Scott Aaronson David Chen

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Generating Random Stabilizer States in Matrix Multiplication Time: A Theorem in Search of an Application. Scott Aaronson David Chen. Stabilizer States. n-qubit quantum states that can be produced from |00 by applying CNOT, Hadamard, and gates only. - PowerPoint PPT Presentation

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Generating Random Stabilizer States in Matrix Multiplication Time:A Theorem in Search of an ApplicationScott AaronsonDavid ChenStabilizer Statesn-qubit quantum states that can be produced from |00 by applying CNOT, Hadamard, and gates onlyBy the celebrated Gottesman-Knill Theorem, such states are classically describable using 2n2+n bits:The X and Z matrices must satisfy: (1) XZT is symmetric (2) (XZ) (considered as an n2n matrix) has rank nHow Would You Generate A classical description of a Uniformly-Random Stabilizer State?Our original motivation: Generating random stabilizer measurements, in order to learn an unknown stabilizer stateObvious approach: Build up the stabilizer group, by repeatedly adding a random generator independent of all the previous generatorsTakes O(n4) timeor rather, O(n+1), where 2.376 is the exponent of matrix multiplicationMore clever approach: O(n3) timeOur algorithm is a consequence of a new Atomic Structure Theorem for stabilizer statesTheorem: Every stabilizer state can be transformed, using CNOT and Pauli gates only, into a tensor product of the following four stabilizer atoms:(And even the fourth atomwhich arises because of a peculiarity of GF(2)can be decomposed into the first three atoms, using the second or third atoms as a catalyst)With the Atomic Structure Theorem in hand, we can easily generate a random stabilizer state as follows:Generate a random tensor product | of stabilizer atoms (and weve explicitly calculated the probabilities for each of the poly(n) possible tensor products)Generate a random circuit C of CNOT gates, by repeatedly choosing an nn matrix over GF(2) until you find one thats invertibleApply the circuit C to | (using [A|B][AC|BC-T])Choose a random sign (+ or -) for each stabilizerThe running time is dominated by steps 2 and 3, both of which take O(n) timeOpen ProblemsFind the killer app for fast generation of random stabilizer states!Find another application for our Atomic Structure Theorem!Is it possible to generate a random invertible matrix over GF(2) (i.e., a random CNOT circuit) in less than n time?******

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