Improved Simulation of Stabilizer Circuits

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Improved 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)

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Quantum 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?

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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

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Gates 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
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Pauli 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?
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Gottesman-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
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Goal 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
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Our 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

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I 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
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Hadamard 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
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State 00?10?
00
0 0 1 00 1 0 0
ZIIX
Destabilizers
00
1 0 0 00 0 0 1
XIIZ
Stabilizers
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CNOT 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
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State 00?11?
00
0 0 1 00 1 0 0
ZIIX
Destabilizers
00
1 1 0 00 0 1 1
XXZZ
Stabilizers
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Phase 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
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State 00?i11?
00
0 0 1 00 1 0 1
ZIIY
Destabilizers
00
1 1 0 10 0 1 1
XYZZ
Stabilizers
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Measurement 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
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Random 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
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• 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

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CHP An interpreter for quantum assembly
language programs that implements our scoreboard
algorithm
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Performance 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
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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 arent even
universal for classical computation! Simulating
?L with stabilizer circuits Obvious Simulating
stabilizer circuits with ?L Harder
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Fun 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
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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)