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Quantum adiabatic algorithm and scaling of gaps at first-order quantum phase transitions
C R Laumann1, R Moessner, A Scardicchio
1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA.
Physical Review Letters
|August 7, 2012
Summary
The quantum adiabatic algorithm (QAA) can succeed across first-order transitions, even with algebraic gaps. However, it may fail on simple problems, highlighting limitations in quantum computing for certain tasks.
Area of Science:
- Quantum physics
- Condensed matter theory
- Quantum computation
Background:
- The quantum adiabatic algorithm (QAA) is a quantum computation method.
- QAA performance is often limited by the Hamiltonian gap at phase transitions.
- First-order transitions typically exhibit exponentially small gaps, hindering QAA.
Purpose of the Study:
- To investigate the scaling of Hamiltonian gaps at quantum first-order transitions.
- To explore conditions under which the QAA can be successful or fail.
- To identify simple systems where QAA might be inefficient.
Main Methods:
- Analysis of a quantum antiferromagnetic Ising chain in a staggered field.
- Construction of a classical, translationally invariant 1D Hamiltonian with nearest-neighbor interactions.
- Theoretical examination of Hamiltonian gap scaling at quantum first-order transitions.
Main Results:
- A quantum antiferromagnetic Ising chain exhibits a first-order transition with an algebraically small gap.
- A simple 1D classical Hamiltonian shows an exponential gap at a topological quantum first-order transition.
- Demonstration that QAA can succeed across first-order transitions, but also fail on simple problems.
Conclusions:
- The QAA's success is not universally precluded by first-order transitions.
- Simple classical models can possess features that challenge the efficiency of QAA.
- Understanding Hamiltonian gap scaling is crucial for optimizing quantum algorithms.
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