Parameterized Quantum Circuits
A quantum circuit that includes gates with tunable parameters is known as a parameterised quantum circuit (PQC). A key component of quantum computing, these circuits combine quantum and traditional methods to tackle challenging issues. The main concept is to optimise the circuit’s tunable parameters using a conventional computer and a quantum circuit for computation. The circuit’s versatility and ability to adjust to different applications, including machine learning, optimisation, and quantum simulation, are made possible by this hybrid quantum-classical method.
Because variational quantum algorithms are made to work with the present generation of noisy intermediate-scale quantum (NISQ) devices, PQCs are especially crucial. These algorithms minimise a cost function that reflects the issue being solved by iteratively updating the parameters. PQCs operate similarly to trainable models in traditional machine learning as a result of this procedure. One crucial element is the interface for importing classical data into these quantum systems, and PQCs are an important approach for this work.
Structure and Implementation
- Architecture and Design of Circuits Several important choices must be made while designing a PQC:
- Architecture Selection: Selecting a suitable circuit structure.
- Choosing the kind and quantity of parameterised gates, such as rotation gates (RX, RY, RZ) and controlled rotation gates (CRX, CRY, CRZ), is known as gate selection.
- Qubit Connectivity: Identifying the connections between the qubits.
- Circuit Depth: The number of gate layers, which might improve the circuit’s performance but may also have an impact on how trainable it is.
These include hardware-efficient architectures based on the native gates and connectivity of a particular quantum device, layered architectures with repeating gate patterns, problem-inspired architectures such as the Quantum Approximate Optimisation Algorithm (QAOA), and tensor network-based architectures.
- The Quantum-Classical Hybrid Structure PQCs work in a hybrid system in which a conventional machine modifies the circuit’s parameters while a quantum machine provides predictions. Typical steps in a workflow include:
- State Preparation: Classical data is encoded into a quantum state via an embedding circuit, also known as a state preparation circuit.
- Computation: The PQC processes the quantum state using its movable parameters, also referred to as variational angles.
- Measurement: To create a histogram of output states, the circuit is run several times (referred to as “shots”).
- Evaluation of Cost Functions: A cost function measures the discrepancy between expected and actual results.
- Classical Optimisation: To minimise the cost function, a classical optimiser (such as the Adam optimiser or gradient descent) modifies the circuit’s parameters. Until convergence is achieved, this process is repeated.
Important Qualities: Trainability and Expressivity
The two primary characteristics of expressivity and trainability have a significant impact on a PQC’s success.
Expressivity
- Definition: The ability of a PQC to represent a broad variety of quantum states and functions is known as expressivity. More intricate patterns and dependencies in data can be captured by a circuit with high expressivity.
- Influential Factors: The type and configuration of parameterised gates, circuit depth, and qubit count all have an impact on expressivity.
- Quantification: The distribution of quantum states produced by the PQC can be compared to the uniform (Haar random) distribution across the whole Hilbert space in order to quantify expressivity. For this comparison, a smaller value of the Kullback-Leibler (KL) divergence denotes higher expressivity.
- Association with Accuracy: Research on quantum machine learning tasks has shown a moderate to strong association between expressibility and classification accuracy. This implies that a circuit’s potential performance can be accurately predicted by its capacity to evenly explore the Hilbert space.
Trainability
- Definition: Trainability is the ease and effectiveness with which the settings of a Parameterised Quantum Circuit (PQC) can be adjusted.
- Challenges: “Barren plateaus,” or areas in the cost function landscape where the gradient disappears and optimisation becomes useless, are among the issues that make training PQCs infamously challenging. The availability of several fake local minima that could trap the optimiser is another difficulty. The majority of PQC learning has been limited to traditional optimisation techniques, which may have drawbacks such as gradient vanishing.
- Enhancing Trainability: There are a number of methods to improve trainability, such as:
- Parameter Initialisation: Bare plateaus can be avoided with the right approach, such as lowering the initial domain of parameters. One layer at a time, parameters are optimised through layerwise training. Adding penalty terms to the cost function is known as regularisation.
The Trainability-Expressivity Relationship Trade-off: These two attributes frequently have to be traded off. Trainability may suffer and the optimisation problem may become more difficult if a circuit’s expressivity is increased (for example, by adding more layers). Making thoughtful design decisions is necessary to balance both factors.
Uses
Many near-term quantum algorithms and applications are based on parameterised quantum circuits:
- Quantum Machine Learning: For classification and regression applications, PQCs are employed as quantum neural networks. It may be possible to increase accuracy by using Hilbert space’s large dimensionality as a feature space.
- Variational quantum algorithms: PQCs play a crucial role in algorithms such as the Quantum Approximate Optimisation Algorithm (QAOA), which solves optimisation issues, and the Variational Quantum Eigensolver (VQE), which determines the ground states of quantum systems.
- State Preparation: One of the core functions of quantum computing is the preparation of certain quantum states, which is accomplished by PQCs.
The Natural Parameterised Quantum Circuit (NPQC), which can be initialised with a Euclidean quantum geometry to expedite the initial training of variational algorithms, is one of the improved variants of normal PQCs that have been proposed to solve some of the inherent problems of these circuits.
