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Pseudointeraction ¯ V$_{ij}$ is given by substituting P$_{ij}$ of the above equation into eq. (4). 

To make a comparison between the new measurement methods and the usual ones, we do simulation in the following three cases: 

(a) Usual MC method with usual measurements. 

(b) SCO method with usual measurements. 

(c) SCO method with measurements by using the for- mulae derived in § III. 

In all the three cases, simulated annealing method is used. The system is gradually cooled from an initial tem- perature T = 1 . 3 D to 0 . 3 D in steps of Δ T = 0 . 05 D . The system is kept at each temperature for 2 × 10 6 MC steps. The first 10 6 MC steps are for equilibration and the fol- lowing 10 6 MC steps are for measurement. The size N is 10 $^{3}$. The size is not so large because we do simulation not only with the SCO method but also with a usual method which requires O ( N $^{2}$) computational time per one MC step. 

In Fig. 1, internal energy E and heat capacity C mea- sured in the three cases are plotted as a function of tem- perature. We see that all the data nicely collapse into a single curve, showing the validity of the formulae de- rived in § III. We also see that heat capacity has a peak around T/D = 0 . 55. This result is consistent with pre- vious works which show the existence of a phase transi- tion around this temperature [20, 21]. Figure 2 shows the average number ¯ n of potentials per site that sur- vive as ¯ V$_{ij}$ in the potential switching process. Though ¯ n increases with decreasing temperature, ¯ n ≈ 70 even at the lowest temperature. This means that more than 90 percent of interactions are cut off by being switched to ˜ V$_{ij}$ = 0. It is worth pointing out that the SCO method becomes more efficient with increasing the size. In fact, in the study of two-dimensional magnetic dipo- lar system with dipolar interactions and ferromagnetic 

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exchange interactions [11], it has been found that the in- crease of ¯ n with size is very slow and ¯ n is 22 . 5 even when N = 256 2 = 65 , 536. 

## B. Statistical error of the new measurement methods 

We notice from Fig. 1 (II) that the data in the case (c), i.e. , those which are measured with eq. (25), fluc- tuate more than the other data. To get some insights of this behavior, we consider estimating statistical er- ror of observables. We suppose that an observable Q is successively measured N times in a MC simulation to estimate the average 1 N ∑ N µ $_{=1}$Q$_{µ}$ . We assume that the measurement is done every δt steps. Although this average value is close to the thermal average value 〈 Q 〉 , they are slightly different because the number of the mea- surement N is finite. We hereafter call the difference δQ ≡ 1 N ∑ µ $_{=1}$Q$_{µ}$ − 〈 Q 〉 statistical error . When the pe- riod of the measurement Nδt is much larger than the correlation time τ$_{Q}$ of the observable Q , the expectation value of the square of the statistical error 〈 ( δQ ) $^{2}$〉 is ap- proximately evaluated as [22, 23] 

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The factor (1+2 δt ) in the right hand side of the equation comes from the fact that Q$_{µ}$ ’s measured successively are correlated with each other. From this equation, we can estimate the relative statistical error as 

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