Quantum algorithm for preparing thermal Gibbs states - detailed analysis

In a recent work [10], Poulin and one of us presented a quantum algorithm for preparing thermal Gibbs states of interacting quantum systems. This algorithm is based on Grovers's technique for quantum state engineering, and its running time is dominated by the factor D/Z(\beta), where D and Z(\beta) denote the dimension of the quantum system and its partition function at inverse temperature \beta, respectively. We present here a modified algorithm and a more detailed analysis of the errors that arise due to imperfect simulation of Hamiltonian time evolutions and limited performance of phase estimation (finite accuracy and nonzero probability of failure). This modfication together with the tighter analysis allows us to prove a better running time by the effect of these sources of error on the overall complexity. We think that the ideas underlying of our new analysis could also be used to prove a better performance of quantum Metropolis sampling by Temme et al. [12].

[1]  P. Wocjan,et al.  Quantum algorithm for approximating partition functions , 2008, 0811.0596.

[2]  D. Poulin,et al.  Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. , 2009, Physical review letters.

[3]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[4]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

[5]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

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

[7]  Jeroen van de Graaf,et al.  Cryptographic Distinguishability Measures for Quantum-Mechanical States , 1997, IEEE Trans. Inf. Theory.

[8]  Pawel Wocjan,et al.  Thermalizing quantum systems and evaluating partition functions with a quantum computer , 2009 .

[9]  Gilles Brassard,et al.  Tight bounds on quantum searching , 1996, quant-ph/9605034.

[10]  Raymond Laflamme,et al.  An Introduction to Quantum Computing , 2007, Quantum Inf. Comput..

[11]  Yong Zhang,et al.  Fast amplification of QMA , 2009, Quantum Inf. Comput..

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

[13]  Seiji Miyashita,et al.  in Quantum Monte Carlo methods in equilib - rium and nonequilibrium systems , 1987 .

[14]  D. DiVincenzo,et al.  Problem of equilibration and the computation of correlation functions on a quantum computer , 1998, quant-ph/9810063.

[15]  F. Verstraete,et al.  Quantum Metropolis sampling , 2009, Nature.

[16]  Christof Zalka.  Simulating quantum systems on a quantum computer , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.