Title: Improved Simulation of Stabilizer Circuits
1Improved Simulation of Stabilizer Circuits
00
0 0 1 00 1 0 0
ZIIX
00
1 0 0 00 0 0 1
XIIZ
- Scott Aaronson (UC Berkeley)
- Joint work with Daniel Gottesman (Perimeter)
2Quantum ComputingNew Challenges for Computer
Architecture
- Cant speculate on a measurement and then roll
back - Cache coherence protocols violate no-cloning
theorem
- How do you design and debug circuits that you
cant even simulate efficiently with existing
tools?
3Our 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
4Gates Allowed In Stabilizer Circuits
1. Controlled-NOT
00??00?, 01??01?, 10??11?, 11??10?
2. Hadamard
0??(0?1?)/?21?? (0?-1?)/?2
H
1 0 0 i
3. Phase
0??0?, 1??i1?
P
4. Measurement of a single qubit
5Pauli Matrices Collect Em All
1 0 0 1
1 0 0 -1
0 1 1 0
0 -i i 0
I
Z
X
Y
X2Y2Z2I XYiZ YZiX ZXiY XZ-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?1? Y stabilizes 0?i1? -Y stabilizes
0?-i1? Z stabilizes 0? -Z stabilizes 1?
6Gottesman-Knill Theorem
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
7Goal Using a classical computer, simulate an
n-qubit CNOT/Hadamard/Phase computer. Gottesman
Knills solution Keep track of n generators of
the stabilizer groupEach generator uses 2n1
bits 2 for each Pauli matrix and 1 for the sign.
So n(2n1) bits total Example But
measurement takes O(n3) steps by Gaussian
elimination
CNOT(1?2)
01?11?
01?10?
XX-ZZ
XI-IZ
Updating stabilizers takes only O(n) steps
8Our Faster, Easier-To-Implement Tableau Algorithm
- Idea Instead of n(2n1) bits, store 2n(2n1)
bits - n stabilizers S1,,Sn, 2n1 bits each
- n destabilizers D1,,Dn
Together generate full Pauli group
- Maintain the following invariants
- Dis commute with each other
- Si anticommutes with Di
- Si commutes with Dj for i?j
9I xij0, zij0 phase ri0X xij1, zij0 -
phase ri1Y xij1, zij1Z xij0, zij1
State 00?
xij bits
zij bits
ri bits
00
1 0 0 00 1 0 0
XIIX
D1D2
Destabilizers
00
0 0 1 00 0 0 1
ZIIZ
S1S2
Stabilizers
10Hadamard on qubit a For all i?1,,2n, swap
xia with zia, and setri ri ? xiazia
State 00?
00
1 0 0 00 1 0 0
XIIX
Destabilizers
00
0 0 1 00 0 0 1
ZIIZ
Stabilizers
11State 00?10?
00
0 0 1 00 1 0 0
ZIIX
Destabilizers
00
1 0 0 00 0 0 1
XIIZ
Stabilizers
12CNOT from qubit a to qubit b For all
i?1,,2n, set xib xib ? xia andzia zia ?
zib
State 00?10?
00
0 0 1 00 1 0 0
ZIIX
Destabilizers
00
1 0 0 00 0 0 1
XIIZ
Stabilizers
13State 00?11?
00
0 0 1 00 1 0 0
ZIIX
Destabilizers
00
1 1 0 00 0 1 1
XXZZ
Stabilizers
14Phase on qubit a For all i?1,,2n, set ri
ri ? xiazia, then setzia zia ? xia
State 00?11?
00
0 0 1 00 1 0 0
ZIIX
Destabilizers
00
1 1 0 00 0 1 1
XXZZ
Stabilizers
15State 00?i11?
00
0 0 1 00 1 0 1
ZIIY
Destabilizers
00
1 1 0 10 0 1 1
XYZZ
Stabilizers
16Measurement of qubit a If xia0 for all
i?n1,,2n, then outcome will be deterministic.
Otherwise 0 with ½ probability and 1 with ½
probability.
State 00?i11?
00
0 0 1 00 1 0 1
ZIIY
Destabilizers
00
1 1 0 10 0 1 1
XYZZ
Stabilizers
17Random outcomePick a stabilizer Si such that
xia1 and set DiSi. Then set SiZa 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 dont commute with Si by Di
State 11?
00
1 1 0 10 1 0 1
XYIY
Destabilizers
10
0 0 1 00 0 1 1
-ZIZZ
Stabilizers
18- 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 cis, then by summing corresponding
Shs we learn sign of Za. Now - So just have to check if Di commutes with Za, or
equivalently if xia1
19CHP An interpreter for quantum assembly
language programs that implements our scoreboard
algorithm
20Performance of CHP
Average time needed to simulate a measurement
after applying ßnlogn random unitary gates to n
qubits, on a 650MHz Pentium III with 256MB RAM
21Simulating 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 arent even
universal for classical computation! Simulating
?L with stabilizer circuits Obvious Simulating
stabilizer circuits with ?L Harder
22Fun With Stabilizers
How many n-qubit stabilizer states are there?
Can we represent mixed states in the stabilizer
formalism?
YES
Can we efficiently compute the inner product
between two stabilizer states?
YES
23Future 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)