markowitz portfolio optimization with a quantum annealer · 31 conclusions • quantum annealing is...
TRANSCRIPT
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ORNL is managed by UT-Battelle, LLC for the US Department of Energy
Markowitz Portfolio Optimization with a Quantum Annealer
Erica Grant, Travis Humble
Oak Ridge National Laboratory
University of Tennessee
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Portfolio Selection
Goals:• Maximize returns
• Minimize risk
• Stay within budget
Inputs:• Uniform random
historical price data
• Budget
• Risk tolerance
Output:• A portfolio
representing a list of investments and the expected return Investment Price Expected
Return/Day1= buy0 = pass
Stock A $150 + $0.10 0Stock B $200 + $0.25 1Stock C $250 + $0.50 0
Budget: $200 Risk Tolerance: Low
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Portfolio Selection as Binary Integer Programming
𝑥 𝜖 {0, 1}
𝑖 = asset number
𝑝+= price
𝑟+ = expected return
𝑏 = budget
max1
∑+ 𝑟+𝑥+
𝑠. 𝑡. ∑+ 𝑝+𝑥+ ≤ 𝑏
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Unconstrained Markowitz Formulation𝑚𝑎𝑥1
𝑓(𝑥)
𝑓 𝑥 = 𝜃=>+
𝑥+𝑟++𝑥+ − 𝜃@ >+
𝑥+𝑝+𝑥+ − 𝑏@
− 𝜃A>+,B
𝑥+𝑐𝑜𝑣 ℎ+, ℎB 𝑥B
• 𝑏 = 𝑏𝑢𝑑𝑔𝑒𝑡, 𝑝+ = 𝑝𝑟𝑖𝑐𝑒, 𝑟+= 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛, 𝑥+ 𝜖 0, 1
• Weights: 𝜃=, 𝜃@, 𝜃A ≥ 0 s.t. 𝜃= + 𝜃@ + 𝜃A = 1
• ℎ+ = vector of historical price data
• 𝑐𝑜𝑣 ℎ+, ℎB =∑NOPQ (RS,NTURS)(RV,NTRV)
WT=where m= number of price points
𝒙𝒊 = 𝟏 → 𝒃𝒖𝒚 𝒙𝒊 = 𝟎 → 𝒅𝒐𝒏c𝒕 𝒃𝒖𝒚
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Unconstrained Markowitz Formulation𝑚𝑎𝑥1
𝑓(𝑥)
𝑓 𝑥 = 𝜃=>+
𝑥+𝑟++𝑥+ − 𝜃@>+
𝑥+𝑝+𝑥+ − 𝑏 @ − 𝜃A>+,B
𝑥+𝑐𝑜𝑣 ℎ+, ℎB 𝑥B
• 𝑏 = 𝑏𝑢𝑑𝑔𝑒𝑡, 𝑝+ = 𝑝𝑟𝑖𝑐𝑒, 𝑟+= 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛, 𝑥+ 𝜖 0, 1
• Weights: 𝜃=, 𝜃@, 𝜃A ≥ 0 s.t. 𝜃= + 𝜃@ + 𝜃A = 1
• ℎ+ = vector of historical price data
• 𝑐𝑜𝑣 ℎ+, ℎB =∑NOPe (RS,NTURS)(RV,NTRV)
fwhere 𝑛 = #of assets
𝒙𝒊 = 𝟏 → 𝒃𝒖𝒚 𝒙𝒊 = 𝟎 → 𝒅𝒐𝒏c𝒕 𝒃𝒖𝒚
Expected Return
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Unconstrained Markowitz Formulation𝑚𝑎𝑥1
𝑓(𝑥)
𝑓 𝑥 = 𝜃=>+
𝑥+𝑟++𝑥+ − 𝜃@>+
𝑥+𝑝+𝑥+ − 𝑏 @ − 𝜃A>+,B
𝑥+𝑐𝑜𝑣 ℎ+, ℎB 𝑥B
• 𝑏 = 𝑏𝑢𝑑𝑔𝑒𝑡, 𝑝+ = 𝑝𝑟𝑖𝑐𝑒, 𝑟+= 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛, 𝑥+ 𝜖 0, 1
• Weights: 𝜃=, 𝜃@, 𝜃A ≥ 0 s.t. 𝜃= + 𝜃@ + 𝜃A = 1
• ℎ+ = vector of historical price data
• 𝑐𝑜𝑣 ℎ+, ℎB =∑NOPe (RS,NTURS)(RV,NTRV)
fwhere 𝑛 = #of assets
𝒙𝒊 = 𝟏 → 𝒃𝒖𝒚 𝒙𝒊 = 𝟎 → 𝒅𝒐𝒏c𝒕 𝒃𝒖𝒚
Budget Penalty
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Unconstrained Markowitz Formulation𝑚𝑎𝑥1
𝑓(𝑥)
𝑓 𝑥 = 𝜃=>+
𝑥+𝑟++𝑥+ − 𝜃@>+
𝑥+𝑝+𝑥+ − 𝑏 @ − 𝜃A>+,B
𝑥+𝑐𝑜𝑣 ℎ+, ℎB 𝑥B
• 𝑏 = 𝑏𝑢𝑑𝑔𝑒𝑡, 𝑝+ = 𝑝𝑟𝑖𝑐𝑒, 𝑟+= 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛, 𝑥+ 𝜖 0, 1
• Weights: 𝜃=, 𝜃@, 𝜃A ≥ 0 s.t. 𝜃= + 𝜃@ + 𝜃A = 1
• ℎ+ = vector of historical price data
• 𝑐𝑜𝑣 ℎ+, ℎB =∑NOPe (RS,NTURS)(RV,NTRV)
fwhere 𝑛 = #of assets
𝒙𝒊 = 𝟏 → 𝒃𝒖𝒚 𝒙𝒊 = 𝟎 → 𝒅𝒐𝒏c𝒕 𝒃𝒖𝒚
Diversification
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Fitting to Unconstrained Problem
• Assets can be divided into any desired fraction based on the budget.
• Normalize purchase price to the budget.
• Use Binary Fractional series: fractions=12q,12=,12@, … ,
12f Special thanks to Benjamin Stump
for the binary fractional series idea
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Quantum Annealing for Unstructured Search
• Quantum annealing is a model of quantum computing tailored to unconstrained search
• QA operates by preparing a uniform superposition of all possible binary combinations.
• The initial state is then evolved toward a Hamiltonian that defines the objective function.
• Measurement samples the prepared probability distribution, which should concentrate at the global extrema.
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Quantum Annealing for Unstructured Search
• D-Wave Systems provides a fourth-generation quantum annealer, 2000Q– Programmable superconducting integrated circuit– 2048-qubit register in a 2D Chimera layout– EM shielding, UHV, cooled to 14mK
• This is a special-purpose optimization solver that finds the energetic minimum of an Ising model– Search uses quantum annealing to find minima
• Non-zero temperature, flux noise, and other fluctuations complicate device physics
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QUBO to Quantum Ising Hamiltonian
𝑓 𝑥 = −𝜃=>+
𝑥+𝑟++𝑥+ + 𝜃@>+
𝑥+𝑝+𝑥+ − 𝑏 @ + 𝜃A>+,B
𝑥+𝑐𝑜𝑣 𝑝+, 𝑝B 𝑥B
𝑓 𝑥 = 𝑄+,B∼ 𝑥+𝑥B + 𝑞+𝑥+𝑞+ = 𝑄++ 𝑎𝑛𝑑 𝑄+,B∼ = 𝑄+,B 𝑖 ≠ 𝑗
Ising: 𝑦 𝜖 {−1, 1}
QUBO: 𝑥 𝜖 0, 1
𝐽+,B =14𝑄+,B∼ ℎ+ =
𝑞+2+>
B
𝐽+,B
𝛾 = =|∑+,B 𝑄+,B +
=@∑+ 𝑞+
}𝐻 =>+,B
𝐽+,B�̂�+�̂�B +>+
ℎ+ �𝑥+ + 𝛾
𝑚𝑖𝑛1𝑓(𝑥)
𝑓 𝑥 = >+
ℎ+𝑦+ +>+,B
𝐽+,B 𝑦+𝑦B + 𝛾
Coupler Strengths
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QUBO to Quantum Ising Hamiltonian
𝑓 𝑥 = −𝜃=>+
𝑥+𝑟++𝑥+ + 𝜃@>+
𝑥+𝑝+𝑥+ − 𝑏 @ + 𝜃A>+,B
𝑥+𝑐𝑜𝑣 𝑝+, 𝑝B 𝑥B
𝑓 𝑥 = 𝑄+,B∼ 𝑥+𝑥B + 𝑞+𝑥+𝑞+ = 𝑄++ 𝑎𝑛𝑑 𝑄+,B∼ = 𝑄+,B 𝑖 ≠ 𝑗
Ising: 𝑦 𝜖 {−1, 1}
QUBO: 𝑥 𝜖 0, 1
𝐽+,B =14𝑄+,B∼ ℎ+ =
𝑞+2+>
B
𝐽+,B
𝛾 = =|∑+,B 𝑄+,B +
=@∑+ 𝑞+
}𝐻 =>+,B
𝐽+,B�̂�+�̂�B +>+
ℎ+ �𝑥+ + 𝛾
𝑚𝑖𝑛1𝑓(𝑥)
𝑓 𝑥 = >+
ℎ+𝑦+ +>+,B
𝐽+,B 𝑦+𝑦B + 𝛾
Qubit Weights
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QUBO to Quantum Ising Hamiltonian
𝑓 𝑥 = −𝜃=>+
𝑥+𝑟++𝑥+ + 𝜃@>+
𝑥+𝑝+𝑥+ − 𝑏 @ + 𝜃A>+,B
𝑥+𝑐𝑜𝑣 𝑝+, 𝑝B 𝑥B
𝑓 𝑥 = 𝑄+,B∼ 𝑥+𝑥B + 𝑞+𝑥+𝑞+ = 𝑄++ 𝑎𝑛𝑑 𝑄+,B∼ = 𝑄+,B 𝑖 ≠ 𝑗
Ising: 𝑦 𝜖 {−1, 1}
QUBO: 𝑥 𝜖 0, 1
𝐽+,B =14𝑄+,B∼ ℎ+ =
𝑞+2+>
B
𝐽+,B
𝛾 = =|∑+,B 𝑄+,B +
=@∑+ 𝑞+
𝑚𝑖𝑛1𝑓(𝑥)
𝑓 𝑥 = >+
ℎ+𝑦+ +>+,B
𝐽+,B 𝑦+𝑦B + 𝛾
Constant }𝐻 =>+,B
𝐽+,B�̂�+�̂�B +>+
ℎ+ �𝑥+ + 𝛾
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QUBO to Quantum Ising Hamiltonian
𝑓 𝑥 = −𝜃=>+
𝑥+𝑟++𝑥+ + 𝜃@>+
𝑥+𝑝+𝑥+ − 𝑏 @ + 𝜃A>+,B
𝑥+𝑐𝑜𝑣 𝑝+, 𝑝B 𝑥B
𝑓 𝑥 = 𝑄+,B∼ 𝑥+𝑥B + 𝑞+𝑥+𝑞+ = 𝑄++ 𝑎𝑛𝑑 𝑄+,B∼ = 𝑄+,B 𝑖 ≠ 𝑗
Ising: 𝑦 𝜖 {−1, 1}
QUBO: 𝑥 𝜖 0, 1
𝐽+,B =14𝑄+,B∼ ℎ+ =
𝑞+2+>
B
𝐽+,B
𝛾 = =|∑+,B 𝑄+,B +
=@∑+ 𝑞+
𝑚𝑖𝑛1𝑓(𝑥)
𝑓 𝑥 = >+
ℎ+𝑦+ +>+,B
𝐽+,B 𝑦+𝑦B + 𝛾
Ising Form
}𝐻 =>+,B
𝐽+,B�̂�+�̂�B +>+
ℎ+ �𝑥+ + 𝛾
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QUBO to Quantum Ising Hamiltonian
𝑓 𝑥 = −𝜃=>+
𝑥+𝑟++𝑥+ + 𝜃@>+
𝑥+𝑝+𝑥+ − 𝑏 @ + 𝜃A>+,B
𝑥+𝑐𝑜𝑣 𝑝+, 𝑝B 𝑥B
𝑓 𝑥 = 𝑄+,B∼ 𝑥+𝑥B + 𝑞+𝑥+𝑞+ = 𝑄++ 𝑎𝑛𝑑 𝑄+,B∼ = 𝑄+,B 𝑖 ≠ 𝑗
Ising: 𝑦 𝜖 {−1, 1}
QUBO: 𝑥 𝜖 0, 1
𝐽+,B =14𝑄+,B∼ ℎ+ =
𝑞+2+>
B
𝐽+,B
𝛾 = =|∑+,B 𝑄+,B +
=@∑+ 𝑞+
𝑚𝑖𝑛1𝑓(𝑥)
𝑓 𝑥 = >+
ℎ+𝑦+ +>+,B
𝐽+,B 𝑦+𝑦B + 𝛾
Quantum Ising
}𝐻 =>+,B
𝐽+,B�̂�+�̂�B +>+
ℎ+ �𝑥+ + 𝛾
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Solving on D-Wave 2000QD-Wave 2000Q:• 2048 qubits
• Anneal times
• Embeddings
• Reverse Annealing
Parameters:
• 𝜃=, 𝜃@, 𝜃A• Number of assets
• Random historical Price Data
Outputs:• Portfolio:
[-1, 1, 1, 1, -1, -1…] à
[0, 1, 1, 1, 0, 0…]
• Energy of the portfolio
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Benchmarking against Brute Force Solver
• Probability of success:
𝑃𝑂𝑆 = ���
• 𝑙 = # of ground state energies found by the D-Wave
• 𝑁� = number of samples/anneals
• Average Probability of success
𝑃𝑂𝑆 = ∑+�� ��� +
��• 𝑁� = # problems• 𝑖 indicates problem
• POS ratio: ��� ���� �
• 𝑃𝑂𝑆 � for reverse annealing• 𝑃𝑂𝑆 � for forward annealing
• Energy Binning
Energy à Energy Level
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Best Fit for Probability of Success
𝑦 = 1.2245 𝑒Tq.��=|∗1�.����
• Embedding = Find_embedding()
• Slices = 0
• Anneal time = 𝟓𝝁𝒔
• Forward annealing
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Forward Annealing Controls
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Spin Reversal Control
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Example Program Embedding in Hardware
find_embedding() Embedding
Clique Embedding
Problem size = 20
• 23 unit cells • 15 unit cells
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Clique vs Find Embedding
Forward Annealing:
• Embeddings:• find_embedding()• Clique
• Anneal Time = 15𝜇𝑠• Spin Reversal = 0
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Clique vs Find Embedding
Anneal time = 15 𝜇𝑠Number of anneals = 1,000
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Forward Annealing: Clique vs Find Embedding
Anneal time = 15 𝜇𝑠Number of anneals = 1,000
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Reverse Annealing
Parameters:
• Initial State• Reinitialize • S• Pause time
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Reverse Annealing Sweep
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Reverse Annealing Sweep
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Reverse Annealing Sweep
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Reverse Annealing Sweep
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Overall Probability of Success
Reverse Annealing:
• S = .9• Pause = 15𝜇𝑠• I.S. = Forward• Reinitialize = False
Forward Annealing:
• Embeddings:• find_embedding()• Clique
• Anneal Time = 15𝜇𝑠• Spin Reversal = 0
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Conclusions• Quantum annealing is a method for solving QUBO problems and
Markowitz Portfolio optimization is an example which demonstrates real-world application.
• Various parameters both in the QUBO and the D-Wave computer can be controlled/fine-tuned to yield better results.
• Forward annealing reveals a sub-exponential decrease in probability of success as problem size increases.
• Spin reversal transform improves variance.
• Clique Embedding improves probability of success over the find_embedding() method.
• Reverse annealing reveals a better probability of success and a better average solution if initial state is close to ground state.
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Future Work
• Measure efficiency of using reverse annealing with forward annealing.
• Compare quantum annealing to classical heuristics.
• Investigate using real market data.
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ORNL is managed by UT-Battelle, LLC for the US Department of Energy
Questions
University of Tennessee, Knoxville
Oak Ridge National Laboratory
Quantum Computing Institute
In collaboration with Khalifa University
Nada Elsokkary, Faisal Khan
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Asseti
350
375
312
300
330
Asseti
1.0606
1.1363
.94545
.90909
1
Asseti, 0
1.0606
1.1363
.94545
.90909
1
Asseti, 1
.53030
.56815
.47273
.45450
.5
Asseti, 2
.26515
.28408
.23636
.22727
.25
𝒑𝒌𝒑𝒎 𝒂𝒊 → 𝒂𝒊,𝒌
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Probability of Success
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Optimality Gap
Optimality Gap:
𝑎𝑏𝑠𝐸¦§�𝐸¨1�
− 1