Asymptotic Bound for Heat-Bath Algorithmic Cooling

Raeisi, Sadegh; Mosca, Michele

doi:10.1103/physrevlett.114.100404

Cited by 45 publications

(58 citation statements)

References 35 publications

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“…We found the asymptotic limit by solving the evolution equation with the results supported by numerical simulation [7]. A proof has been reported by Raeisi and Mosca [21].…”

Section: Introductionmentioning

confidence: 69%

“…Furthermore, we explain how to get the number of steps needed to have a certain polarization ∞ 1l − δ (we give the exact solution for n = 3, and upper bound for n > 3). (See also [21]. )…”

Section: Resultsmentioning

confidence: 99%

“…We now show that starting in the totally mixed state and applying steps of HBAC, the system will asymptotically go to a state that satisfies the equality in (S7) ( [21] gives an alternative argument). To show this, we will prove that the target qubit (the spin−1/2) is cooled after each iteration of HBAC, and the reset qubit keeps extracting entropy from the system (cooling the system) after each iteration.…”

Section: Increasing Puritymentioning

confidence: 99%

“…If we start with a maximally mixed state, it is possible to show that (see supplemental material, a proof can be found in [21])…”

Section: Maximally Mixed Initial Statementioning

confidence: 99%

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Achievable Polarization for Heat-Bath Algorithmic Cooling

Rodríguez-Briones

Laflamme

2016

Phys. Rev. Lett.

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Pure quantum states play a central role in applications of quantum information, both as initial states for quantum algorithms and as resources for quantum error correction. Preparation of highly pure states that satisfy the threshold for quantum error correction remains a challenge, not only for ensemble implementations like NMR or ESR but also for other technologies. Heat-Bath Algorithmic Cooling is a method to increase the purity of a set of qubits coupled to a bath. We investigated the achievable polarization by analysing the limit when no more entropy can be extracted from the system. In particular we give an analytic form for the maximum polarization achievable for the case when the initial state of the qubits is totally mixed, and the corresponding steady state of the whole system. It is however possible to reach higher polarization while starting with certain states, thus our result provides an achievable bound. We also give the number of steps needed to get a specific required polarization.PACS numbers: 03.67.Pp INTRODUCTIONPurification of quantum states is essential for applications of quantum information science, not only for many quantum algorithms but also as a resource for quantum error correction. The need to find a scalable way to reach approximate pure states is a challenge for many quantum computation modalities, especially the ones that relies on ensembles such as NMR or ESR [1].A potential solution is algorithmic cooling (AC), a protocol which purifies qubits by removing entropy of a subset of them, at the expense of increasing the entropy of others [2, 3]. An explicit way to implement this idea in ensemble quantum computers was given by Schulman et al. [4]. They showed that it is possible to reach polarization of order unity using only a number of qubits which is polynomial in the initial polarization. This idea was improved by adding contact with a heat-bath to extract entropy from the system [5], a process known as Heat-Bath Algorithmic Cooling (HBAC). 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. We found the asymptotic limit by solving the evolution equation with the results su...

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“…We found the asymptotic limit by solving the evolution equation with the results supported by numerical simulation [7]. A proof has been reported by Raeisi and Mosca [21].…”

Section: Introductionmentioning

confidence: 69%

Section: Resultsmentioning

confidence: 99%

Section: Increasing Puritymentioning

confidence: 99%

“…If we start with a maximally mixed state, it is possible to show that (see supplemental material, a proof can be found in [21])…”

Section: Maximally Mixed Initial Statementioning

confidence: 99%

See 2 more Smart Citations

Achievable Polarization for Heat-Bath Algorithmic Cooling

Rodríguez-Briones

Laflamme

2016

Phys. Rev. Lett.

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“…Algorithmic cooling, experimentally implemented in this work, is a method that might contribute to both scopes. On the one hand, it was originally suggested as a method for increasing the qubits' purification level [5][6][7][8][9][10], as qubits in a highly pure state are required both for initialization and for fault tolerant [11,12] quantum computing. On the other hand, the suggested novel usage of data compression may potentially be found useful for increasing the signal to noise ratio of liquid-state NMR and in vivo magnetic resonance spectroscopy [6,13,14].…”

Section: Introductionmentioning

confidence: 99%

Algorithmic cooling in liquid-state nuclear magnetic resonance

et al. 2016

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Algorithmic cooling is a method that employs thermalization to increase qubit purification level, namely it reduces the qubit-system's entropy. We utilized gradient ascent pulse engineering (GRAPE), an optimal control algorithm, to implement algorithmic cooling in liquid state nuclear magnetic resonance. Various cooling algorithms were applied onto the three qubits of 13 C2-trichloroethylene, cooling the system beyond Shannon's entropy bound in several different ways. In particular, in one experiment a carbon qubit was cooled by a factor of 4.61. This work is a step towards potentially integrating tools of NMR quantum computing into in vivo magnetic resonance spectroscopy.

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