Contents [hide]
- 1 The Adiabatic Theorem and its Application to Computation
- 2 How an Adiabatic Quantum Computer Works
- 3 Encoding Problems into Hamiltonians
- 4 Comparison with Gate-Based Quantum Computing
- 5 Advantages of AQC
- 6 Adiabatic Quantum Computing Challenges
- 7 Difference between Adiabatic Quantum Computing and Quantum Annealing
Adiabatic quantum computing (AQC) is a model of quantum computation solving computational tasks by use of the adiabatic theorem of quantum mechanics. Farhi, Goldstone, Gutmann, and Sipser presented it initially in the early 2000s.
Especially for the solution of optimization issues, AQC is a possible substitute for gate-based quantum computing. Research is still under progress to fulfill its possibilities even if it has difficulties preserving the adiabatic condition and resolving decoherence. It provides a distinct paradigm for quantum computation, therefore enabling more general research of quantum information processing.
The Adiabatic Theorem and its Application to Computation
The basic idea of AQC is found in the adiabatic theorem, which holds that a quantum system will remain in the ground state of the developing Hamiltonian if it begins in the ground state of a Hamiltonian and the Hamiltonian is gradually modified enough.
In AQC, a problem is embodied in the ground state of a Hamiltonian (HP). The computation begins with a simpler starting Hamiltonian (HB) whose ground state is readily obtained. Under a time-dependent Hamiltonian H(t), the system then develops gradually interpolating between HB and HP. Should this development be sufficiently slow, the system will remain in its ground state and at the conclusion of the evolution it will be in the ground state of HP, thereby encoding the answer to the issue.
The Hamiltonian interpolation is often expressed as:
H(t) = (1 – f(t/T))*HB + f(t/T)HP
- where f(t/T) is a smooth, monotonic function that varies from 0 to 1, over a time period T
- HB is the beginning Hamiltonian
- HP is the problem Hamiltonian
How an Adiabatic Quantum Computer Works
Encoding the Problem: The first step in an Adiabatic Quantum Computer, is the problem to be solved in terms of a cost function . This cost function is then mapped onto a problem Hamiltonian , where the ground state of the optimal solution to the problem.
Initialization: The quantum computer is prepared in the ground state of the beginning Hamiltonian HB. This initial state is chosen to be easily preparable.
Adiabatic Evolution: The system’s state is evolved using the time-dependent Hamiltonian H(t). The Hamiltonian changes slowly, smoothly transitioning from HB to the problem Hamiltonian HP over a time T. This evolution is designed to maintain the system in its instantaneous ground state.
Measurement: Finally, a measurement in the computational basis is performed. If the adiabatic evolution is slow enough and meets the conditions of the adiabatic theorem, the final state will be close to the ground state of the problem Hamiltonian HP, which is the solution to the encoded problem.
Encoding Problems into Hamiltonians
The key challenge in AQC is finding ways to encode the solution to a given problem as the ground state of a problem Hamiltonian HP. In many cases, this involves mapping the problem to a combinatorial optimization problem where the goal is to find the minimum of a cost function. Any local combinational search problem can be formulated in such a way. A cost function f: {0,1}N → R, for which the global minimum x needs to be found, can be encoded into the Hamiltonian. Assuming that this global minimum is unique, AQC can be used to find it.
Comparison with Gate-Based Quantum Computing
- Computational Model: Gate-based quantum computing uses quantum gates to operate qubits. AQC, relies on the adiabatic evolution of a quantum system.
- Algorithm Design: In gate-based computing, algorithms are designed using sequences of quantum gates. In AQC, the problem is encoded into the Hamiltonian.
- Universality: The gate model is considered to be a universal model for quantum computation meaning that it can efficiently simulate any other model of quantum computation. It is an open question whether AQC is universal.
Advantages of AQC
Robustness to Noise: AQC is inherently robust to certain types of noise and decoherence, as the computation is performed in the ground state of the system, which is less sensitive to environmental perturbations.
Natural Mapping: Many optimization problems can be naturally expressed as finding the ground state of a Hamiltonian. This makes AQC particularly well-suited for solving combinatorial optimization problems.
Simplicity of Implementation: The hardware implementation of AQC requires maintaining a single quantum state rather than complex sequences of quantum gates, as in the circuit model.
Adiabatic Quantum Computing Challenges
Despite its potential, AQC faces challenges:
- Adiabatic Condition: The adiabatic theorem requires that the system evolves slowly, which can make the computation time longer. If the evolution is not sufficiently slow, the system may not remain in the ground state.
- “Gap” Issue: The speed of the computation is limited by the “gap” between the ground state and the first excited state of the Hamiltonian. If the gap becomes too small during the evolution, the system can jump to the excited state, and the computation can fail.
- Universality: AQC can solve any problem that can be solved on a quantum computer, whether AQC is equivalent to gate-based quantum computing is still under research.
- Decoherence: Quantum systems are vulnerable to decoherence, which can destroy the superposition and entanglement properties needed for AQC. The system needs to be isolated from the environment to preserve its coherent state.
- Hardware Limitations: Building large-scale, fault-tolerant adiabatic quantum computers is still a challenge. Current machines have a limited number of qubits.
Current status of AQC has seen experimental implementations. D-Wave Systems has built superconducting quantum annealers that use AQC principles. These machines have been applied to optimization problems.
Difference between Adiabatic Quantum Computing and Quantum Annealing
| Feature | Adiabatic Quantum Computing (AQC) | Quantum Annealing (QA) |
| Core Principle | Relies on the adiabatic theorem, where a system remains in its ground state during slow changes in the Hamiltonian. | A metaheuristic approach that explores the energy landscape of a problem to find the minimum energy state, often by simulating a physical process of cooling. |
| Computational Process | Starts with a simple initial Hamiltonian and slowly evolves it into a problem Hamiltonian. If done slowly enough, the system remains in its ground state, which represents the problem’s solution. | Starts with a set of qubits and gradually changes the energy encountered by the system until the problem parameters are defined by a Hamiltonian. The goal is to find the lowest energy state. |
| Problem Encoding | Problems are encoded into the ground state of a problem Hamiltonian. The system evolves from a simple starting Hamiltonian to this problem Hamiltonian. | Problems are often formulated as Quadratic Unconstrained Binary Optimization (QUBO) problems or Ising models, which the annealer tries to minimize. |
| Error Correction | Ideally requires full error correction, similar to gate-based quantum computing, to prevent deviations from the ground state. | Often requires less error correction than universal quantum computers. |
| Universality | It is believed that AQC can solve any problem a quantum computer can solve. The equivalence of AQC to gate-based quantum computing is still under research. | QA is generally considered to be a limited form of quantum computation that is not computationally universal, unlike gate-based quantum computers. |
| Control | Achieved by a carefully designed time dependent Hamiltonian | Achieved through an energy landscape or “cost function” where the system is intended to settle in a low-energy state. |
| Hardware | Can be implemented using various technologies, including superconducting circuits, trapped ions, and Rydberg atoms. | Primarily implemented using superconducting quantum annealers, like those from D-Wave. Other hardware types such as trapped ions and Rydberg atoms may also be used. |
| Use Cases | Well-suited for optimization and simulation problems, and can potentially solve general computational problems if universal. | Best suited for optimization problems where finding the absolute best solution among many is crucial, such as traffic flow optimization, financial modeling, and logistics. |
| Computation Speed | The adiabatic theorem requires slow evolution, making computation time potentially longer. Speed is limited by the energy gap between ground and excited states. | Can be faster in finding approximate solutions to certain optimization problems, but may not always reach the true global minimum. |
| Relationship to QA | QA can be seen as a heuristic implementation of AQC, with less strict adherence to the adiabatic theorem. | Often seen as a subset of AQC or an approximation, as it does not always follow the conditions of the adiabatic theorem closely. |