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 2n 2 +n bits: The X and Z matrices must satisfy: (1) XZ T is symmetric (2) (XZ) (considered as an n 2n 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(n 4 ) timeor rather, O(n +1 ), where is the exponent of matrix multiplication More clever approach: O(n 3 ) time Our Result: Can generate a random stabilizer state in O(n ) 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 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: 1.Generate a random tensor product | of stabilizer atoms (and weve explicitly calculated the probabilities for each of the poly(n) possible tensor products) 2.Generate a random circuit C of CNOT gates, by repeatedly choosing an n n matrix over GF(2) until you find one thats invertible 3.Apply the circuit C to | (using [A|B] [AC|BC -T ]) 4.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?