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Background
In this project, you will investigate problems related to the
representation of quantum circuits using polynomials. This approach has
been extensively studied in the field of quantum computing, with
foundational contributions from [DHH+05] and further advancements by
[Mon17].
Consider quantum circuits composed of gates such as \(\mathrm{H}\), \(\mathrm{Z}\), \(\mathrm{CZ}\), and \(\mathrm{CCZ}\). In [Mon17], it is
demonstrated that for any \(n\)-qubit
quantum circuit \(C\) containing \(h\) Hadamard gates, there exists a degree-3
polynomial \(f_C\) over the finite
field of order two. This polynomial satisfies the relationship: \[\begin{equation*} \bra{0^n} C \ket{0^n}
= \frac{\mathrm{gap}(f_C)}{2^{n+h/2}}, \end{equation*}\] where
\(\mathrm{gap}(f_C)\) represents the
difference between the number of solutions (\(x\) such that \(f_C(x)=0\)) and the number of non-solutions
(\(f_C(x)=1\)) of the polynomial \(f_C\).
Problems
You are asked to work on two problems: the first is relatively
straightforward, while the second is research-level.
Can you establish a similar result for circuits using the gate
set of \(\mathrm{H}\), \(\mathrm{T}\), and \(\mathrm{CNOT}\)? Additionally, can you
provide general criteria for gate sets that can be represented in a
similar manner?
The transformation from a circuit \(C\) to its corresponding function \(f_C\) is efficient, and each circuit \(C\) uniquely determines \(f_C\). However, multiple distinct circuits
can correspond to the same function \(f\) when given as input. Develop algorithms
to find circuits \(C\) that minimize
the number of qubits required for a given input function \(f\). Alternatively, can you establish any
hardness results for this problem?
The challenge lies in the fact that both \(\mathrm{H}\) and \(\mathrm{CZ}\) gates correspond to quadratic
terms in the function \(f\), making it
unclear whether a term arises from a Hadamard gate or not. The goal is
to maximize the use of Hadamard gates, as this reduces the number of
qubits needed to represent the polynomial.
Ziyuan Wang (zy-wang23@mails.tsinghua.edu.cn)
Longxiang Yuan (ylx23@mails.tsinghua.edu.cn)
Zhengfeng Ji (jizhengfeng@mail.tsinghua.edu.cn)
References
[Mon17] A. Montanaro, Quantum circuits and low-degree polynomials
over F2, J. Phys. A: Math. Theor., vol. 50, no. 8, p. 084002, 2017.
[DHH+05] C. M. Dawson, A. Hines, H. Haselgrove et al, Quantum
computing and polynomial equations over the finite field Z2, Quantum
Info. Comput., vol. 5, no. 2, p. 102-112, 2005.