A New Scheduling Problem Motivated by Quantum Computation
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Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration
under contract DE-AC04-94AL85000.
A New Scheduling Problem Motivated by Quantum
ComputationRobert CarrAnand Ganti
Cynthia A. PhillipsSandia National Laboratories
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Slide 2
Quantum Computation
Use a machine motivated by quantum mechanics to solve problems that are difficult for traditional computers
Known benefits include faster:• Factoring• Search• Simulating quantum physics
To date, theoretical algorithms and a few early physical experiments
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Slide 3
Sandia National Laboratories Project
• Sandia basic quantum information sciences– Advanced computing architectures– Future engineered systems will require increased
understanding of quantum effects.
• Current three-year project to– Build physical qubit
•Will test current understanding of quantum mechanics
– Design a logical qubit•There are scheduling problems critical for quantum
architecture design
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Slide 4
Quantum Bits
• Classical bits: 0 or 1• Quantum bits (qubits):
• Superposition
• Measurement destroys superposition, makes
0 or 1
0 1
, complex numbers
2 2 1
2 * Probability of finding in state 0
2 * Probability of finding in state 1
0 or 1
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Slide 5
Gates (examples)
1-bit gates:
2-bit gates:
preparation - create 0 or 1
X (not) : X 0 1 , X 1 0
Z : Z 0 0 , Z 1 1
Y : Y 0 i1 , Y 1 i 0measurement
swap : s xy yx
CNOT : c xy c x y x
if x 0 , y unchanged
if x 1 , y flipped
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Slide 6
Quantum Errors
Interaction with environment decoherence
Errors act like X,Y,Z gates
Errors are continuous
X bit flip 0 1 1 0Z phase flip 0 1 0 1Y phase and bit flip
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Slide 7
Quantum Error Correction
• Consider just flip errors• Idea similar to classical error correction
– Encode a single bit with more bits– Define a set of legal codewords– Ensure that all illegal codewords that result from a single
error are closest to unique legal codeword• Simple example:
• Use majority to correct any single flip error.• Real Example Steane [7,3,3], Calderbank-Shor-Steane codes
0 000
1 111
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Slide 8
Quantum Complication 1
• Have to encode as without knowing or .
– Only 2 of the 8 possible states have positive probability• This circuit creates the appropriate (entangled) states:
0 1
000 111
0 1
0
0
000 111}
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Slide 9
Quantum Complication 2
• Measurement destroys information• Ancilla bits
– Interact with real qbits– Pattern of ancilla values encodes single errors uniquely– Measure the ancilla
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Slide 10
Quantum Error Correction
• Critical for quantum computing– Cannot completely isolate qubits from the world (e.g.
components of the computer itself)• Error correction happens often
– Essentially after every operation– Error correction vastly dominates operations
• Error correction is worth doing quickly/well– Throughput– Error threshold
• Burn error correction into silicon, kind of like microcode• The precise nature depends on
– General quantum architecture– Precise code
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Slide 11
Our Architecture: Bilinear Array
Hollenberg et al
Rail
} Gate
Gate entry node
Gate node
Measurement Gate
= location that can hold a qubit/information
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Slide 12
Bilinear Array: Legal Movement
• Move wherever there is an edge, including across gate• Multiple possible transport mechanisms such at CTAP (teleportation)• One edge per step (full to empty)• Bits cannot pass through each other
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Slide 13
Error Correction is a Program
Three types of operations• Single bit• 2 bit• Measurement
PREPAREPLUS 7CNOT(7,9)MEASUREX 8MEASUREZ 9CNOT (0,3)CNOT (3,8)…
} Executed in gates
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Slide 14
Scheduling Problem
• Select initial placement (cyclic)• Schedule location and timing of operations• Schedule legal movements• Obey precedence constraints
– (Usually) two operations that share a bit done serially• Possible parallelism limits
• Minimize makespan• Avoid unnecessary movement
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Slide 15
Example
• 3 encoding bits, 2 ancilla• 4 measurements, 4 CNOTs (2-bit gates)
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Slide 16
Example
m
Step 0
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Slide 17
Example
Step 1m
CNOT
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Slide 18
Example
Step 2
CNOT
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Slide 19
Example
Step 3
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Slide 20
Example
CNOT
Step 4
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Slide 21
Example
CNOT
Step 5
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Slide 22
Example
Step 6
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Slide 23
Example
Step 7
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Slide 24
Example
Step 8
m
m
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Slide 25
Integer Programming Variables
• xbnt, binary, 1 if bit b in node n at start of time t• y(1)
git binary, 1 if 1-bit instruction i executes in gatenode g, time t
• y(2)git binary, 1 if 2-bit instruction i executes at full gate g,
time t• y(2f)
git same as y(2)git but flip control bit top to bottom
• y(m)mit binary, 1 if measurement instruction executes in
measurement gate m at time t• fbvwt implicit binary flow variables. Bit b moves v->w during
time t
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Slide 26
Some simple Special Ordered Sets
• Bit locations (0 is empty)• Performing all operations
xbntn 1, b0,t
xbntb 1, n, t
ygit(m ) 1, i Im
gt
ygit(1)
gt 1, i I1
ygit(2) ygit
(2 f ) gt , i I2
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Slide 27
Movement Control
• Flow conservation• Full->empty• Cyclic
fbuvtu,v E fbvvt xbv,t1 v, t
fbuvtu,v E fbuut xbut u, t
fbuvtb0u,v E
x0vt v, t
xbnT xbn1 b,n
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Slide 28
Precedence Constraints
• 9 sets depending on i,j in I1, I2, Im• = minimum time between operations (usually 1)• Enforce only for nearest neighbors• EST = earliest start time• LAST = last start time
ygjEST j
t
g ygi
EST j
min(LAST (i),t )
g
i, j I1 : i j,EST j t LAST j
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Slide 29
Matching Computation with Transportation
• ci = control bit• di = data bit• g1 = top gatenode of gate g• g2 = bottom gatenode of gate g
ygit(1) xd i gt g,i I1, t
ymit(m ) xd imt m,i Im,t
ygit(2) xci g1t
g,i I2, t
ygit(2) xd i g2t
g,i I2, t
ygit(2 f ) xd i g1t
g,i I2, t
ygit(2 f ) xci g2t
g,i I2,t
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Slide 30
Stronger Transportation/Computation Coupling• If a bit is not in a gatenode at the proper time, none of the
associated gate-firing variables can be 1.
• Over 20x faster
(similar constraints for bottom gates and measurement gates)
ygit(1) ygit
(2)
g=ci
bd i
ygit(2 f )
gd i
xbgt b,g topgates, t
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Slide 31
Objective
Generally none.Can add a relaxation variable z, relaxing all coupling
constraints:
Minimize z
Strange phenomenon: When z is integral, cplex 11 can require 4x as long to solve as when z and y’s are continuous.
When y’s are integral, having no z is better (tiny examples)
ygit(1) ygit
(2)
g=ci
bd i
ygit(2 f )
gd i
xbgt - z b,g topgates,t
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Slide 32
LP cheating
• Half-bits can pass each other
m
m
CNOTCNOT
Steps 0 and 3 Steps 1 and 2
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Slide 33
Comments and Issues
• LP example motivates forcing initial placements– Considerably faster– Have to enumerate over placements
•Need to understand structure• How to determine time? Number of rails
– Recursive doubling– Better to understand/compute bounds– LP time grows quickly with both
• Heuristics– LP based?– Constraint programming?
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Slide 34
Extra Slides
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Slide 35
Error Corrected Logical Qubit
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Slide 36
Example
m
m
CNOT CNOT
Step 0 Step 1 Step 2
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Slide 37
Example
CNOT CNOT
Step 3 Step 4 Step 5
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Slide 38
Example
Step 6 Step 7 Step 8
m
m