Cyclic cooling algorithm

Abstract

We introduce a scheme to perform the cooling algorithm, first presented by Boykin et al. in 2002, for an arbitrary number of times on the same set of qbits. We achieve this goal by adding an additional SWAP-gate and a bath contact to the algorithm. This way one qbit may repeatedly be cooled without adding additional qbits to the system. By using a product Liouville space to model the bath contact we calculate the density matrix of the system after a given number of applications of the algorithm.PACS numbers: 03.67.Lx, Algorithmic cooling is a method to obtain highly polarized spins in a spin system without cooling down the environment. It may for example be used for medical magnetic resonance imaging to improve the resolution by cooling down a subset of nuclear spins of a patient without cooling of the patient himself, or for the preparation of the ground state of a quantum computer by means of the computer itself, that means that no external cooling mechanism would have to be attached to the system [1,2].The spin to be cooled down (a nuclear spin for example) has to couple weakly to the environment. In addition one uses some rapid relaxing spins to transport energy out of the system. The transportation of energy from the cooled spin to the others is achieved in a strictly non classical way by applying a quantum algorithm to the system, therefore the spins are further referred to as qbits.Recently, Boykin et al. [3] have developed a quantum algorithm to cool down a single qbit with the aid of two auxiliary qbits. Initially the system is prepared in an equilibrium state with all spins at the same inverse temperature β(0). (Note that we label all quantities belonging to the n th application of the algorithm by (n), so the initial state and the temperature of the bath, introduced later on, are labeled (0).) By applying several quantum gate operations one spin is cooled down by transferring energy to the others. The algorithm itself consists of a controlled NOT (CNOT) gate and a controlled swap gate (CSWAP) [4]. The CSWAP is a 3 qbit gate which swaps qbit 1 with qbit 3 if qbit 2 is |0 , otherwise it does nothing. This leads to an increase of the inverse temperature β(1) of qbit 1 to approximately 3/2 β(0) (cf. [3,5]). Having applied the algorithm once, the initial state is recovered by two further applications of the algorithm. However, by cooling down two other qbits by applying the same algorithm as described above to two additional sets of 3 qbits allows a second application of * Electronic address: Florian.Rempp@itp1.uni-stuttgart.de 00 00 11 11

Local effective dynamics of quantum systems: A generalized approach to work and heat

“…Recently an algorithm published by Boykin et al [12] has found experimental realization by Baugh et al [13] and its thermodynamic properties where investigated by Rempp et al [14]. Of course, not all qubits [i. e. two level systems (TLS) used for quantum computing] are cooled down this way, some, the so called auxiliary qubits, are heated up as well and thus have to be discarded or coupled to the environment to work again as auxiliaries.…”

Semioptimal practicable algorithmic cooling

“…Last but not least, in real life, the reset spins do not relax infinitely faster than the computation spins (see [36][37][38][39][40][41] for viewing AC as a novel type of heat-engine). Let R denote the ratio between the relaxation time of the computation spins and the relaxation time of the reset spins, R = T 1 (comp.…”

Algorithmic cooling in liquid-state nuclear magnetic resonance

“…We bypassed Shannon's bound in three different processes. The current optimal control methods (GRAPE), and better ones such as a second order GRAPE [42] and Krotov based optimization [43] could enable various applications of AC in magnetic resonance spectroscopy [13,14,44] and maybe also other potential applications [45][46][47][48][49][50][51][52][53][54]. ln (4) (see [23]), where ε C,eq is the carbons' equilibrium polarization.…”