Generating Random Stabilizer States in Matrix Multiplication Time:

A Theorem in Search of an Application

Scott AaronsonDavid Chen

Stabilizer Statesn-qubit quantum states that can be produced from |0…0 by applying CNOT, Hadamard, and gates only

i0

01

By the celebrated Gottesman-Knill Theorem, such states are classically describable using 2n2+n bits:

ZX

IZZ

IXX

XII

110

000

000

000

110

001

2

1100

2

10

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) timeOur 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”:

2

11100100,

2

10,

2

10,0

i

(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:

1.Generate a random tensor product | of stabilizer atoms (and we’ve 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 nn matrix over GF(2) until you find one that’s 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 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?