Download presentation

Presentation is loading. Please wait.

Published byNatalie Stewart Modified over 4 years ago

1
Generating Random Stabilizer States in Matrix Multiplication Time: A Theorem in Search of an Application Scott Aaronson David Chen

2
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

3
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 2.376 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

4
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)

5
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

6
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?

Similar presentations

OK

Intro to Quantum Algorithms SUNY Polytechnic Institute Chen-Fu Chiang Fall 2015.

Intro to Quantum Algorithms SUNY Polytechnic Institute Chen-Fu Chiang Fall 2015.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google