Hamiltonian simulation

Hamiltonian simulation (also referred to as quantum simulation) is a problem in quantum information science that attempts to find the computational complexity and quantum algorithms needed for simulating quantum systems. Hamiltonian simulation is a problem that demands algorithms which implement the evolution of a quantum state efficiently. The Hamiltonian simulation problem was proposed by Richard Feynman in 1982, where he proposed a quantum computer as a possible solution since the simulation of general Hamiltonians seem to grow exponentially with respect to the system size. [1]

Problem statement

In the Hamiltonian simulation problem, given a Hamiltonian ( hermitian matrix acting on qubits), a time and maximum simulation error , the goal is to find an algorithm that approximates such that , where is the ideal evolution and is the spectral norm.[2] A special case of the Hamiltonian simulation problem is the local Hamiltonian simulation problem. This is when is a k-local Hamiltonian on qubits where and acts non-trivially on at most qubits instead of qubits. [3] The local Hamiltonian simulation problem is important because most Hamiltonians that occur in nature are k-local. [3]

Techniques

Product Formulas

Also known as Trotter formulas or Trotter-Suzuki decompositions, Product formulas simulate the sum-of-terms of a Hamiltonian by simulating each one separately for a small time slice. [4] If , then for a large ; where is the number of time steps to simulate for. The large the , the more accurate the simulation.

If the Hamiltonian is represented as a Sparse matrix, the distributed edge coloring algorithm can be used to decompose it into a sum of terms; which can then be simulated by a Trotter-Suzuki algorithm. [5]

Taylor Series

by the Taylor series expansion. [6] This says that during the evolution of a quantum state, the Hamiltonian is applied over and over again to the system with a various number of repetitions. The first term is the identity matrix so the system doesn't change at first, but in the second term the Hamiltonian is applied once. For practical implementations, the series has to be truncated () where the bigger the , the more accurate the simulation. [7]

Quantum Walk

In the quantum walk, a unitary operation whose spectrum is related to the Hamiltonian in implemented then the Quantum phase estimation algorithm is used to adjust the eigenvalues. This makes it unnecessary to decompose the Hamiltonian into a sum-of-terms like the Trotter-Suzuki methods. [6]

Quantum signal processing

The quantum signal processing algorithm works by transducing the eigenvalues of the Hamiltonian into an ancilla qubit, transforming the eigenvalues with single qubit rotations and finally projecting the ancilla. [8] It has been proved to be optimal in query complexity when it comes to Hamiltonian simulation. [8]

Complexity

The table of the complexities of the Hamiltonian simulation algorithms mentioned above. The Hamiltonian simulation can be studied in two ways. This depends on how the Hamiltonian is given. If it is given explicitly, then gate complexity matters more than query complexity. If the Hamiltonian is described as an Oracle (black box) then the number of queries to the oracle is more important than the gate count of the circuit. The following table shows the gate and query complexity of the previously mentioned techniques.

Technique Gate complexity Query complexity
Product formula 1st order [7] [9]
Taylor Series [7] [6]
Quantum walk [7] [5]
Quantum Signal Processing [7] [8]

Where is the largest entry of .

References

  1. Richard P Feynman (1982). "Simulating physics with computers". International Journal of Theoretical Physics. 21 (6): 467–488. Bibcode:1982IJTP...21..467F. doi:10.1007/BF02650179. Retrieved 2019-05-04.
  2. Stuart Hadfield, Anargyros Papageorgiou (2018). "Divide and conquer approach to quantum Hamiltonian simulation". New Journal of Physics. 20 (4): 043003. Bibcode:2018NJPh...20d3003H. doi:10.1088/1367-2630/aab1ef.CS1 maint: uses authors parameter (link)
  3. Lloyd, S. (1996). "Universal quantum simulators". Science. 273 (5278): 1073–8. Bibcode:1996Sci...273.1073L. doi:10.1126/science.273.5278.1073. PMID 8688088.
  4. Barthel, Thomas; Zhang, Yikang (2019). "Optimized Lie-Trotter-Suzuki decompositions for two and three non-commuting terms". arXiv:1901.04974 [quant-ph].
  5. Berry, Dominic; Childs, Andrew; Kothari, Robin (2015). "Hamiltonian simulation with nearly optimal dependence on all parameters". 2015 IEEE 56th Annual Symposium on Foundations of Computer Science. pp. 792–809. arXiv:1501.01715. Bibcode:2015arXiv150101715B. doi:10.1109/FOCS.2015.54. ISBN 978-1-4673-8191-8.
  6. Berry, Dominic; Childs, Andrew; Cleve, Richard; Kothari, Robin; Rolando, Somma (2015). "Simulating Hamiltonian dynamics with a truncated Taylor series". Physical Review Letters. 114 (9): 090502. arXiv:1412.4687. Bibcode:2015PhRvL.114i0502B. doi:10.1103/PhysRevLett.114.090502. PMID 25793789.
  7. Childs, Andrew; Maslov, Dmitri; Nam, Yunseong (2017). "Toward the first quantum simulation with quantum speedup". Proceedings of the National Academy of Sciences. 115 (38): 9456–9461. arXiv:1711.10980. Bibcode:2017arXiv171110980C. doi:10.1073/pnas.1801723115. PMC 6156649. PMID 30190433.
  8. Low, Guang Hao; Chuang, Isaac (2017). "Optimal Hamiltonian Simulation by Quantum Signal Processing". Physical Review Letters. 118 (1): 010501. arXiv:1606.02685. Bibcode:2017PhRvL.118a0501L. doi:10.1103/PhysRevLett.118.010501. PMID 28106413.
  9. Kothari, Robin (Dec 8, 2017). Quantum algorithms for Hamiltonian simulation: Recent results and open problems (Youtube). United States: IBM Research.
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