Classiq Library

The library provides implementations of several hybrid quantum-classical frameworks for Quantum Machine Learning (QML). These algorithms demonstrate how parameterized quantum circuits can be integrated with classical optimization and deep-learning tools (such as PyTorch) to perform tasks like classification, generative modeling, and data compression.

Key implemented algorithms include:

• Hybrid Quantum Neural Networks (QNN): Incorporates quantum layers into classical neural networks using parameterized quantum circuits.
• Quantum Generative Adversarial Networks (GANs): A quantum analogue of classical GANs where classical neural networks are replaced by quantum neural networks (parameterized quantum circuits) to generate data.
• Quantum Support Vector Machine (QSVM): Uses a quantum circuit to implement a feature map and quantum measurements to evaluate the kernel matrix for data classification.
• Quantum Autoencoder: A quantum program trained to reduce the memory required to encode data, often used for anomaly detection.

The quantum_state_preparation module provides advanced techniques for preparing quantum states, which are essential for quantum simulation and optimization. It focuses on two primary methodologies:

1. ADAPT-VQE (Adaptive Derivative-Assembled Pseudo-Trotter Variational Quantum Eigensolver): A hybrid variational algorithm that extends the standard VQE framework. It constructs problem-specific solutions adaptively, resulting in shallower circuits compared to standard VQE by increasing the number of measurements.

2. Gibbs State Preparation: A primitive used for solving semidefinite programs, Boltzmann sampling, and Metropolis-type algorithms. It prepares a quantum thermal (Gibbs) state by implementing a block-encoding of the Lindbladian superoperator. This process simulates open-system dynamics (a system coupled to a thermal bath) to drive the initial state toward thermal equilibrium. The implementation utilizes an operator Fourier transform and mid-circuit weak measurements, leveraging the quantum Zeno effect for efficiency.

Classiq provides several advanced methods for modeling and analyzing physical systems through Hamiltonian simulation. These methods are categorized into two main approaches:

1. Product-formula decompositions: Includes standard methods like Suzuki-Trotter decomposition and qDrift.
2. Block-encoding–based techniques: Advanced methods that utilize block-encoding for efficient quantum time evolution. These include:
   • GQSP (Generalized Quantum Signal Processing): Implements time evolution using only one auxiliary qubit and no amplitude amplification.
   • QSVT (Quantum Singular Value Transformation): Implements time evolution via interleaved signal-processing rotations.
   • Qubitization: Implements time evolution as a Linear Combination of Unitaries (LCU) over Chebyshev-polynomial block-encodings of the walk operator.

All these methods are implemented using Classiq's built-in functions.

The library provides several computational models for addressing linear algebraic problems ($A |\mathbf{x}\rangle = |\mathbf{b}\rangle$) using quantum computing frameworks. These models include:

• Adiabatic linear solvers: Maps the solution to the ground state of a corresponding Hamiltonian, using a quantum adiabatic protocol.
• HHL (Harrow-Hassidim-Lloyd): A fundamental algorithm that prepares the state $|x\rangle$ in polynomial time relative to the number of qubits, offering exponential speedup for evaluating observables of the form $\langle \mathbf{x}| M| \mathbf{x}\rangle$.
• QSVT (Quantum Singular Value Transformation) matrix inversion: A general framework that uses block-encoding to embed a classical matrix as a quantum function. It provides a concise implementation for matrix inversion when a block-encoding and the matrix's condition number are known.
• VQLS (Variational Quantum Linear Solver) with LCU (Linear Combination of Unitaries): A hybrid quantum-classical algorithm. It uses LCU to block-encode a matrix $A$ within a unitary, then uses classical optimization to find an approximate solution. This approach is well-suited for NISQ (Noisy Intermediate-Scale Quantum) devices due to limited hardware requirements.

The algorithms directory contains implementations of canonical and recent quantum algorithms. These implementations are organized into specialized subject areas to help researchers and developers find specific algorithmic logic.

Key subject areas include:

• Amplitude Amplification and Estimation: Variants of Grover's search and amplitude estimation.
• Foundational: Fundamental algorithms demonstrating query complexity advantages (e.g., Deutsch-Jozsa, Simon).
• Hamiltonian Simulation: Methods like QSP, qubitization, and QSVT for simulating physical systems.
• Number Theory and Cryptography: Algorithms for prime factorization and discrete logarithms (e.g., Shor).
• Quantum Machine Learning: Hybrid quantum-classical algorithms for classification and generative modeling.
• Quantum Differential Equation Solvers: Mapping PDEs to linear systems for scientific simulation.
• Quantum Linear Solvers: Algorithms for solving linear systems (e.g., HHL, QSVT-based inversion).
• Quantum Phase Estimation: Extracting eigenvalues via controlled unitary evolution.
• Quantum Primitives: Modular building blocks like the Hadamard test, Swap test, and GQSP.
• Quantum State Preparation: Techniques like ADAPT-VQE and Gibbs state generation.
• Quantum Walks: Discrete quantum walk implementations.
• Search and Optimization: Unstructured search (Grover) and combinatorial optimization (QAOA, DQI).

The quantum_differential_equations_solvers module provides quantum algorithms for solving both stationary and time-dependent differential equations. These algorithms map linear systems and dynamical evolution problems to quantum circuits, leveraging block-encoding techniques and quantum linear system methods.

Key implementations include:

• Discrete Poisson solver: A quantum solver for the discrete Poisson equation (a PDE). It reformulates the equation as a system of linear equations and solves it using the HHL algorithm. It utilizes quantum cosine and sine transforms to enable concise implementations that can be generalized to higher dimensions.
• Time marching: A method for solving linear differential equations by integrating dynamics in small discrete steps. For an equation of the form $\frac{d|\psi(t)\rangle}{dt} = A(t) |\psi(t)\rangle$, the algorithm uses a block-encoding of the time-dependent matrix $A(t)$ to solve for $|\psi(t)\rangle$.