Alladi Ramakrishnan Hall

On scalable quantum circuit representation of unitary matrices

Rohit Sarma Sarkar

IIT--Kharagpur

In this talk, we provide a brief introduction to the design of scalable quantum circuits for various unitary matrices that lie in the heart of evolution of many-body quantum systems. First, we present a scalable
quantum neural network based framework for the approximation of any target unitary matrix for n-qubit systems. We demonstrate that such a quantum circuit is achieved by proposing a parametric representation
of unitary matrices of dimension d through the use of a novel unitary Hermitian basis for the algebra of d × d complex matrices known as Recursive Block Basis (RBB). Then we utilize a Lie group theoretic approach to propose an optimization algorithm capable of approximating any target unitary matrix via this representation. Consequently, we employ these results to propose a new unitary Hermitian traceless basis for the algebra of 2^n × 2^n complex matrices, called the Standard Recursive Block Basis (SRBB) which serves as an alternative to the Pauli-string basis and is used to develop an optimization-based approximation algorithm for a scalable quantum neural network representation of a target unitary for n-qubit systems. Next, we shed light on
the construction of quantum circuits for the exponential of scaled Pauli strings. The proposed circuits are
implementable on low-connected quantum hardware and are scalable in nature. Applying these circuit models,
we approximate the unitary evolution for several classes of one-dimensional Hamiltonian operators. We also consider diverse noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers. Subsequently, we showcase scalable qutrit quantum circuits for discrete-time three state quantum walks on Cayley graphs defined by Dihedral groups and the cycle graph. In the proposed circuits for DTQWs, we demonstrate the use of generalized qutrit Toffoli gates and their decomposition into single qutrit
rotation gates, qutrit-X gates, and two-qutrit controlled-X gates. We explore the circuit complexity of the proposed quantum circuit model, and examine the performance of the proposed circuit simulations on nearterm quantum computers through the incorporation of various noise models. We also provide a brief overview
of discrete-time quantum walks (DTQWs) on various graphs, their fundamental properties like periodicity and localization, along with presenting a combinatorial generalization of the Grover matrix that is often used as the quantum coin operator in DTQWs. Finally, we deliver a concise overview of our current works and
present new algorithms for decomposing arbitrary square matrices into linear sum of Pauli strings along with
its application to construction of scalable quantum circuits for continuous-time quantum walks.

Keywords: Scalable quantum circuits, Pauli strings, Hamiltonian, Quantum walks, Periodicity, Localization, Grover matrix, Cayley graphs.