Generating Random Stabilizer States in Matrix Multiplication Time: A Theorem in Search of an Application

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Generating Random Stabilizer States in Matrix
Multiplication TimeA Theorem in Search of an
Application

• Scott Aaronson
• David Chen

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Stabilizer States
n-qubit quantum states that can be produced from
00? by applying CNOT, Hadamard, and
gates only
By the celebrated Gottesman-Knill Theorem, such
states are classically describable using 2n2n
bits
The X and Z matrices must satisfy(1) XZT is
symmetric(2) (XZ) (considered as an n?2n matrix)
has rank n
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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) timeor rather, O(n?1),
where ??2.376 is the exponent of matrix
multiplication More clever approach O(n3) time
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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)
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• 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 n?n matrix over GF(2)
  until you find one thats invertible
• Apply the circuit C to ?? (using
  AB?ACBC-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
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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?