Quantum Gaussian Processes - Gawel Kus

1. 1
Quantum Gaussian
Processes
Gaweł Kuś
Master student at Novel Aerospace Materials (NovAM) group,
Faculty of Aerospace Engineering
Delft University of Technology
Dr.ir. M.A. (Miguel) Bessa
Assistant Professor in Materials Science and Engineering
Delft University of Technology
Prof.Dr.ir. S. (Sybrand) van der Zwaag
Chairholder of Novel Aerospace Materials (NovAM) group,
Faculty of Aerospace Engineering
Delft University of Technology
for computational design of materials

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Example material design problem
Unit cell of super-compressible metamaterial
Objective:
• Optimize for absorbed energy: Eabs
• Optimize for buckling load: Pcrit

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Parameter Description
1. D1 /D2 Ratio of diameters
2. P Pitch
3. Ixx Moment of inertia of longeron around x
4. Ixx Moment of inertia of longeron around y
5. J Torsional stiffness of longeron
6. A Cross-section of longeron
7 G/E Shear modulus/Young’s modulus
Objective:
• Optimize for absorbed energy: Eabs
• Optimize for buckling load: Pcrit
Example material design problem
Unit cell of super-compressible metamaterial

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Framework for data-driven design

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Gaussian processes regression
● Inference from noisy data
● Uncertainty quantification
● Bayesian Machine learning
GP applications:
● Optimization under uncertainty
● Modelling of imperfection
sensitive phenomena

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Gaussian processes regression
GP assumes a prior distribution
Fully specified by:
● Mean:
● Covariance matrix: K
given by kernel
function, e.g. RBF

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Kernel function
Radial basis function (RBF) as a measure of similarity

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Gaussian processes regression
Conditioning the prior with training data
By definition of GP, y and f*
are jointly distributed

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Gaussian processes regression
...to convert the prior to posterior
Marginalizing f* results in
posterior distribution:

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● Matrix inversion scales as:
● Practical limitation: ~10 000
training points
Gaussian processes regression
In practice: solving system of linear equations

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The curse of dimensionality
In design of materials
High-dimensional design spaces, due to
phenomena at many different levels:
● Nanoscale:
-chemical composition
-crystalline structure, phases
-imperfections
● Microscale:
-microstructural
parameters
● Macroscale
-macro-architecture
parameters
Big data

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Idea 1: PCA approximation
Eigendecomposition of Knn and select m<n eigenmodes:
But, eigendecomposition ~O(n3)
Overcoming the limitations of GP
How to improve scalability?

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Idea 1: PCA approximation
Eigendecomposition of Knn and select m<n eigenmodes:
But, eigendecomposition ~O(n3)
Idea 2: Nystrom approximation
Approximate m eigenfunctions to construct a low-rank approximation:
Inversion complexity:
Overcoming the limitations of GP
How to improve scalability?

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Nystrom approximation
Constructing low-rank kernel matrix

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Sparse Gaussian processes
Constructing approximation by introducing m inducing points:

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Sparse Gaussian processes
Number of inducing points (m) affects the approximation quality

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Sparse Gaussian processes
Improvement in scaling

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Sparse Gaussian processes
Potential improvement

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Quantum computing and machine learning use similar formulation
(linear algebra)
Common trend in QML: replacing BLAS with qBLAS
Quantum algorithm Function Application in ML
QPE eigendecomposition - qPCA
HHL Solving systems of
linear equations
- SVM
- Least squares
- Kernel methods
- Gaussian
Processes
Quantum machine learning

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u vA
[1] Z.Zhao et al., Quantum assisted Gaussian process regression (2015)
Quantum Gaussian processes
Concept: speed-up the matrix inversion with HHL algorithm[1]

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u vA
1. Find b = A-1v with HHL algorithmb = A-1v
[1] Z.Zhao et al., Quantum assisted Gaussian process regression (2015)
Quantum Gaussian processes
Concept: speed-up the matrix inversion with HHL algorithm[1]

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u vA
1. Find b = A-1v with HHL algorithm
2. Apply a measurement operator to
find the dot product: u.b = uA-1v
b = A-1v
u.b = u.A-1.v
u
[1] Z.Zhao et al., Quantum assisted Gaussian process regression (2015)
Quantum Gaussian processes
Concept: speed-up the matrix inversion with HHL algorithm[1]

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Quantum Gaussian processes
Concept: speed-up the matrix inversion with HHL algorithm[1]
u vA
1. Find b = A-1v with HHL algorithm
2. Apply a measurement operator to
find the dot product: u.b = uA-1v
Exponential speed-up due to HHL:
[1] Z.Zhao et al., Quantum assisted Gaussian process regression (2015)

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find b = A-1v dot product:
u.b = uA-
1v
Initialize u and v
in superposition
Quantum Gaussian processes
Circuit representation

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QGP implementation
How to program Quantum computer?
Qiskit: IBM’s open source quantum
computing framework:
○ Quantum SDK (Python)
○ Compilers (QASM)
○ Backends
■ Simulators
■ Real devices (5-20 qubits)
Software development
+
Execution

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Approximations in QGP
Eigendecomposition approximation with QPE
QPE is controlled with 2 parameters:
● r - number of time slices
-controls the matrix
exponentiation
● k - size of the eigenvalue register
- controls the eigenspectrum
discretization
- min. resolvable eigenvalue:

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QGP as low-rank approximation
Approximate eigendecomposition with QPE allows to induce a low-
rank approximation
● The approximation
is similar to the
classical Sparse
GP
● Exploiting this
feature reduces
the computational
cost of QGP
algorithm

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QGP application example
Design of metamaterial: inference of a 1D relationship - Eabs vs Ixx

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Conclusions
● Discovered a mechanism for inducing the low-rank approximation, analog to
that in classical sparse Gaussian processes
● Demonstrated application of QC in materials engineering