# Scott Aaronson David Chen

Post on 14-Jan-2016

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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…TRANSCRIPT

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 |0…0 by applying CNOT, Hadamard, and gates only
By 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 n
How 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 state
Obvious approach: Build up the stabilizer group, by repeatedly adding a random generator independent of all the previous generators
Takes O(n4) time—or rather, O(n+1), where 2.376 is the exponent of matrix multiplication
More clever approach: O(n3) time
Our algorithm is a consequence of a new “Atomic Structure Theorem” for stabilizer states…
Theorem: 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 “atom”—which 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 we’ve 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 that’s invertible
Apply the circuit C to | (using [A|B][AC|BC-T])
Choose a random sign (+ or -) for each stabilizer
The running time is dominated by steps 2 and 3, both of which take O(n) time
Open Problems
Find 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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