Pranav Singh

Can quantum computers tackle problems beyond the reach of classical simulation?

 

Hamiltonian simulation is one of the most promising frontiers where quantum computing companies are pushing to demonstrate quantum advantage. My group designs quantum algorithms utilizing Trotterization, VQAs, Lie group methods and qubitization for simulating the dynamics of many-body quantum systems, including those driven by time-dependent fields (i.e. non-autonomous).

 

Our focus: developing accurate algorithms with shorter circuit depths, real viability on near-term quantum devices and better preservation of physical properties.

 

Areas: quantum algorithms, Hamiltonian simulation, variational methods, many-body systems

The solution to a linear PDE

u' = A u

such as the Schrödinger equation can be expressed explicitly in terms of the exponential of A,

u(t) = exp(t A) u(0).

We may approximate the exponential by truncating the Taylor expansion. These are polynomial approximations. The first such truncation is the Forward Euler method, which suffers from stability issues. Krylov subspace methods are among the most flexible polynomial approximations and have superlinear convergence.
 

However, the Ritz values (which are the Krylov approximation of the eigenvalues) start accumulating at the eigenvalues from the outer part of the spectrum - the animation shows how Ritz values accumulate for a random matrix. This means, as the norm of the matrix grows, we either need a large number of Krylov iterations or very small time-steps.
 

I am interested in the design of rational approximations, exponential splittings and Krylov methods that can allow us to take much larger steps and lead to efficient approximation of the exponential.
 

Areas: rational approximation, Krylov methods, exponential splittings 

Symmetries in a physical system lead to conservation laws such as conservation of mass, momentum, angular momentum and energy.

 

This animation shows the solution of the Schrödinger equation with radially symmetric initial conditions. The radial symmetry of the system results in conservation of angular momentum. Do the numerical methods we use for simulating this system conserve the angular momentum?

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Geometric Numerical Integration is about the design and analysis of methods that can conserve some of these properties exactly or approximately.

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Areas: Lie algebras/groups, exponential splittings, convergence analysis.