Optimization Algorithms on
a Quantum Computer
A New Paradigm for Technical Computing
Richard H. Warren, PhD
richard.warren@villanova.edu
© Copyright 2011 Lockheed Martin Corporation
1
Lockheed Martin Buys Quantum Computer
In 2011 Lockheed Martin Corp. purchased a
quantum computer from D-Wave Systems, a
Canadian technology firm.
2
Lockheed Martin Buys Quantum Computer
In 2011 Lockheed Martin Corp. purchased a
quantum computer from D-Wave Systems, a
Canadian technology firm.
Lockheed Martin planned uses:
• software validation and verification methods
3
Lockheed Martin Buys Quantum Computer
In 2011 Lockheed Martin Corp. purchased a
quantum computer from D-Wave Systems, a
Canadian technology firm.
Lockheed Martin planned uses:
• software validation and verification methods
• complex systems engineering
4
Lockheed Martin Buys Quantum Computer
In 2011 Lockheed Martin Corp. purchased a
quantum computer from D-Wave Systems, a
Canadian technology firm.
Lockheed Martin planned uses:
• software validation and verification methods
• complex systems engineering
• algorithms and computer science
5
Lockheed Martin Buys Quantum Computer
In 2011 Lockheed Martin Corp. purchased a
quantum computer from D-Wave Systems, a
Canadian technology firm.
Lockheed Martin planned uses:
• software validation and verification methods
• complex systems engineering
• algorithms and computer science
• enhanced machine learning
6
Ising Function for Quantum Annealing
Quantum bits (qubits) can achieve an optimal state of
low energy when super cooled. i and j are qubits
7
Ising Function for Quantum Annealing
Quantum bits (qubits) can achieve an optimal state of
low energy when super cooled. i and j are qubits
Inputs:
hi = the energy bias for qubit i
Jij = the coupling energy between qubits i, j
8
Ising Function for Quantum Annealing
Quantum bits (qubits) can achieve an optimal state of
low energy when super cooled. i and j are qubits
Inputs:
hi = the energy bias for qubit i
Jij = the coupling energy between qubits i, j
Output:
si = the state of qubit i, either 0 or 1
9
Quantum Machine Hardware
• Qubit is a loop of superconducting wire made
of niobium
10
Quantum Machine Hardware
• Qubit is a loop of superconducting wire
• Coupling between qubits is magnetic wiring
11
Quantum Machine Hardware
• Qubit is a loop of superconducting wire
• Coupling between qubits is magnetic wiring
• Temperature is close to 0 degrees kelvin
12
Quantum Machine Hardware
•
•
•
•
Qubit is a loop of superconducting wire
Coupling between qubits is magnetic wiring
Temperature is close to 0 degrees kelvin
Quantum machine has 128 qubits arranged in
blocks of 8 qubits. All qubits in a block are
coupled. Limited coupling between qubits in
different blocks. 32 qubits were inoperative.
– Result: Maximum number of qubits that can be
logically coupled is 13
13
Adiabatic Quantum Optimization
• Physics inspired approach
• Hamiltonian changes over time
– After the computation is complete
• Non-zero probability the system is in its lowest energy
state which encodes a global minimization of the
problem being solved
14
Programming Minimization Problem
1. Convert variables to binary 0, 1
15
Programming Minimization Problem
1. Convert variables to binary 0, 1
2. Design objective function
16
Programming Minimization Problem
1. Convert variables to binary 0, 1
2. Design objective function
3. Convert constraints to penalty functions
17
Programming Minimization Problem
1.
2.
3.
4.
Convert variables to binary 0, 1
Design objective function
Convert constraints to penalty functions
Combine object function & penalty functions
into one expression that has only linear and
quadratic terms
18
Programming Minimization Problem
1.
2.
3.
4.
Convert variables to binary 0, 1
Design objective function
Convert constraints to penalty functions
Combine object function & penalty functions
into one expression that has only linear and
quadratic terms
5. Map #4 to quantum machine hardware
19
Programming Minimization Problem
1.
2.
3.
4.
Convert variables to binary 0, 1
Design objective function
Convert constraints to penalty functions
Combine object function & penalty functions
into one expression that has only linear and
quadratic terms
5. Map #3 to quantum machine hardware
6. Interface to quantum machine through
processor, such as MATLAB or Python
20
Programming Minimization Problem (cont’d)
• si are the variables
– Hardware limited to 13 fully coupled variables
• Hamiltonian = at most 13 X 13 matrix for full coupling
– Diagonal elements are hi
– Off diagonal elements are Jij
21
Symmetric Traveling Salesman Problem (TSP)
A salesman needs to visit n cities once and return to
the starting city. The distance dij between each pair of
cities i, j is know. Symmetric means dij = dji
What is a route that minimizes the distance that the
salesman travels?
22
Symmetric Traveling Salesman Problem (TSP)
A salesman needs to visit n cities once and return to the starting city. The
distance dij between each pair of cities i, j is know. Symmetric means dij = dji
What is a route that minimizes the distance that the salesman travels?
Our binary variables are xij for the road from city i to
city j for i < j.
The number of variables is n(n – 1)/2 which is 10 for 5
cities and 15 for 6 cities.
23
Symmetric Traveling Salesman Problem (TSP)
A salesman needs to visit n cities once and return to the starting city. The
distance dij between each pair of cities i, j is know. Symmetric means dij = dji
What is a route that minimizes the distance that the salesman travels?
Our binary variables are xij for the road from city i to city j for i < j.
The number of variables is n(n – 1)/2 which is 10 for 5 cities and 15 for 6
cities.
Our objective function is Σdijxij for i < j.
24
Symmetric Traveling Salesman Problem (TSP)
A salesman needs to visit n cities once and return to the starting city. The
distance dij between each pair of cities i, j is know. Symmetric means dij = dji
What is a route that minimizes the distance that the salesman travels?
Our binary variables are xij for the road from city i to city j for i < j.
The number of variables is n(n – 1)/2 which is 10 for 5 cities and 15 for 6
cities.
Our objective function is Σdijxij for i < j.
The constraints are xij = xji , xij2 = xij and constraints to
ensure the salesman uses exactly 2 roads for each city,
one into the city and different one out of the city. For
city 1 in a 4 city problem, our constraint is C(x12, x13, x14)
= (x12 + x13 + x14 – 2)2
25
Constraint C(x12, x13, x14) = (x12 + x13 + x14 – 2)2
ensures one route into city 1 and one route out of city 1
• None of the routes x12, x13, x14 are selected
Then C(x12, x13, x14) = (0 + 0 + 0 - 2)2 = 4
26
Constraint C(x12, x13, x14) = (x12 + x13 + x14 – 2)2
ensures 1 route into city 1 and 1 route out of city 1
• None of the routes x12, x13, x14 are selected
Then C(x12, x13, x14) = (0 + 0 + 0 - 2)2 = 4
• One of the routes is selected, say x12
Then C(x12, x13, x14) = (1 + 0 + 0 - 2)2 = 1
27
Constraint C(x12, x13, x14) = (x12 + x13 + x14 – 2)2
ensures 1 route into city 1 and 1 route out of city 1
• None of the routes x12, x13, x14 are selected
Then C(x12, x13, x14) = (0 + 0 + 0 - 2)2 = 4
• One of the routes is selected, say x12
Then C(x12, x13, x14) = (1 + 0 + 0 - 2)2 = 1
• Two of the routes are selected, say x12 and x13
Then C(x12, x13, x14) = (1 + 1 + 0 - 2)2 = 0
28
Constraint C(x12, x13, x14) = (x12 + x13 + x14 – 2)2
ensures 1 route into city 1 and 1 route out of city 1
• None of the routes x12, x13, x14 are selected
Then C(x12, x13, x14) = (0 + 0 + 0 - 2)2 = 4
• One of the routes is selected, say x12
Then C(x12, x13, x14) = (1 + 0 + 0 - 2)2 = 1
• Two of the routes are selected, say x12 and x13
Then C(x12, x13, x14) = (1 + 1 + 0 - 2)2 = 0
• All three of the routes are selected
Then C(x12, x13, x14) = (1 + 1 + 1 - 2)2 = 1
29
Ensuring result is a tour for 4 city TSP
Case for city 1
use C(x12, x13, x14) = (x12 + x13 + x14 – 2)2
• Consider solution (x12 x21) and (x34 x43)
Then C(x12, x13, x14) = (1 + 0 + 0 - 2)2 = 1
30
Ensuring result is a tour for 4 city TSP
Case for city 1
use C(x12, x13, x14) = (x12 + x13 + x14 – 2)2
• Consider solution (x12 x21) and (x34 x43)
Then C(x12, x13, x14) = (1 + 0 + 0 - 2)2 = 1
• Consider solution (x23 x34 x42)
Then C(x12, x13, x14) = (0 + 0 + 0 - 2)2 = 4
31
Ensuring result is a tour for 4 city TSP
Case for city 1
use C(x12, x13, x14) = (x12 + x13 + x14 – 2)2
• Consider solution (x12 x21) and (x34 x43)
Then C(x12, x13, x14) = (1 + 0 + 0 - 2)2 = 1
• Consider solution (x23 x34 x42)
Then C(x12, x13, x14) = (0 + 0 + 0 - 2)2 = 4
• Consider solution (x12 x23 x34 x41)
Then C(x12, x13, x14) = (1 + 0 + 1 - 2)2 = 0
32
Compute Hamiltonian for 4 City TSP
Step 1: Reduce constraint function for each city
For city 1:
Let a = x12 , b = x13 and c = x14
Then (x12 + x13 + x14 – 2)2 = a2 + b2 + c2 + 2(ab
+ bc + ac) – 4(a + b + c) + 4
Applying a2 = a, b2 = b and c2 = c
Then (x12 + x13 + x14 – 2)2 = 2(ab + bc + ac) –
3(a + b + c) + 4
33
Compute Hamiltonian for 4 City TSP
Step 1: Reduce constraint function for each city
For city 1:
Let a = x12 , b = x13 and c = x14
Then (x12 + x13 + x14 – 2)2 = a2 + b2 + c2 + 2(ab + bc +
ac) – 4(a + b + c) + 4
Applying a2 = a, b2 = b and c2 = c
Then (x12 + x13 + x14 – 2)2 = 2(ab + bc + ac) – 3(a + b + c)
+4
Step 2: Add coefficients of linear terms to diagonal
Add coefficients of quadratic terms to off diagonal
34
Compute Hamiltonian for 4 City TSP
Step 1: Reduce constraint function for each city
For city 1:
Let a = x12 , b = x13 and c = x14
Then (x12 + x13 + x14 – 2)2 = a2 + b2 + c2 + 2(ab + bc +
ac) – 4(a + b + c) + 4
Applying a2 = a, b2 = b and c2 = c
Then (x12 + x13 + x14 – 2)2 = 2(ab + bc + ac) – 3(a + b + c)
+4
Step 2: Add coefficients of linear terms to diagonal
Add coefficients of quadratic terms to off diagonal
Step 3: Add coefficients of objective function Σdijxij to
diagonal
35
Compute Hamiltonian for 4 City TSP
Step 1: Reduce constraint function for each city
For city 1:
Let a = x12 , b = x13 and c = x14
Then (x12 + x13 + x14 – 2)2 = a2 + b2 + c2 + 2(ab + bc +
ac) – 4(a + b + c) + 4
Applying a2 = a, b2 = b and c2 = c
Then (x12 + x13 + x14 – 2)2 = 2(ab + bc + ac) – 3(a + b + c)
+4
Step 2: Add coefficients of linear terms to diagonal
Add coefficients of quadratic terms to off diagonal
Step 3: Add coefficients of objective function Σdijxij to
diagonal
Hamiltonian is input to quantum computer
36
Job Shop Problem on One Processor
There are n jobs. Each job has an earliest time it can be started,
latest time it can be finished, length of time to complete it, and
setup time to prepare processor. The number of start times is
small.
A value vj is assigned for completing job j. Maximize the value by
completing jobs.
37
Job Shop Problem on One Processor
There are n jobs. Each job has an earliest time it can be started, latest time it can
be finished, length of time to complete it, and setup time to prepare processor. The
number of start times is small.
A value vj is assigned for completing job j. Maximize the value by completing jobs.
Our binary variables are xjt where j designates a job and
t designates a start time for job j.
38
Job Shop Problem on One Processor
There are n jobs. Each job has an earliest time it can be started, latest time it can
be finished, length of time to complete it, and setup time to prepare processor. The
number of start times is small.
A value vj is assigned for completing job j. Maximize the value by completing jobs.
Our binary variables are xjt where j designates a job and t designates a start time for
job j.
Our objective function is -Σvjxjt over all jobs j and all start
times t for job j.
39
Job Shop Problem on One Processor
There are n jobs. Each job has an earliest time it can be started, latest time it can
be finished, length of time to complete it, and setup time to prepare processor. The
number of start times is small.
A value vj is assigned for completing job j. Maximize the value by completing jobs.
Our binary variables are xjt where j designates a job and t designates a start time for
job j.
Our objective function is -Σvjxjt over all jobs j and all start times t for job j.
Constraints model:
At most one job can start at time t
A job can start at most once
Two jobs cannot be running at the same time.
40
Job Shop Problem on One Processor
There are n jobs. Each job has an earliest time it can be started, latest time it can be
finished, length of time to complete it, and setup time to prepare processor. The
number of start times is small.
A value vj is assigned for completing job j. Maximize the value by completing jobs.
Our binary variables are xjt where j designates a job and t designates a start time for job
j.
Our objective function is -Σvjxjt over all jobs j and all start times t for job j.
Constraints model:
At most one job can start at time t
A job can start at most once
Two jobs cannot be running at the same time.
Constraints introduce new variables that limit job variables xjt to 7
for quantum processor. These are slack variables similar to those
in linear programming.
41
Other Inputs to Quantum Processor
• LaGrange multipliers – balance objective
function and constraints. Very sensitive.
42
Other Inputs to Quantum Processor
• LaGrange multipliers – balance objective
function and constraints. Very sensitive.
• Cooling time – time to allow processor to
reach near 0 degrees kelvin
43
Other Inputs to Quantum Processor
• LaGrange multipliers – balance objective
function and constraints. Very sensitive.
• Cooling time – time to allow processor to
reach near 0 degrees kelvin
• Annealing time – time to allow processor to
reach minimum energy state, after cooled
44
Other Inputs to Quantum Processor
• LaGrange multipliers – balance objective function
and constraints. Very sensitive.
• Cooling time – time to allow processor to reach
near 0 degrees kelvin
• Annealing time – time to allow processor to reach
minimum energy state, after cooled
• Iterations – number of times same problem is
sampled. We found optimal 99% of time for 100
iterations of 4 city TSP.
45
Other Inputs to Quantum Processor
• LaGrange multipliers – balance objective function
and constraints. Very sensitive.
• Cooling time – time to allow processor to reach
near 0 degrees kelvin
• Annealing time – time to allow processor to reach
minimum energy state, after cooled
• Iterations – number of times same problem is
sampled. We found optimal 99% of time for 100
iterations of 4 city TSP.
• Loads – number of times to iterate
46
Advantage of Quantum Annealing
• Speed of solving intractable problems
– Experimental evidence shows several orders of
magnitude faster than current digital computers
– Not known whether polynomial or exponential
47
Limitations of Quantum Annealing
• Limited number of qubits restricts number of
variables hi
• Limited connectivity between qubits restricts
which parameters Jij can be nonzero
• Precision affected by temperature and analog
nature of parameters hi and Jij in hardware
Techniques
Limitations
Techniquesthat
thatOvercome
Mitigate Limitations
48
Comparison to Linear Programming
Linear Programming
•
•
•
•
•
Optimization technique
Linear objective function
Variables ≥ 0
Linear constraints
Solution - Simplex Method
Quantum Computing
•
•
•
•
•
Optimization technique
Quadratic objective function
Variables are 0 or 1
Quadratic constraints
Solution - Adiabatic Annealing
49
Comparison to Linear Programming
Linear Programming
•
•
•
•
•
•
Optimization technique
Linear objective function
Variables ≥ 0
Linear constraints
Solution - Simplex Method
Runs in polynomial time
Quantum Computing
•
•
•
•
•
•
Optimization technique
Quadratic objective function
Variables are 0 or 1
Quadratic constraints
Solution - Adiabatic Annealing
Extremely fast on intractable
problems
50
Comparison to Linear Programming
Linear Programming
•
•
•
•
•
•
Optimization technique
Linear objective function
Variables ≥ 0
Linear constraints
Solution - Simplex Method
Runs in polynomial time
• Optimal solution from 1
execution
Quantum Computing
•
•
•
•
•
•
Optimization technique
Quadratic objective function
Variables are 0 or 1
Quadratic constraints
Solution - Adiabatic Annealing
Extremely fast on intractable
problems
• Near optimal solution from
many executions
51
References
1. D. Aharonov et al., Adiabatic quantum computation is equivalent to
standard quantum computation, Proceedings of the 45th Annual IEEE
Symposium on Foundations of Computer Science, 42-51, (2004).
2. G. A. Kochenberger et al., A unified modeling and solution framework for
combinatorial optimization problems, OR Spectrum, 26, 237-250, (2004)
3. K. Karimi and 8 other D-Wave employees, Investigating the performance of
an adiabatic quantum optimization processor, Quantum Information
Processing, published on line, (2011)
52
Summary
• Inputs to adiabatic quantum computer are
integers that are interpreted as analog
• Adiabatic quantum computer optimizes all
solutions simultaneously
• Output from adiabatic quantum computer is
0, 1 assignment to variables with extremely
high probability of being a minimization
53
Terminology
• Adiabatic – describes a process in which there
is no loss or gain of heat
• Quantum annealing – hardware is cooled to a
very low temperature which approximates the
minimum energy function in the Ising model
54