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Quantum adiabatic computation with a constant gap is not useful in one dimension
1Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, California 93106, USA.
Physical Review Letters
|October 2, 2009
Summary
Classical computers can efficiently simulate one-dimensional quantum systems with constant spectral gaps using matrix product states. This finding suggests adiabatic quantum computation is not universally powerful under these specific conditions.
Area of Science:
- Quantum Computing
- Computational Physics
- Condensed Matter Theory
Background:
- Adiabatic quantum computation leverages the adiabatic theorem for solving complex problems.
- The efficiency of quantum simulations is often limited by the complexity of the quantum system's state.
- Area laws in quantum systems constrain the entanglement and complexity of ground states.
Purpose of the Study:
- To investigate the classical simulation feasibility of one-dimensional adiabatic quantum evolution with a constant spectral gap.
- To determine the implications of efficient classical simulation for the power of adiabatic quantum computation.
Main Methods:
- Utilizing a recently proven area law for one-dimensional quantum systems.
- Employing matrix product state (MPS) representations for ground states.
- Developing algorithms to update MPS as the Hamiltonian changes during adiabatic evolution.
Main Results:
- Demonstrated efficient classical simulation of adiabatic evolution for 1D quantum systems with constant spectral gaps.
- Showed that matrix product states provide an efficient representation for simulating these systems.
- Established that adiabatic evolution with constant spectral gaps in 1D is not sufficient for universal quantum computation.
Conclusions:
- Adiabatic quantum computation is not universally powerful for one-dimensional systems with constant spectral gaps.
- Universal quantum computation via adiabatic algorithms likely requires a vanishing spectral gap or higher dimensions.
- The computational power of adiabatic simulation in higher dimensions with constant spectral gaps remains an open question.
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