Simulating Sparse Hamiltonians with Star Decompositions

We present an efficient algorithm for simulating the time evolution due to a sparse Hamiltonian. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian H acts for time t, this algorithm uses (d2(d+log* N) ∥Ht∥)1+o(1) queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d4(d+log* N) ∥Ht∥)1+o(1) To achieve this, we decompose a general sparse Hamiltonian into a small sum of Hamiltonians whose graphs of non-zero entries have the property that every connected component is a star, and efficiently simulate each of these pieces.

[1]  Edward Farhi,et al.  A Quantum Algorithm for the Hamiltonian NAND Tree , 2008, Theory Comput..

[2]  Andrew M. Childs,et al.  The quantum query complexity of implementing black-box unitary transformations , 2009 .

[3]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[4]  Andrew M. Childs,et al.  Quantum information processing in continuous time , 2004 .

[5]  Andrew M. Childs.  On the Relationship Between Continuous- and Discrete-Time Quantum Walk , 2008, 0810.0312.

[6]  R. Cleve,et al.  Efficient Quantum Algorithms for Simulating Sparse Hamiltonians , 2005, quant-ph/0508139.

[7]  R. Feynman.  Simulating physics with computers , 1999 .

[8]  Andrew V. Goldberg,et al.  Parallel symmetry-breaking in sparse graphs , 1987, STOC.

[9]  Daniel A. Spielman,et al.  Exponential algorithmic speedup by a quantum walk , 2002, STOC '03.

[10]  Richard Cole,et al.  Deterministic Coin Tossing with Applications to Optimal Parallel List Ranking , 2018, Inf. Control..

[11]  Amnon Ta-Shma,et al.  Adiabatic quantum state generation and statistical zero knowledge , 2003, STOC '03.

[12]  M. Sipser,et al.  Quantum Computation by Adiabatic Evolution , 2000, quant-ph/0001106.

[13]  Andrew M. Childs,et al.  Black-box hamiltonian simulation and unitary implementation , 2009, Quantum Inf. Comput..

[14]  Alessandro Panconesi,et al.  Some simple distributed algorithms for sparse networks , 2001, Distributed Computing.

[15]  Edward Farhi,et al.  Analog analogue of a digital quantum computation , 1996 .

[16]  Andrew M. Childs,et al.  Limitations on the simulation of non-sparse Hamiltonians , 2009, Quantum Inf. Comput..