Transcript The Learnability of Quantum States

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  1 0  gates only
0

i 
By the celebrated Gottesman-Knill Theorem, such states
are classically describable using 2n2+n bits:
 XII
  1 0 0  0 0 0 



0 1
00  11

  IXX    0 1 1  0 0 0 
2
2
 IZZ
  0 0 0  0 1 1 
 
X
The X and Z matrices must satisfy:
(1) XZT is symmetric
(2) (XZ) (considered as an n2n matrix) has rank n
Z
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 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”:
0,
0 1
2
,
0 i 1
2
,
00  01  10  11
2
(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 nn 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 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?