# Gate Model Quantum Computing versus Quantum Annealing - the practical and mathematical differences

January 9, 2023

Quantum computers and quantum annealers are two types of quantum computing devices that use quantum mechanics to perform calculations. While both types of devices have the potential to solve certain problems much faster than classical computers, they use different approaches and are best suited for different types of tasks.

One key difference between gate model quantum computers and quantum annealers is the type of quantum operations they can perform. A gate model quantum computer is a universal quantum device, which means it can perform any quantum operation that can be represented as a sequence of quantum gates. This makes gate model quantum computers very flexible and powerful, but also more complex and harder to control. In contrast, a quantum annealer is a specialized quantum device that is designed to solve a specific type of problem known as an optimization problem. Quantum annealers use a process called quantum annealing to find the minimum (or maximum) of a function, which can be used to solve optimization problems. Check out our introductory post on QUBOs and how quantum annealing can be used to solve those.

Another difference between the two types of devices is their architecture. Gate model quantum computers typically use a circuit-based architecture, where quantum bits (qubits) are connected together in a way that allows them to perform arbitrary quantum operations. Quantum annealers, on the other hand, use a lattice-based architecture, where qubits are arranged in a lattice and interact with each other through interactions that are native to the lattice. This makes quantum annealers simpler and easier to control than gate model quantum computers, but also limits their flexibility and the types of problems they can solve.

In terms of performance, gate model quantum computers are generally considered to be more powerful than quantum annealers. This is because they can perform a wider range of quantum operations, which allows them to tackle a broader range of problems. However, quantum annealers have the advantage of being able to solve certain types of problems much faster than classical computers, even if they are less powerful than gate model quantum computers. This includes mostly optimization problems at this stage.

Currently, quantum annealers such as the latest D-Wave Advantage with 5000 qubits are powerful enough to solve interesting proof of concept use cases such as optimizing paths of forklifts in a small warehouse. Gate model quantum computers such as the latest IBM machine on the other hand still struggle with such use cases.

In conclusion, gate model quantum computers and quantum annealers are two types of quantum computing devices that are best suited for different types of tasks. While gate model quantum computers are more powerful and flexible, quantum annealers are simpler and more specialized, and are particularly good at solving optimization problems.

Mathematical differences

In addition to the differences in their architectures and the types of quantum operations they can perform, there are also some mathematical differences between gate model quantum computers and quantum annealers.

One key difference between gate model quantum computers and quantum annealers is the type of quantum states that are used to represent the computation. In a gate model quantum computer, quantum states are typically represented using the quantum circuit model, where quantum states are represented as quantum circuits made up of quantum gates. This means that a quantum state in a gate model quantum computer can be represented using a sequence of quantum gates acting on some initial state. For example, consider the quantum state |ψ⟩ represented by the following quantum circuit:

|0⟩
──H──
|ψ⟩

Here, the quantum state |ψ⟩ is obtained by applying the Hadamard gate H to the initial state |0⟩.

In contrast, in a quantum annealer, quantum states are represented using the quantum adiabatic model, where quantum states are represented as the ground state of a time-dependent Hamiltonian. This means that a quantum state in a quantum annealer can be represented as the lowest-energy state of a Hamiltonian that depends on time. For example, consider the quantum state |ψ⟩ represented by the following Hamiltonian:

H(t) = (1 - t/T)H_0 + (t/T)H_1

Here, H_0 and H_1 are time-independent Hamiltonians, and T is the total annealing time. The quantum state |ψ⟩ is obtained by starting in the ground state of H_0 and slowly evolving the Hamiltonian from H_0 to H_1 over a period of time T. This process is known as quantum annealing. Hamiltonians can be modeled very easily by QUBOs, which can be used to solve optimization problems on a quantum annealiner in practical terms, as explained in our introductory post on QUBOs.

Another difference is the type of quantum algorithms that can be used to solve problems on these devices. On a gate model quantum computer, a wide variety of quantum algorithms can be used, including algorithms based on quantum gates, quantum walks, and quantum machine learning. For example, the quantum algorithm for factoring integers, known as Shor's algorithm, is based on quantum gates and can be implemented on a gate model quantum computer.

In contrast, quantum annealers are limited to using quantum annealing algorithms, which are based on the adiabatic evolution of quantum states as described above. These algorithms are specifically designed to solve optimization problems by finding the minimum (or maximum) of a function. For example, consider the optimization problem of finding the minimum of the following function:

f(x) = x^2 - 3x + 4

This problem can be solved using a quantum annealer by encoding the function f(x) into the Hamiltonian H(t) and using quantum annealing to find the minimum value of the function.

Finally, there are also some differences in the types of problems that can be solved using these two types of devices. As mentioned earlier, quantum annealers are specialized for solving optimization problems, and are particularly good at finding the minimum (or maximum) of a function. In contrast, gate model quantum computers are more general-purpose devices that can be used to solve a wide range of problems, including optimization problems, but also many other types of problems such as factorization, search, and machine learning.

Overall, while both gate model quantum computers and quantum annealers are quantum computing devices that use quantum mechanics to perform calculations, they have some important mathematical differences that make them better suited for different types of tasks.