Cooling Algorithms Based on the 3-bit Majority

Abstract

Algorithmic cooling is a potentially important technique for making scalable NMR quantum computation feasible in practice. Given the constraints imposed by this approach to quantum computing, the most likely cooling algorithms to be practicable are those based on simple reversible polarization compression (RPC) operations acting locally on small numbers of bits. Several different algorithms using 2-and 3-bit RPC operations have appeared in the literature, and these are the algorithms I consider in this note. Specifically, I show that the RPC operation used in all these algorithms is essentially a majority vote of 3 bits, and prove the optimality of the best such algorithm. I go on to derive some theoretical bounds on the performance of these algorithms under some specific assumptions about errors.KEY WORDS: algorithmic cooling; NMR quantum computation; reversible polarization compression; error analysis of cooling algorithms; cooling algorithms based on the 3-bit majority.

Asymptotic Bound for Heat-Bath Algorithmic Cooling

“…The original idea of algorithmic cooling was developed by Schulman and Vazirani in [25] which uses a technique for Schumacher's quantum data compression [26,27]. Later it was proposed to use a heat-bath to enhance the cooling beyond the compression bounds [23,28].…”

“…The idea is that after the entropy transfer, the heat-bath refreshes the hot qubits and then the entropy transfer can be repeated. Different iterative methods were developed based on this idea [27,29,30]. All of these methods are referred to as "Heat-Bath Algorithmic Cooling".…”

Achievable Polarization for Heat-Bath Algorithmic Cooling

“…Based on this work, many cooling algorithms have been designed [6][7][8][9][10][11]. HBAC is not only of theoretical interest, experiments have already demonstrated an improvement in polarization using this protocol with a few qubits [12][13][14][15][16][17][18], where a few rounds of HBAC were reached; and some studies have even included the impact of noise [19].…”

“…Based on this work, many cooling algorithms have been designed [6][7][8][9][10][11]. HBAC is not only of theoretical interest, experiments have already demonstrated an improvement in polarization using this protocol with a few qubits [12][13][14][15][16][17][18], where a few rounds of HBAC were reached; and some studies have even included the impact of noise [19].Through numerical simulations, Moussa [7] and Schulman et al [8] observed that if the polarization of the bath ( b ) is much smaller than 2 −n , where n is the number of qubits used, the asymptotic polarization reached will be ∼ 2 n−2 b ; but when b is greater than 2 −n , a polarization of order one can be reached. Inspired also by the work of Patange [20], who investigated the use of algorithmic cooling on spins bigger than 1 2 (using NV center where the defect has an effective spin 1), we investigate the case of cooling a qubit using a general spin l, and extra qubits which get contact with a bath.…”

Semioptimal practicable algorithmic cooling

“…The optimal algorithms, the PPA and all-bonacci, need about half the spins (more precisely, J +2 spins) in order to cool (asymptotically) the coldest spin to 2 J ; we thus see that mPAC is semi-optimal in the sense that it needs twice as much spins to reach nearly the same optimal bias. Section III discusses simple variants of the Fibonacci algorithm: δ-Fibonacci [21,23], and mFib, which fixes the number of cycles at each recursive level. Section IV compares the new algorithms, mPAC and mFib, to previous cooling algorithms, including the PPA, Fibonacci, allbonacci, and PAC algorithms.…”