124 lines
6.8 KiB
TeX
124 lines
6.8 KiB
TeX
% vim: ft=tex
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\chapter{Conclusion and Outlook}
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As seen in \ref{ref:performance} simulation using stabilizers is exponentially
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faster than simulating with dense state vectors. The graphical representation
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for stabilizer states is in realistic cases more efficiently than the
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stabilizer tableaux \cite{andersbriegel2005}. In particular one can simulate
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more qbits while only applying Clifford gates.
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This is considerably useful when working on quantum error correcting strategies
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as they often include many qbits; the smallest quantum error correcting
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stabilizer code requires $5$ qbits to encode one logical qbit
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\cite{nielsen_chuang_2010}. Several layers of data encoding increase the
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number of required qbits exponentially.
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Simulating in the stabilizer formalism is uninteresting from a physical
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point of view as basically no physically interesting simulations can be
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performed: As shown in \ref{ref:meas_stab} probability amplitudes have to be
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$0, \frac{1}{2}, 1$; this leaves very few points in time that could be
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simulated by applying a transfer matrix. Algorithms like the quantum fourier
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transform also require non-Clifford gates for qbit counts $n \neq 2, 4$.
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The basic idea of not simulating a state but (after imposing some conditions on
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the Hilbert space) other objects that describe the state is
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interesting for physics as often the exponentially large or infinitely large
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Hilbert spaces cannot be mapped to a classical (super) computer. One key idea
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to take from the stabilizer formalism is to simulate the Hamiltonian instead of
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the state:
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\begin{equation} H := -\sum\limits_{S^{(i)}} S^{(i)} \end{equation}
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The stabilizer state $\ket{\psi}$ as defined in \ref{ref:stab_states} is the
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ground state of this Hamiltonian.
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While trying to extend the stabilizer formalism one inevitably hits the
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question:\\ \textit{Why is there a constraint on the $R_\phi$ angle? Why is it
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$\frac{\pi}{2}$?}\\ The answer to this question can be found by taking a look
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at the Clifford group. Recalling Definition \ref{def:clifford_group} the
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Clifford group is not defined to be generated by $H, S, CZ$, but by its property
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of normalizing the multilocal Pauli group. Storing and manipulating the
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multilocal Pauli group is only so efficient (or possible) because it is the
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tensor product of Pauli matrices. A general unitary on $n$ qbits would be
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a $2^{n} \times 2^{n}$ matrix which requires more space than a dense state
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vector. The Clifford group is a group preserving this tensor product property.
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%{{{
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%When lifting the constraint that $\ket{\varphi}$ is stabilized by the multilocal Pauli
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%group but using $n$ arbitrary commuting hermitians $\langle h_1, ..., h_n
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%\rangle$ that are the tensor product of $2\times 2$ hermitians one quickly
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%realizes that one could apply any single-qbit gate to the $h_i$ and preserve
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%the tensor product property. Applying the $CX$ gate however will break this
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%property in general.
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%
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%Writing $h_j = \bigotimes\limits_{i=1}^{n} h_{j,i}$,
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%$A := \left(\bigotimes\limits_{l<j} I\right)$
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%and $B := \left(\bigotimes\limits_{l>i} I\right)$ this can be seen easily
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%by transforming a general $h_k$ with $CX_{i,j}$, $i = j+1$:
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%
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%
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%\begin{equation}
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%\begin{aligned}
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% CX_{i,j} h_k CX_{i,j}^\dagger &= \left( A\otimes |1\rangle\langle 1| \otimes X \otimes B
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% + A \otimes |0\rangle\langle 0| \otimes I \otimes B\right)\\
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% &h_k\\
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% &\left(A \otimes |1\rangle\langle 1| \otimes X \otimes B
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% + A \otimes |0\rangle\langle 0| \otimes I \otimes B\right) \\
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% &= h_{k,A} \otimes h_{k,j,11} |1\rangle\langle 1| \otimes Xh_{k,i}X \otimes h_{k,B}\\
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% &+ h_{k,A} \otimes h_{k,j,00}|0\rangle\langle 0| \otimes Ih_{k,i}I \otimes h_{k,B}\\
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% &+ h_{k,A} \otimes h_{k,j,01}|0\rangle\langle 1| \otimes Ih_{k,i}X\otimes h_{k,B}\\
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% &+ h_{k,A} \otimes h_{k,j,10}|1\rangle\langle 0| \otimes Xh_{k,i}I\otimes h_{k,B}\\
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%\end{aligned}
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%\end{equation}
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%
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%Searching for hermitians $h_1, h_2$ that fulfill
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%
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%\begin{equation}
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%CX_{1,2} (h_1 \otimes h_2) CX_{1,2} = h_1' \otimes h_2'
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%\end{equation}
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%
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%and
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%
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%\begin{equation}
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%CX_{2,1} (h_1 \otimes h_2) CX_{2,1} = h_1'' \otimes h_2''
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%\end{equation}
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%
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%might be a good step to find new classes of states that can be simulated
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%efficiently using this method. This property has to be fulfilled by all
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%elements of a group generated by such hermitian matrices. How computations and
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%measurements would work using this method is not clear at the moment as many
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%basic properties of the stabilizers are lost. One important property is that the
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%stabilization: The simulated state is the $+1$ eigenstate of the stabilizers.
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%This is another property that will have to be fulfilled by the hermitians as it
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%is a key property used in \ref{ref:dynamics_stabilizer}. To ensure that the
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%state is well defined one will have to demand that the eigenvalues fulfill
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%$\lambda_1 = 1$ and $\lambda_2 < 1$.
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%
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%One should also note that the sabilizer states do not form a Hilbert (sub)space.
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%Linear combinations of stabilizer states are (in general) no stabilizer states.
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%The superposition principle is quite essential to many quantum algorithms and
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%quantum physics which limits the use of the stabilizer formalism drastically.
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%}}}
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The stabilizer formalism as introduced in \ref{ref:stab_states} has since been
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generalized to normalizers of a finite Abelian group over the Hilbert space
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\cite{bermejovega_lin_vdnest2015}\cite{bermejovega_vdnest2018}\cite{vandennest2019}\cite{vandennest2018}.
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This allows to simulate more classes of circuits efficiently on classical
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computers including the Quantum Fourier Transforms which is often believed to
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be responsible for exponential speedups. One must note that in the definition
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of the QFTs as in \cite{vandennest2018} the QFT depends on the Abelian group it
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acts on. In particular the QFT on the group that decomposes the Hilbert space
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as seen in this paper ($Z_2^n$) is just the tensor product of the $H$ gates.
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The QFT as used in \ref{ref:quantum_algorithms} for the phase estimation
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however normalizes the group $Z_{2^n}$ \cite{vandennest2018}. This yields two
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interesting results: It is not the tensor product property of the multilocal
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Pauli group that makes computations efficient but the normalization property of
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the Clifford group \cite{vandennest2018}.
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The exponential speedup of quantum computing is often attributed to
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entanglement, superposition and interference effects
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\cite{uwaterloo}\cite{21732}\cite{vandennest2018}. Stabilizer states show
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entanglement, superposition and interference effects; as do computations using
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general normalizers \cite{vandennest2018}. The question why quantum computing
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can speed up computations exponentially is non-trivial.
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