Tratamientos digitales

In this section, and in the rest of the chapter, we will be focusing on the version of the belief propagation algorithm discussed in section 2.5.2 called sliding window belief prop- agation. To understand the error behavior of the algorithm physically, we will consider a one-dimensional chain with nearest neighbor interactions H = P

ihi,i+1. The thermal Gibbs state at inverse temperature β is given by ρ = e−βH_{/Z(β). The reduced state of} spins 2,3, . . . , N is obtained by tracing out the first spin of the chain, i.e.,

ρ2,...,N = Tr1(ρ). (3.1) We can formally define the effective thermal Hamiltonian Heff acting on sites 2, . . . , N up to normalization by

ρ2,...,N ∝e−βHeff. (3.2) In other words,Heff is the traceless part of−1_βLog(ρ2,...N) where Log is the principal matrix logarithm. Similar effective Hamiltonians have also been studied in [45] in the context of reduced density matrices of fermionic and bosonic ground states. We also define theeffective

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50

100

150

β

= 1

β

= 20

j

j

β

= 1

β= 20

β

= 10

β

= 10

Figure 3.1. (Color online) Upper bound on||V_effj ||(left) and correlations (right) of the Ising chain with critical transverse field. These exact values are obtained from a Jordan-Wigner transform. Note that _||V∞j|| decays much faster than the correlations (notice the different length-scales). The values for β= 10 are in a different color as an aid to the eye.

thermal potential, V =Heff− N−1 X i=2 hi,i+1, (3.3)

as the term added to the bare Hamiltonian on sites 2 toN due to the presence of site 1. At high temperatures, the effective thermal potential is short ranged. Consider for instance the one-dimensional Ising model with transverse field on an infinite chain

H= ∞

X

i=−∞

σ_izσz_i+1+Bσ_ix. (3.4)

At zero temperature, this model exhibits a phase transition at the critical transverse field valueB = 1. Figure 3.1 shows the value of the effective potential,

Veff =− 1

βLogTr−∞,...,0(e

−βH_)−H

obtained from cutting the critical Ising chain in half, i.e., tracing out spins−∞ to 0 from the thermal state of an infinite chain. What is plotted is (an upper bound to) the operator norm of the cumulants ofVeff:

V_effj = Trj+1,...,∞(Veff − j−1 X k=1 V_effk), (3.6) withV1

eff = Tr2,...,∞(Veff). We see thatVeff has a very short range, in fact much shorter than the correlation length in the system.

When the window size in the belief propagation algorithm is sufficiently large to support the effective thermal potential, the messages contain all the information required to calculate the states of the rest of the system. In this case, we can use the sliding window belief propagation algorithm as discussed in section 2.5.2.

However, the effective thermal potential Veff cannot be computed exactly because of computational limitations. Instead, we can only estimate its value on a cluster of finite size. Thus, the main source of error in our method is due to the truncation ofVeff. We can assess the error by evaluating the portion of Veff we discard. On a chain for instance, the error caused on the estimate of the beliefs is

1 Z(β)(e −βHeff _−e−β(Heff−Pj>lV j eff))≈βX j>l hV_effj _i (3.7)

for βV_effj 1. Making the assumption (see figure 3.1) that _kV_effj _k decreases exponentially withj, we estimate this quantity by

β_hVl+1 eff i ≈βhV l effi hVl effi hV_effl−1i ! , (3.8)

which is our final error estimate. With the same reasoning, similar estimates can be derived for the error associated with different observables.

Figure 3.2 compares this error estimate to the true error produced by BP for the eval- uation of the energy density of the critical one-dimensional Ising chain. Clearly, the error estimate accurately bounds the true error. This figure also illustrates the power of BP by comparing the accuracy with which BP estimates the energy density of an infinite chain to what is achieved by brute-force diagonalization with equivalent computational resources.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10!10 10!8 10!6 10!4 10!2 Error in energy T

Exact diagonalization error

BP error estimate BP error

iTEBD error

Figure 3.2. Error on the energy density as a function of temperature for the critical Ising chain. For QBP with window size l= 10, we show the error estimate of (3.8) and the true error obtained by comparison with analytical solution. Also shown is the finite size error for exact diagonalization of a 11-site chain and error caused by iTEDB [14] with parameters δT = 0.001 and χ= 150, both of which require equivalent computational resources.

BP largely outperforms diagonalization down to temperatures of order 0.1. This behavior is expected because, as seen on figure 3.1 (red line), the range of the effective thermal potential becomes larger than the window size (l= 10) at this temperature.