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I added comments to the sandbox page. My comments are in the grey boxes underlined. I mostly just made some recommendations for clarifying the article, but as always its up to you the final details to include. Good luck :) |
Doodleflip42 (talk | contribs) Added peer review comments. |
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When looking at a set of quantum gates that are exactly universal the complexity of the quantum circuit only varies linearly. When looking at a set of quantum gates that are approximately universal the complexity of the quantum circuit varies by factors that are bounded polylogarithmically.<ref name=":1" />
This section needs more work and clarification.
'''''I agree with the other peer reviewer -- more clarification is needed. What does it mean to "approximately apply quantum gates?" You may also want to insert a parenthetical clarification on what it means for quantum gates to be "exactly universal".'''''
=== Simulating Quantum Circuits ===
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. First the amplitudes associated with the <math>S(n)</math> qubits must be accounted for. Therefore, <math>2^{S(n)}</math>amplitudes must be accounted for with a <math>2^{S(n)}</math> dimensional complex vector which it 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> Next the application of the <math>T(n)</math> quantum gates on <math>2^{S(n)}</math> amplitudes must be accounted for. Therefore, <math>O(T(n))</math> bits of precision will be required required 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. 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 all of the quantum gates, the state vector must be multiplied by a <math>2^{S(n)}\times2^{S(n)}</math> sparse matrix for every quantum gate. 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 can 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.'''''
== Prime Factorization ==
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