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[DSC Europe 22] The nuts and bolts of music source separation - Stipe Kabic

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Quantum machine learning is one of the promising application domains of quantum computing, which is expected to improve and accelerate the most resource-intensive machine learning calculations. This talk will explain how we implement quantum machine learning algorithms, what are the limits and challenges, and how these challenges can be addressed.

- 1. Quantum Machine Learning Sasha Lazarevic, Qiskit Advocate LZRVC.com www.linkedin.com/in/LZRVC
- 2. Exploiting Quantum Phenomena Superposition Quantum Measurement Entanglement Interference |Y⟩ = a|0⟩ + b|1⟩
- 3. Universal Quantum Computer Architecture of the quantum computer Quantum Processor Analog instructions Microarchitecture Runtime Transpiler Compiler Programming language Algorithm Problem representation Mathematical model Software code Firmware Hardware (quantum) 1 qubit can represent simultaneously 2 bits 2 qubits can represent 4 bits or 2^2 10 qubits can represent 2^10 bits or 128 bytes 30 qubits can represent 2^30 bits or 128 MB 40 qubits can represent 2^40 bits or 128 GB 50 qubits can represent 2^50 bits or 128 TB … 1. Exploit superposition to keep more information in quantum states 2. Exploit entanglement to parallelize operations on these quantum states 3. Evolve the overall state to get expected results 4. Measure multiple times to get the probability distribution of these results Bit flip Phase flip Hadamard CNOT Toffoli Meaasurement Quantum gates Design of a quantum algorithm Quantum circuit Qubits Classical bits (results) Initialization Classical computer
- 4. High-Value Use Cases for Finance VaR / CVaR Risk and scenario analysis CVA and XVA Derivative pricing Portfolio optimization Optimal arbitrage opportunities Collateral optimization Trade settlement Combinatorial auction Fraud detection Customer scoring Forecasting Risk estimation Dimensionality reduction Synthetic data generation Simulation Optimization Machine Learning Quantum amplitude estimation (AE) Quantum Kernel Estimation (QKE) Variational Quantum Classification (VQC) Quantum PCA (qPCA) Quantum Circuit Born Machines (QCBM) Quantum Boltzmann Machines (QBM) Convex optimization: Quantum Semidefinite Programming (QSDP) Combinatorial optimization: QUBO -> Variational Quantum Eigensolver (VQE) / Quantum Approximate Optimization Algorithm (QAOA) Mixed-Binary optimization: 3-ADMM-H -> QUBO with VQE/QAOA, CPLEX
- 5. • Linear regression has been implemented using QPE and HHL • Ridge regression, where QPE and QFT help find optimal value of the reg. parameter • Perceptron can be implemented with Grover's search • Support Vector Machine has been implemented with HHL algorithm • k-means clustering and k-nearest neighbours classification can be improved with Grover's search • Principal component analysis has been implemented with QPE and VQA • Autoencoders have been implemented with VQA • Restricted Boltzmann Machines have been implemented as Var-QBM • Deep neural networks can be implemented as hybrid models with VQA • GANs training has been done in a way that Generator or Discriminator use associative QBM-based NNs • Synthetic data has been generated with QCBM • Model-based RL can be enhanced with Grover's algorithm to find rewarding action-sequences • RL Q-learning has been successfully tested with QBM, which generates values of the Q-value function • Monte-Carlo sampling, implemented with QAE, can be used for policy evaluation in RL Quantum Algorithms : • QPE – Quantum Phase Estimation • QAE – Quantum Amplitude Estimation • QFT – Quantum Fourier Transform • HHL - Harrow-Hassidim-Lloyd • Grover’s search • VQA/E – Variational Quantum Algorithm / Eigensolver • QAOA – Quantum Approximate Optimization Algorithm • QCBM – Quantum Circuit Born Machine • QBM – Quantum Boltzmann Machines Quantum-enhanced Machine Learning Error-free NISQ QPE ✓ QAE ✓ QFT ✓ HHL ✓ Grover’s ✓ VQA/E ✓ QAOA ✓ QCBM ✓ QBM ✓
- 6. Data Encoding Basis 𝑥 = 0.1, −0.8, 1.0 First, convert to binary numbers (up to a precision, lets’ say 𝜏=5) 𝑥 = 00011, 11001, 11111 Add a sign bit (0 for +, 1 for -): 𝑥 = 000011, 111001, 011111 Our state can be written as : | ⟩ 𝜓 = | ⟩ 000011111001011111 Next, initialize qubits and use bit flip to create this state. Advantage: Easy to create Disadvantage: Requires 𝜏+1*(#𝑥) qubits Not suitable for NISQ computers Amplitude 𝑥 = 0.1, −0.8, 1.0 First, normalize it : 𝑥 = 0.12 + −0.82 +12= 1.2845 𝑥 = 0.1 1.2845 , −0.8 1.2845 , 1 1.2845 Pad it to log2(#𝑥) elements: 𝑥 = 0.078, −0.623, 0.779, 0 Our state can be written as : | ⟩ 𝜓 = 0.078| ⟩ 00 - 0.623| ⟩ 01 + 0.779| ⟩ 10 + 0| ⟩ 11 Next, make a custom operator to create this state. Advantage: Uses log2N qubits to encode N features Disadvantage: Depth of encoding circuit Not suitable for NISQ computers Angle 𝑥 = 0.1, −0.8, 1.0 Rescale the data to 0-2𝜋 radians: 𝑥 = 3.45575, 0.62832, 6.28319 Use X rotation for each data point: ⟩ |𝜓1 = 𝑅𝑥(3.45575) ⟩ |𝜓2 = 𝑅𝑥(0.62832) ⟩ |𝜓3 = 𝑅𝑥 6.28319 Advantage: Can be used for NISQ Higher-order Angle Repeat angle encoding circuit multiple times This can encode complex frequencies, as the frequency spectrum increases linearly with the number of repetitions Intertwine encoding with data processing In Qiskit implemented with PauliFeatureMap and ZZFeatureMap as rotation around Z axis
- 7. QML Training Process Training procedure : 1. Data Encoding • Basis • Amplitude • Angle 2. Variational Circuit • Expressive power 3. Measurements (readouts) 4. Cost Function 5. Analytic Gradient Descent 6. Update parameters 7. Run again until convergence - Parameter shift rule and Linear combination - Natural gradient • Higher entangling capability means being better able to capture non-trivial correlations in the input data • Expressivity increases with stacking up several layers Labels extraction : • based on parity or • measuring only the first qubit Beware of the barren plateau ! How to implement non-linearity ? • QFT • repeat-until-success schemes • Use stochastic nature of measurement
- 8. QML – Neural Networks Source: S. Mangini et al 2021 EPL 134 1000
- 9. QML - Kernel Methods Quantum Kernels - SVM Feature maps help resolve the data non-linearity problem Dual formulation reduces the dimensions of the search space through a function that implicitly encodes the feature map But to ensure quantum speedup, the quantum kernel function Φ(𝑥) has to be hard to estimate clasically. Idea: Design quantum kernels to exploit the group structure in data. Example is DLOG kernel Sources: Global Qiskit Summer School 2021 https://arxiv.org/pdf/2010.02174.pdf
- 10. QML – Coding Examples • Building hybrid quantum-classical classifier • Hybrid quantum-classical neural network for MNIST recognition • 6 • Quantum Support Vector Machine https://github.com/LZRVC/Quantum-Computing
- 11. Your next steps https://www.meetup.com/Quantum-Serbia https://www.linkedin.com/groups/9024908/ email: quantumserbia@gmail.com https://qiskit.org/advocates/ Join : Learn : From Books • Qiskit Textbook (https://qiskit.org/textbook) by IBM • Quantum Computation and Quantum Information by M. Nielson, I.Chuang • Quantum Programming Illustrated by Aleksandar Radovanovic • Introduction to Quantum Information Science by Vlatko Vedral • Quantum Computing for Computer Scientists by N. Yanofski • Machine Learning with Quantum Computers by Maria Schuld Online Courses • CaltechX, DelftX (2018) Quantum Cryptography (edX, online) • Keio (2018) Understanding Quantum Computers (futurelearn.com) IBM Qiskit Summer Schools (in July, online) • TUDelft (2021) The Hardware of a Quantum Computer (edX, online)
- 12. Thank you! Sasha Lazarevic Qiskit Advocate Sasha.Lazarevic@gmail.com www.linkedin.com/in/lzrvc

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