Spectral Gap Amplification

@article{Somma2013SpectralGA,
  title={Spectral Gap Amplification},
  author={Rolando D. Somma and Sergio Boixo},
  journal={SIAM J. Comput.},
  year={2013},
  volume={42},
  pages={593-610}
}
Many problems can be solved by preparing a specific eigenstate of some Hamiltonian $H$. The generic cost of quantum algorithms for these problems is determined by the inverse spectral gap of $H$ for that eigenstate and the cost of evolving with $H$ for some fixed time. The goal of spectral gap amplification is to construct a Hamiltonian $H'$ with the same eigenstate as $H$ but a bigger spectral gap, requiring that constant-time evolutions with $H'$ and $H$ are implemented with nearly the same… 

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References

SHOWING 1-10 OF 67 REFERENCES

Complexity of Stoquastic Frustration-Free Hamiltonians

TLDR
The Cook-Levin theorem proving NP-completeness of the satisfiability problem is generalized to the complexity class MA (Merlin-Arthur games)—a probabilistic analogue of NP.

Fast quantum algorithms for traversing paths of eigenstates

TLDR
This work gives ``digital'' methods for performing the transformation of Hamiltonians that require no assumption on path continuity or differentiability other than the absence of large jumps, and demonstrates that path length and the gap are the primary parameters that determine the complexity of state transformation along a path.

Eigenpath traversal by phase randomization

TLDR
The quantum simulated annealing algorithm for solving combinatorial optimization problems provides a quadratic speed-up in the gap of the stochastic matrix over its classical counterpart implemented via Markov chain Monte Carlo.

The complexity of stoquastic local Hamiltonian problems

TLDR
It is proved that LH-MIN for stoquastic Hamiltonians belongs to the complexity class AM -- a probabilistic version of NP with two rounds of communication between the prover and the verifier, and that any problem solved by adiabatic quantum computation using stoquian Hamiltonians is in PostBPP.

Analog analogue of a digital quantum computation

We solve a problem, which while not fitting into the usual paradigm, can be viewed as a quantum computation. Suppose we are given a quantum system with a Hamiltonian of the form $E|w〉〈w|$ where $|w〉$

Quantum Computation by Adiabatic Evolution

We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that

Quantum annealing in the transverse Ising model

We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. Quantum fluctuations cause transitions between

The Power of Quantum Systems on a Line

TLDR
The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Some illegal configurations cannot be ruled out by local checks, and are instead ruled out because they would, in the future, evolve into a state which can be seen locally to be illegal.

Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

We present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse Hamiltonian H over a given time t in terms of a procedure for computing the matrix entries of H.

Universal quantum walks and adiabatic algorithms by 1D Hamiltonians

We construct a family of time-independent nearest-neighbor Hamiltonians coupling eight-state systems on a 1D ring that enables universal quantum computation. Hamiltonians in this family can achieve
...