Simulation and Design of Stabilizer Quantum Circuits

1
Simulation and Design of Stabilizer Quantum
Circuits
ZIIX
0 0 1 11 1 0 0
?
XXZZ
1 0 0 00 0 0 1

• Scott Aaronson and Boriska Toth
• CS252 Project
• December 10, 2003

2
Quantum Computing New Challenges for Architecture

• If you speculate on a measurement, rollback will
  not happen
• Cache coherence protocols violate no-cloning
  theorem

• How do you design and debug circuits that you
  cant even simulate efficiently?

3
Our Approach Start With A Subset of Quantum
Computations

• Stabilizers (Gottesman 1996) Beautiful formalism
  that captures much (but not all) of quantum
  weirdness
• Quantum linear error-correcting codes
• Teleportation
• Dense quantum coding
• GHZ (Greenberger-Horne-Zeilinger) paradox
• What We Did Invented new algorithms for
  simulating and designing quantum circuits
  described by the stabilizer formalism.
  Implemented and tested an efficient simulator
  with possible practical value.

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Quantum Gates We Allow
1. Controlled-NOT (CNOT) Replaces a,b by a,b?a
00??00?, 01??01?, 10??11?, 11??10?
1 1 1 -1
2. Hadamard Applies /?2 to single
qubit
0??(0?1?)/?21?? (0?-1?)/?2
H
1 0 0 i
3. Phase Applies to single qubit
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 an
abelian group Theorem ?? can be produced from
the all-0 state by just CNOT, Hadamard, and phase
gates, iff ?? is stabilized by 2n tensor
products of Pauli matrices or their opposites
(where n number of qubits)In that case, ?? is
uniquely determined by these stabilizers
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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 as we
discovered when we tried to implement,
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 Solution
Scoreboarding

• Idea Instead of n(2n1) bits, store 2n(4n1)
  bits
• n stabilizers, 2n1 bits each
• n destabilizers
• A 2n?2n scoreboard, that stores how to write
  XIIII,,IIIIX, ZIIII,,IIIIZ as products of the
  stabilizers and destabilizers

Together generate full Pauli group
XI IX ZI IZ
XIIX
1 0 0 00 1 0 0
Initial State00?
Destabilizers
ZIIZ
0 0 1 00 0 0 1
Stabilizers
Scoreboard
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Our Faster, Easier-to-Implement Solution
Scoreboarding

• Idea Instead of n(2n1) bits, store 2n(4n1)
  bits
• n stabilizers, 2n1 bits each
• n destabilizers
• A 2n?2n scoreboard, that stores how to write
  XIIII,,IIIIX, ZIIII,,IIIIZ as products of the
  stabilizers and destabilizers

Together generate full Pauli group
XI IX ZI IZ
ZIIX
0 0 1 00 1 0 0
Hadamard the 1st qubit00?10?
Destabilizers
Swap
XIIZ
1 0 0 00 0 0 1
Stabilizers
Scoreboard
9
Our Faster, Easier-to-Implement Solution
Scoreboarding

• Idea Instead of n(2n1) bits, store 2n(4n1)
  bits
• n stabilizers, 2n1 bits each
• n destabilizers
• A 2n?2n scoreboard, that stores how to write
  XIIII,,IIIIX, ZIIII,,IIIIZ as products of the
  stabilizers and destabilizers

Together generate full Pauli group
XI IX ZI IZ
ZIIX
0 0 1 11 1 0 0
CNOT into the 2nd qubit00?11?
Destabilizers
?
XXZZ
1 0 0 00 0 0 1
Stabilizers
Scoreboard
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Advantages

• Because we force each instruction to tell the
  scoreboard what it did, measuring a state (and
  updating it after the measurement) can be done in
  only O(n2) steps.No Gaussian elimination needed!
• Recently measured observables are automatically
  cachedmeasuring them again takes only O(n)
  steps.

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CHP An interpreter for quantum assembly
language programs that implements our scoreboard
algorithm
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Performance of CHP
650MHz Pentium III, 256MB RAM Compiler
optimizations made it 50 slower!
20000 gates
15000 gates
10000 gates
5000 gates

• Randomly-generated circuits with equal mix of
  CNOT, Hadamard, phase, and measurement gates
• Updating the state after measurements with
  random outcomes dominates the running time.
  Amdahls Law suggests this is what we should
  optimizeand we have ideas!

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Other Stuff We Did

• Proved that any stabilizer quantum circuit can
  be simulated using only CNOT gatesIn theory
  jargon Simulating stabilizer circuits is
  ?L-complete
• Proved that any stabilizer circuit has an
  equivalent circuit with at most O(n2/log n)
  gates, saturating the Shannon lower boundBuilds
  on work by an architecture group at U.
  MichiganK. Patel, I. Markov, and J. Hayes
  (quant-ph/0302002)who showed this for CNOT
  circuits

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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)