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 between--say 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 5-Local Hamiltonian is QMA-complete.
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.
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