I expanded the section on simulating quantum gates; specifing more carefully each part. I also put in more expantation to this section where comments sugested
There is no known way to efficiently simulate a quantum computational model with a classical computer. This means that a classical computer cannot simulate a quantum computational model in polynomial time. However, a [[quantum circuit]] of <math>S(n)</math> qubits with <math>O(T(n))</math> quantum gates can be simulated by a classical circuit with <math>O(2^{S(n)}T(n)^3)</math> [[Logic gate|classical gates]].<ref name=":1">{{Citation|last=Cleve|first=Richard|title=An Introduction to Quantum Complexity Theory|date=2000|url=http://dx.doi.org/10.1142/9789810248185_0004|work=Quantum Computation and Quantum Information Theory|volume=|pages=103–127|publisher=WORLD SCIENTIFIC|isbn=978-981-02-4117-9|access-date=October 10, 2020}}</ref> This number of classical gates is obtained by determining how many bit operations are necessary to simulate the quantum circuit. FirstIn order to do this, first the amplitudes associated with the <math>S(n)</math> qubits must be accounted for. Each of the the states of the <math>S(n)</math> qubits can be described by a two-dimensional complex vector, or a state vector. These state vectors can also be described a [[linear combination]] of its [[Euclidean vector|component vectors]] with coefficients called amplitudes. These amplitudes are complex numbers which are normalized to one, meaning the sum of the squares of the absolute values of the amplitudes must be one.<ref name=":1" /> The entries of the state vector are these amplitudes. Which entry each of the amplitudes are correspond to the none-zero component of the component vector which they are the coefficients of in the linear combination description. As an equation this is described as <math>\alpha \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}</math> or <math>\alpha \left \vert 1 \right \rangle + \beta \left \vert 0 \right \rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}</math> using [[Bra–ket notation|Dirac notation]]. The state of the entire <math>S(n)</math> qubit system can be described by a single state vector. This state vector describing the entire system is the tensor product of the state vectors describing the individual qubits in the system. The result of the tensor products of the <math>S(n)</math> qubits is a single state vector which has <math>2^{S(n)}</math> dimensions and entries that are the amplitudes associated with each basis state or component vector. Therefore, <math>2^{S(n)}</math>amplitudes must be accounted for with a <math>2^{S(n)}</math> dimensional complex vector which itis the state vector for the <math>S(n)</math> qubit system.<ref>{{Cite journal|last=Häner|first=Thomas|last2=Steiger|first2=Damian S.|date=2017-11-12|title=0.5 petabyte simulation of a 45-qubit quantum circuit|url=http://dx.doi.org/10.1145/3126908.3126947|journal=Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis|___location=New York, NY, USA|publisher=ACM|doi=10.1145/3126908.3126947|isbn=978-1-4503-5114-0}}</ref> NextIn order to obtain an upper bound for the applicationnumber of thegates <math>T(n)</math>required to simulate a quantum gatescircuit onwe need a sufficient upper bound for the amount data used to specify the information about each of the <math>2^{S(n)}</math> amplitudes. mustTo bedo accounted for. Therefore,this <math>O(T(n))</math> bits of precision willare be required requiredsufficient for encoding each amplitude.<ref name=":1" /> So it takes <math>O(2^{S(n)}T(n))</math> classical bits to account for the state vector of the <math>S(n)</math> qubit system. Next the application of the <math>T(n)</math> quantum gates on <math>2^{S(n)}</math> amplitudes must be accounted for. The quantum gates can be represented as <math>2^{S(n)}\times2^{S(n)}</math> [[Sparse matrix|sparse matrices]].<ref name=":1" /> So to account for the each application of alleach of the <math>T(n)</math> quantum gates, the state vector must be multiplied by a <math>2^{S(n)}\times2^{S(n)}</math> sparse matrix for everyeach of the <math>T(n)</math> quantum gategates. Every time the state vector is multiplied by a <math>2^{S(n)}\times2^{S(n)}</math> sparse matrix, <math>O(2^{S(n)})</math> arithmetic operations must be preformed.<ref name=":1" /> Therefore, there are <math>O(2^{S(n)}T(n)^2)</math> bit operations for every quantum gate applied to the state vector. So <math>O(2^{S(n)}T(n)^2)</math> classical gate are needed to simulate <math>S(n)</math> qubit circuit with just one quantum gate. Therefore, <math>O(2^{S(n)}T(n)^3)</math> classical gates canare needed to simulate a quantum circuit of <math>S(n)</math> qubits with <math>O(T(n))</math> quantum gates.<ref name=":1" />
<u>I don' know how difficult it is to include graphics, but I think perhaps a graphic here be helpful. Personally, anything I read with just equations is immediately disregarded in my mind. Or perhaps if you could give an example here (or a link to an example) that could help. Would the Deutsch algorithm we discuss in class be applicable?</u>
'''''I am a bit confused about the sentence "Therefore, <math>O(T(n))</math> bits of precision will be required required for encoding each amplitude". What is a bit of precision? I agree with the other peer reviewer's suggestion to include an example, if you can find one. That would certainly clear the idea up. I am sure the math and logic here is sound, but the explanation could be improved. Also, there are some little grammatical errors that need to be resolved.'''''
