Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

@article{Berry2005EfficientQA,
  title={Efficient Quantum Algorithms for Simulating Sparse Hamiltonians},
  author={Dominic W. Berry and Graeme Ahokas and Richard Cleve and Barry C. Sanders},
  journal={Communications in Mathematical Physics},
  year={2005},
  volume={270},
  pages={359-371}
}
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. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and ||H|| is bounded by a constant, we may select any positive integer k such that the simulation requires O((log*n)t1+1/2k) accesses to matrix entries of H. We also show that the temporal… 

A quantum algorithm for simulating non-sparse Hamiltonians

A quantum algorithm for simulating the dynamics of Hamiltonians that are not necessarily sparse, based on the input model where the entries of the Hamiltonian are stored in a data structure in a quantum random access memory to achieve poly-logarithmic dependence on precision.

Simulating quantum dynamics on a quantum computer

A range of techniques to simulate Hamiltonians with badly behaved derivatives are proposed, including using adaptive time steps, adapting the order of the integrators, and omitting regions about discontinuities.

Exponential improvement in precision for simulating sparse Hamiltonians

The algorithm is based on a significantly improved simulation of the continuous- and fractional- query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error.

Quantum Algorithm for Simulating Real Time Evolution of Lattice Hamiltonians

This paper studies the problem of simulating the time evolution of a lattice Hamiltonian, and proves a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires (nT) gates in the worst case.

Randomized Algorithms for Hamiltonian Simulation

First a scheme to bound the error of the final quantum state in a randomized algorithm is provided, and then randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement are obtained.

Hamiltonian simulation with nearly optimal dependence on spectral norm

  • G. Low
  • Computer Science
    STOC
  • 2019
This paper presents a quantum algorithm for approximating the real time evolution e−iHt of an arbitrary d-sparse Hamiltonian to error є, given black-box access to the positions and b-bit values of its non-zero matrix entries, which matches the quantum search lower bound of Ω(√d) queries and improves upon prior art of Õ(d2/3) queries.

Optimal Hamiltonian Simulation by Quantum Signal Processing.

It is argued that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation.

Parallel Quantum Algorithm for Hamiltonian Simulation

A novel notion of parallel quantum walk is introduced, based on Childs’ quantum walk, that is applied to simulating three physical models: the Heisenberg model, the Sachdev-Ye-Kitaev model and a quantum chemistry model in second quantization, and it is shown that the total gate depth of the algorithm has a polylog log(1/ǫ) dependence in the parallel setting.

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

A new “Quantum singular value transformation” algorithm is developed that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator.

Exponentially more precise quantum simulation of fermions in the configuration interaction representation

We present a quantum algorithm for the simulation of molecular systems that is asymptotically more efficient than all previous algorithms in the literature in terms of the main problem parameters. As
...

References

SHOWING 1-10 OF 22 REFERENCES

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 information processing in continuous time

Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not

Quantum walk algorithm for element distinctness

  • A. Ambainis
  • Computer Science
    45th Annual IEEE Symposium on Foundations of Computer Science
  • 2004
An O(N/sup k/(k+1)/) query quantum algorithm is given for the generalization of element distinctness in which the authors have to find k equal items among N items.

Adiabatic quantum state generation and statistical zero knowledge

The ASG approach to quantum algorithms provides intriguing links between quantum computation and many different areas: the analysis of spectral gaps and groundstates of Hamiltonians in physics, rapidly mixing Markov chains, statistical zero knowledge, and quantum random walks.

Quantum lower bounds by polynomials

This work examines the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.

The Complexity of the Local Hamiltonian Problem

This paper settles the question and shows that the 2-LOCAL HAMILTONIAN problem is QMA-complete, and demonstrates that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation.

Limit on the Speed of Quantum Computation in Determining Parity

It is shown that any quantum algorithm capable of determining the parity of f contains at least N/2 applications of the unitary operator which evaluates f and quantum computers cannot outperform classical computers.

Quantum computation and quantum information

  • T. Paul
  • Physics
    Mathematical Structures in Computer Science
  • 2007
This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal

Quantum random-walk search algorithm

It will be shown that this algorithm performs an oracle search on a database of N items with $O(\sqrt{N})$ calls to the oracle, yielding a speedup similar to other quantum search algorithms.

Exponential algorithmic speedup by a quantum walk

A black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer is constructed and it is proved that no classical algorithm can solve the problem in subexponential time.