algebra operations. In particular, we develop a variable-time quantum algorithm for matrix inversion using QSVT, where we use quantum eigenvalue discrimination as a subroutine instead of gapped phase estimation. This ensures that substantially fewer ancilla qubits are required for this procedure than prior results. Owing to the generality of the block-encoding framework, our algorithms are applicable to a variety of input models and can also be seen as improved and generalized versions of prior results on standard (non-regularized) quantum least squares algorithms.

@inproceedings{bib_Quad_2022, AUTHOR = {Simon Apers, Shantanav Chakraborty, Leonardo Novo, Jérémie Roland}, TITLE = {Quadratic speedup for spatial search by continuous-time quantum walk}, BOOKTITLE = {Physical Review Letters}. YEAR = {2022}}

Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this article, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analog procedure to perform operations on a state of the form $e^{-tH^2}|\psi\rangle$, for a given Hamiltonian H: it only requires evolving H for time scaling as $\sqrt{t}$. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our work thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.

@inproceedings{bib_Limi_2022, AUTHOR = {Amrita Mandal, Rohit Sarma Sarkar, Shantanav Chakraborty, Bibhas Adhikari}, TITLE = {Limit theorems and localization of three state quantum walks on a line defined by generalized Grover coins}, BOOKTITLE = {Physical Review A}. YEAR = {2022}}

In this article, we undertake a detailed study of the limiting behavior of a three-state discrete-time quantum walk on a one-dimensional lattice with generalized Grover coins. Two limit theorems are proved and consequently we show that the quantum walk exhibits localization at its initial position for a wide range of coin parameters. Finally, we discuss the effect of the coin parameters on the peak velocities of probability distributions of the underlying quantum walks.

@inproceedings{bib_Impr_2021, AUTHOR = {Yosi Atia, Shantanav Chakraborty}, TITLE = {Improved upper bounds for the hitting times of quantum walks}, BOOKTITLE = {Physical Review A}. YEAR = {2021}}

Quantum walks [1–6] are quantum analogs of random walks on graphs and have widespread applications in several areas of quantum information processing [7]. In particular, they are a universal model for quantum computation and are crucial to the design of quantum algorithms for a plethora of problems such as element distinctness [8,9], searching for a marked state in graphs [10–16], matrix product verification [17], triangle finding [18], group commutativity [19], and many others [20–24]. In fact, for some oracular problems, quantum walks exhibit an exponential speedup over their classical counterparts [21,22]. For several of these algorithms, the runtime strongly depends on the time required for the quantum walk on a graph to localize at a vertex inside a marked set of vertices of interest. In the context of classical random walks, the hitting time is defined as the expected time required to hit a marked vertex. As such, the quantum hitting time characterizes the running time of the underlying quantum algorithm [14,15,20,21,25,26]. For continuous-time quantum walk (CTQW), the Hamiltonian governing the dynamics of the walk encodes the (vertex or edge) connectivity of the underlying graph. The dynamics involves evolving some initial state according to this Hamiltonian for time T , followed by a measurement in the basis spanned by the vertices of the graph, thus finding a target state with a high probability. However, Hamiltonian evolutions are unitary, and therefore, unlike classical random walks, quantum walks never converge to a fixed state with time. One method to address this is to evolve the system by H