Quantum Turing machines, quantum circuits, and quantum adiabatic algorithms are polynomially equivalent, in complexity class BQP [1,2]. Concerning quantum annealers, it is unknown whether they offer and speedup relative to classical annealers.

To find a computational model that is in a different complexity class, one has to look for non-universal quantum computers. For example, adiabatic quantum computation of "stoquastic" Hamiltonians [3] (with real nonpositive off-diagonal matrix in the computational basis) is in BStoqP $\subset$ BQP.

Another example, Instantaneous Quantum Polynomial-time (IQP) computation is a model of quantum computation consisting only of commuting two-qubit gates. It is not believed that a classical algorithm that can simulate IQP efficiently [4].

1. A survey of quantum complexity theory, U. Vazirani.

2. Simple proof of equivalence between adiabatic quantum computation and the circuit model, A. Mizel, D.A. Lidar, M. Mitchell.

3. Adiabatic Quantum Computing, T. Albash, D.A. Lidar.

4. Quantum Commuting Circuits and Complexity of Ising Partition Functions, K. Fujii, T. Morimae.

Quantum Turing machines, quantum circuits, and quantum adiabatic algorithms are polynomially equivalent, in complexity class BQP [1,2]. Concerning quantum annealers, it is unknown whether they offer any speedup relative to classical annealers.

To find a computational model that is in a different complexity class, one has to look for non-universal quantum computers. For example, adiabatic quantum computation of "stoquastic" Hamiltonians [3] (with real nonpositive off-diagonal matrix in the computational basis) is in BStoqP $\subset$ BQP.

Another example, Instantaneous Quantum Polynomial-time (IQP) computation is a model of quantum computation consisting only of commuting two-qubit gates. It is believed that no classical algorithm can simulate IQP efficiently [4].

1. A survey of quantum complexity theory, U. Vazirani.

2. Simple proof of equivalence between adiabatic quantum computation and the circuit model, A. Mizel, D.A. Lidar, M. Mitchell.

3. Adiabatic Quantum Computing, T. Albash, D.A. Lidar.

4. Quantum Commuting Circuits and Complexity of Ising Partition Functions, K. Fujii, T. Morimae.

Quantum Turing machines, quantum circuits, and quantum adiabatic algorithms are polynomially equivalent, in complexity class BQP [1,2]. Concerning quantum annealers, it is unknown whether they offer and speedup relative to classical annealers.

To find a computational model that is in a different complexity class, one has to look for non-universal quantum computers. For example, adiabatic quantum computation of "stoquastic" Hamiltonians [3] (with real nonpositive off-diagonal matrix in the computational basis) is in BStoqP $\subset$ BQP.

1. A survey of quantum complexity theory, U. Vazirani.

2. Simple proof of equivalence between adiabatic quantum computation and the circuit model, A. Mizel, D.A. Lidar, M. Mitchell.

3. Adiabatic Quantum Computing, T. Albash, D.A. Lidar.

Quantum Turing machines, quantum circuits, and quantum adiabatic algorithms are polynomially equivalent, in complexity class BQP [1,2]. Concerning quantum annealers, it is unknown whether they offer and speedup relative to classical annealers.

To find a computational model that is in a different complexity class, one has to look for non-universal quantum computers. For example, adiabatic quantum computation of "stoquastic" Hamiltonians [3] (with real nonpositive off-diagonal matrix in the computational basis) is in BStoqP $\subset$ BQP.

Another example, Instantaneous Quantum Polynomial-time (IQP) computation is a model of quantum computation consisting only of commuting two-qubit gates. It is not believed that a classical algorithm that can simulate IQP efficiently [4].

1. A survey of quantum complexity theory, U. Vazirani.

2. Simple proof of equivalence between adiabatic quantum computation and the circuit model, A. Mizel, D.A. Lidar, M. Mitchell.

3. Adiabatic Quantum Computing, T. Albash, D.A. Lidar.

4. Quantum Commuting Circuits and Complexity of Ising Partition Functions, K. Fujii, T. Morimae.