Improved Simulation of Stabilizer Circuits

Scott Aaronson (UC Berkeley)

Joint work with Daniel Gottesman (Perimeter)

00

00

0 0 1 00 1 0 01 0 0 00 0 0 1

+ZI+IX+XI+IZ

Quantum Computing:New Challenges for Computer

Architecture

• Can’t speculate on a measurement and then roll back

• Cache coherence protocols violate no-cloning theorem

• How do you design and debug circuits that you can’t even simulate efficiently with existing tools?

Our Approach: Start With A Subset of Quantum Computations

Stabilizers (Gottesman 1996): Beautiful formalism that captures much (but not all) of quantum weirdness

– Linear error-correcting codes– Teleportation– Dense quantum coding– GHZ (Greenberger-Horne-Zeilinger) paradox

Gates Allowed In Stabilizer Circuits1. Controlled-NOT

|00|00, |01|01, |10|11, |11|10

2. Hadamard |0(|0+|1)/2|1 (|0-|1)/2

H

3. Phase = |0|0, |1i|1

P

1 0

0 i

4. Measurement of a single qubit

AMAZING FACTThese gates are NOT

universal—Gottesman & Knill showed how to

simulate them quickly on a classical computer

To see why we need some group theory…

X2=Y2=Z2=I XY=iZ YZ=iX ZX=iYXZ=-iY ZY=-iX YX=-iZ

Unitary matrix U stabilizes a quantum state | if U| = |. Stabilizers of | form a group

X stabilizes |0+|1 -X stabilizes |0+|1Y stabilizes |0+i|1 -Y stabilizes |0-i|1Z stabilizes |0 -Z stabilizes |1

1 0

0 1

1 0

0 -1I = Z =

0 1

1 0X =

0 -i

i 0Y =

Pauli Matrices: Collect ‘Em All

If | can be produced from the all-0 state by just CNOT, Hadamard, and phase gates, then | is stabilized by 2n tensor products of Pauli matrices or their opposites (where n = number of qubits)

So the stabilizer group is generated by log(2n)=n such tensor products

Indeed, | is then uniquely determined by these generators, so we call | a stabilizer state

Gottesman-Knill Theorem

Goal: Using a classical computer, simulate an n-qubit CNOT/Hadamard/Phase computer.

Gottesman & Knill’s solution: Keep track of n generators of the stabilizer groupEach generator uses 2n+1 bits: 2 for each Pauli matrix and 1 for the sign. So n(2n+1) bits total

Example:

But measurement takes O(n3) steps by Gaussian elimination

+XX-ZZ

|01+|11 |01+|10CNOT(12)

+XI-IZ

Updating stabilizers takes only O(n) steps

OUCH!

Our Faster, Easier-To-Implement Tableau Algorithm

Idea: Instead of n(2n+1) bits, store 2n(2n+1) bits

• n stabilizers S1,…,Sn, 2n+1 bits each

• n “destabilizers” D1,…,Dn

Together generate full Pauli group

Maintain the following invariants:

• Di’s commute with each other

• Si anticommutes with Di

• Si commutes with Dj for ij

00

00

1 0 0 00 1 0 00 0 1 00 0 0 1

Destabilizers

Stabilizers

State: |00

+XI+IX+ZI+IZ

xij bits zij bits ri bits

I: xij=0, zij=0 + phase: ri=0X: xij=1, zij=0 - phase: ri=1Y: xij=1, zij=1Z: xij=0, zij=1

S1

S2

D1

D2

00

00

1 0 0 00 1 0 00 0 1 00 0 0 1

Destabilizers

Stabilizers

State: |00

+XI+IX+ZI+IZ

Hadamard on qubit a:

For all i{1,…,2n}, swap xia with zia, and setri := ri xiazia

00

00

0 0 1 00 1 0 01 0 0 00 0 0 1

Destabilizers

Stabilizers

State: |00+|10

+ZI+IX+XI+IZ

00

00

0 0 1 00 1 0 01 0 0 00 0 0 1

Destabilizers

Stabilizers

State: |00+|10

+ZI+IX+XI+IZ

CNOT from qubit a to qubit b:

For all i{1,…,2n}, set xib := xib xia andzia := zia zib

00

00

0 0 1 00 1 0 01 1 0 00 0 1 1

Destabilizers

Stabilizers

State: |00+|11

+ZI+IX+XX+ZZ

00

00

0 0 1 00 1 0 01 1 0 00 0 1 1

Destabilizers

Stabilizers

State: |00+|11

+ZI+IX+XX+ZZ

Phase on qubit a:

For all i{1,…,2n}, set ri := ri xiazia, then setzia := zia xia

00

00

0 0 1 00 1 0 11 1 0 10 0 1 1

Destabilizers

Stabilizers

State: |00+i|11

+ZI+IY+XY+ZZ

00

00

0 0 1 00 1 0 11 1 0 10 0 1 1

Destabilizers

Stabilizers

State: |00+i|11

+ZI+IY+XY+ZZ

Measurement of qubit a:

If xia=0 for all i{n+1,…,2n}, then outcome will be deterministic. Otherwise 0 with ½ probability and 1 with ½ probability.

10

00

1 1 0 10 1 0 10 0 1 00 0 1 1

Destabilizers

Stabilizers

State: |11

+XY+IY-ZI+ZZ

Random outcome:Pick a stabilizer Si such that xia=1 and set Di:=Si. Then set Si:=Za and output 0 with ½ probability, and set Si:=-Za and output with ½ probability, where Za is Z on ath qubit and I elsewhere. Finally, left-multiply whatever rows don’t commute with Si by Di

Novel part: How to obtain deterministic measurement outcomes in only O(n2) steps, without using Gaussian elimination?

Za must commute with stabilizer, so

for a unique choice of c1,…,cn{0,1}. If we can determine ci’s, then by summing corresponding Sh’s we learn sign of Za. Now

So just have to check if Di commutes with Za, or equivalently if xia=1

1

n

h h ah

c S Z

1 1

mod2n n

i h i h i h h i ah h

c c D S D c S D Z

CHP: An interpreter for “quantum assembly language” programs that implements our

scoreboard algorithm

Example: Quantum Teleportation

H

H

H H

|

|0

|0

|0

|0

|

Alic

e’s

Qu

bit

sB

ob

’s Q

ub

its

Prepare EPR pair

Alice’s part Bob’s part

h 1c 1 2c 0 1h 0m 0m 1c 0 3c 1 4c 4 2h 2c 3 2h 2

CHP Code

Performance of CHP

Average time needed to simulate a measurement after applying βnlogn random unitary gates to n qubits, on a 650MHz Pentium III with 256MB RAM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

200 600 1000 1400 1800 2200 2600 3000

Number of qubits n

Se

co

nd

s p

er

me

as

ure

me

nt

β=1.2

β=1.1

β=1.0

β=0.9

β=0.8

β=0.7

β=0.6

Simulating Stabilizer Circuits is L-Complete

L = class of problems reducible in logarithmic space to evaluating a circuit of CNOT gates

Natural, well-studied subclass of P

Conjectured that L P. So our result means stabilizer circuits probably aren’t even universal for classical computation!

Simulating L with stabilizer circuits: Obvious

Simulating stabilizer circuits with L : Harder

How many n-qubit stabilizer states are there?

Fun With Stabilizers

211/ 2 1

0

2 2 1 2n

o nn n k

k

Can we represent mixed states in the stabilizer formalism?

YES

Can we efficiently compute the inner product between two stabilizer states?

YES

Future Directions• Measurements (at least some) in O(n) steps?

• Apply CHP to quantum error-correction, studying conjectures about entanglement in many-qubit systems…

• Efficient minimization of stabilizer circuits?

• Superlinear lower bounds on stabilizer circuit size?

• Other quantum computations with efficient classical simulations: bounded entanglement (Vidal 2003), “matchgates” (Valiant 2001)…