Physics Proposal
Quantum Hamiltonian Experiments
Yun Xuan Shi (yunxuan2633@hotmail.com)
Advisor: Dr. Chris Pollett
Description:
My thesis is about quantum computation. The first part of my thesis is about
the simulation of the Schrodinger equation; in this specific problem my matrix is called the Hamiltonian.
In my thesis, I will discuss what causes a Hamiltonian matrix existing in nature to be implementable and simulatable on
a quantum computer. I am interested in the ground state energy of the Hamiltonians. I
am also interested in the degenerate eigen energies of the Hamiltonian and what causes them both
mathematically and physically. The second topic I will discuss is how does the amount of error correcting
codes circuits grow due to the growing number of qubits. So in effect I would like to find the
function that defines the amount of error correcting codes as a function of number of qubits. Right now
the smallest scale is 5 qubits, the largest scale is 50 qubits. I would like to research about implementing
some number in betweensay 20 qubits. I will be writing codes that simulate the quantum concepts
on a classical computer for the experiment section of my project, most likely done in Java.
To actually implement
these algorithms for a large number of qubits will depend on being able to create Hamiltonians
with a certain energy gaps needed to perform the final measurements of these algorithms, and
in which the Hamiltonian affords the necessary unitary operations to carry out the algorithms
steps. For general local Hamiltonians, meeting these constraints is known to be computationally realizable.
Overall this thesis is to affirm that quantum computer is indeed realistic, and that
it is possible to be efficient in energy and circuit complexity.
Quantum Hamiltonian Complexity is concerned with the questions of
whether given a Hamiltonian, are there efficient algorithms for approximating its local
properties, to what degree can the ground state of such a Hamiltonian be represented by
an efficient data structure, and given an efficient approximate data structure to what degree
can we use it to make predictions about future properties of the quantum system.
Schedule:
Week 1:
Feb 20, 2018  Presentation on threshold of quantum computation 
Week 2:
Feb 27, 2018  Presentation on threshold of quantum computation 
Week 3:
Mar 6, 2018  Learn how to use JQuantum 
Week 4:
Mar 13, 2018  Prepare a Tutorial about JQuantum 
Week 5:
Mar 20, 2018  Implement Shor's Algorithm to factor 21 
Week 6:
Mar 27, 2018  Read about Deutsch Josza algorithm 
Week 7:
April 3, 2018  Think about Deutsch Josza algorithm 
Week 8:
April 10, 2018  Code Deutsch Josza algorithm 
Week 9:
April 17, 2018  Code Deutsch Josza algorithm 
Week 10:
April 24, 2018  Presentation on Hamiltonian Paper 
Week 11:
May 1, 2018  Presentation on Hamiltonian Paper 
Week 12:
May 8, 2018  Presentation on Hamiltonian Paper 
Week 13:
May 15, 2018  Presentation on Hamiltonian Paper 
Week 14:
Calender_Date_14  10 page report 
Week 15:
Calender_Date_15  10 page report 
Week 16:
Calender_Date_16  10 page report 
Deliverables:
The full project will be done when the second semester work is completed. The following will
be done by the end of the semester:
1. Give a short presentation on the threshold for quantum computation result. Work out what would be p for Shor's. Then figure out what the size of a Shor's circuit would be if the error value was 10^5.
2. Learn how to use jQuantum. Prepare a short tutorial for Dr. Pollett. Implement Shor's algorithm to factor 21.
3. Code in Java a program that takes simple input functions f(truth table given in file) and computes the Deutsch Josza algorithm for f (up to 5 qubits).
4. Create a presentation on Quantum Hamiltonian Complexity
paper. In particular, work out the reduction for some particular concrete examples used to show 5Local Hamiltonian is QMAcomplete.
5. 10 page report on progress by end of semester
References:
1. [M.Nielsen,I.Chuang2010] Quantum Computation and Quantum Information.
M.A.Nielsen & Issac L.Chuang. Cambridge University Press. 2010.
2. [Gharibian2014] Quantum Hamiltonian Complexity. S.Gharibian, Y. Huang, Z. Landau
and S.W.Shin. Foundation and Trends in Theoretical Computer Science. 2014.
