# heat bath algorithm

## heat bath algorithm

In MRS, the nuclear spins of the atoms of the metabolites are required to be with a certain degree of polarization, so the spectroscopy can succeed. This notation denotes a diagonal matrix whose diagonal entries are listed within the parentheses. the state is exactly in the center. ϵ {\displaystyle A_{new},C_{new}} 1 − ⁡ , and applying = {\displaystyle A} = 2 = ρ on average. , 2 0 w And it was an early meeting place for the basic simulation methods as the Metropolis algorithm that we discussed in the lecture or the closely related heat-bath algorithm later on, also for the cluster simulation methods. B and 2 , then the cooling limit is and the goal is to use coins , n In this illustrative description of the algorithm, the boosted bias of qubit | ( , 2 + Infrared lamps, aka heat lamps, are an excellent source of heat that is relatively simple and easy to install. , and 0 = ≪ The result of this operation can be obtained by writing the product state of the 3 qubits, n 0 | c The algorithm operates on an array of equally (and independently) biased qubits. b ϵ {\displaystyle |0\rangle } {\displaystyle \epsilon } ( 1 A {\displaystyle A'} The "cooling limit" (the maximum bias the target qubit can reach) depends on the bias of the bath and the number of qubits of each kind in the system. {\displaystyle \rho } ϵ k {\displaystyle \epsilon _{b}\ll 1} 1 B ρ ϵ ⊗ = . − Â© 2020 Coursera Inc. All rights reserved. | 2 parameter ( w − A bias can be negative (for example o a + The heat-bath algorithm implements a true Markov chain that does approach equilibrium but using a random seed you can be sure that any other configuration will after some time lead to the same sequence of configurations So, we can continue our two simulations here to see that they really stay the same for all times This simply means that the state of our Markov chain depends on the sequence of sites k(t) and random numbers Upsilon(t), but not on the initial configuration. b 2 0 C , where If external work is applied in order to move the partition in a reversible manner, the gas in one compartment is compressed, resulting in higher temperature (and entropy), while the gas in the other is expanding, similarly resulting in lower temperature (and entropy). ϵ ϵ and is pure i The compression step in each iteration is slightly different, but its goal is to sort the qubits in descending order of bias, so that the reset qubit would have the smallest bias (namely, the highest temperature) of all qubits. 1 1 2 A ) 0 | This means that entropy can be transferred from the qubits to an external reservoir and some operations can be irreversible, which can be used for cooling some qubits without heating the others. A more general method, "irreversible algorithmic cooling", makes use of irreversible transfer of heat outside of the system and into the environment (and therefore may bypass the Shannon bound). ϵ But note however that in order to study this phase transition in very large systems it is actually better to use the cluster algorithm that has been presented by Werner in this week's lecture. 0 A Compression: a reversible compression (entropy transfer) is applied on the three qubits. σ ⁡ 1 {\displaystyle {\frac {1+\epsilon }{2}}} If we would return the spin in the state +1 it would have the energy E=-h if instead we would return the spin in the state -1 it would have the energy E=h This leads to the normalized probabilities pi_plus and pi_minus So if a random number Upsilon between 0 and 1 is smaller than the probability pi_plus, we return the spin in the state +1 if on the other hand the number Upsilon is larger than pi_plus we return the spin in the state -1 In Python, this gives the very short program heat-bath_ising.py In this program we keep track of and update the energy E so that we can easily obtain the specific heat c_V of the system as a function of temperature, and we see its characteristic peak which indicates the ferromagnetic-paramagnetic phase transition which takes place at a temperature Tc that we know exactly. b b  In addition, algorithmic cooling can be applied to in vivo magnetic resonance spectroscopy.. For this, we simply pull the spin out of the system and measure the field h at the now empty position. 1 b systems with short-ra n ge in teractions. -biased, which is exactly the state of the qubits before the reversible algorithm is applied. 0 {\displaystyle |0\rangle } . {\displaystyle \epsilon _{new}^{average}={\frac {3}{2}}\epsilon } The best method may depend on your personal living circumstances and the climate you live in. ϵ {\displaystyle B,C} The density matrices The basic procedure is called "Basic Compression Subroutine" or "3 Bit Compression".. ∑ 0 Therefore, it is important to be familiar with both the core principles and the relevant notations. {\displaystyle |1\rangle } r , the maximal polarization that can be obtained is proportional to For example for a system of 100x100 spins how can we check that all the 2^10000 configurations finally merge at a given time? C i 1 2 ϵ a Ensemble computing is a computational model that uses a macroscopic number of identical computers. For this, algorithmic cooling can be used to produce qubits with the desired purity for quantum error correction. ( e + i n A ∈ v e diag Therefore, the process of cooling spins can be thought of as a process of transferring entropy between spins, or outside of the system. − a β Transferring as much entropy as possible away from the whole system (and in particular the reset qubit) and into the bath in the following refresh step. Two typical applications that require a large number of pure qubits are quantum error correction (QEC) and ensemble computing. The only problem was that I was using python 3+ and the programs were written with python 2+. b C 1 ⊕ w For So now let us look at the configurations generated by this algorithm. ) B = {\displaystyle \epsilon _{new}^{average}={\frac {3}{2}}\epsilon }

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