Figure 1: The performance of QA (triangles), and QJA (cir- cles). The reference curve describes the instantaneous Gibbs- Boltzmann factor. point of JE. Even if quantum nature a ff ects the instan- taneous quantum state, the property of JE guarantees that we keep the quantum state expressing the instan- taneous equilibrium state. If one considers to simu- late this procedure in “classical” computers, we have to need the repetition of the pre-determined process to deal with all fluctuations in the nonequilibrium-process average, since we assume an ensemble in equilibrium in the formulation of JE. However, when we implement QJA in “quantum” computation, we operate QJA to a single quantum system in principal since the classical ensemble is mapped to the quantum wave function. We do not need the repetition of the same procedure di ff er- ently from the classical case [12]. Let us take a simple instance to search the minimum from a one-dimensional random potential, which is for- mulated as the Hamiltonian H$_{0}$ = − ∑ N i = $^{$_{1}$V$_{i}$}$| $^{i}$〉〈$^{i}$| . Here V$_{i}$ denotes the potential energy at site i and chosen ran- domly. We employ a linear schedule for tuning the pa- rameter β from 0 to 100. Figure 1 shows the plot for QJA (upper circles), which are fixed along the refer- ence curves (solid curve) representing the instantaneous Gibbs-Boltzmann factor. In contrast, QA (lower tri- angles) can not su ffi ciently find the ground state since we consider a very short annealing is considered in this case. state in a short annealing and implies that we may over- come the di ffi culties in hard optimization problems and solve them in a reasonable time. The present result is nothing but preliminary one. We should address the problem on practical e ffi ciency for several interesting hard problems we wish to solve in the future study [13]. This work was supported by CREST, JST. Acknowledgement References 5. Summary We consider an application of JE to quantum com- putation as QA to solve the optimization problems by using the classical-quantum mapping. As we expected, this protocol keeps the quantum system to express the equilibrium state for the instantaneous inverse temper- ature. The result by QJA shown here gives the ground 3 [1] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness Freeman, (San Fran- cisco, 1979). [2] A. B. Finnila, M. A. Gomez, C. Sebenik, S. Stenson, and J. D. Doll, Chem. Phys. Lett. 219 , 343 (1994). [3] T. Kadowaki and H. Nishimori, Phys. Rev. E 58 , 5355 (1998). [4] S. Morita, and H. Nishimori, J. Math. Phys. 49 , 125210 (2008). [5] S. Suzuki and M. Okada, J. Phys. Soc. Jpn. 74 , 1649 (2005). [6] T. Jorg, F. Krzakala, J. Kurchan, A. C. Maggs, Phys. Rev. Lett. 101 , 147204 (2008). [7] A. P. Young, S. Knysh, and V. N, Smelyanskiy, Phys. Rev. Lett. 104 , 020502 (2010). [8] C. Jarzynski, Phys. Rev. Lett. 78 , 2690 (1997). [9] C. Jarzynski, Phys. Rev. E 56 , 5018 (1997). [10] R. D. Somma, C. D. Batista, and G. Ortiz, Phys. Rev. Lett. 99 , 030603 (2007). [11] P. Wocjan, C. Chiang, D. Nagaj, and A. Abeyesinghe, Phys. Rev. A. 80 , 022340 (2009). [12] M. Ohzeki, work in progress. [13] M. Ohzeki and S. Tanaka, work in progress. 0.02 0.04 0.06 0.08 0.10 t 0.2 0.4 0.6 0.8 1.0 P$_{GS}$