The Asymptotic Cooling of Heat-Bath Algorithmic Cooling
Sadegh Raeisi, Michele Mosca
备注：
11 pages, 4 figures, In the new version we clarified the main proofs and moved parts to the main body of the paper. Elaborated on implications for more general initial states. Additional references added. Minor corrections and changes
The purity of quantum states is a key requirement for many quantum applications. Improving the purity is limited by fundamental laws of thermodynamics. Here we are probing the fundamental limits for a natural approach to this problem, namely heat-bath algorithmic cooling(HBAC). The existence of the cooling limit for HBAC techniques was proved by Schulman et al. in, the limit however remained unknown for the past decade. Here for the first time we find this limit. In the context of quantum thermodynamics, this corresponds to the maximum extractable work from the quantum system.
In a recent paper, PRL 114 100404, 2015, Raeisi and Mosca gave a limit for cooling with Heat-Bath Algorithmic Cooling (HBAC). Here we show how to exceed that limit by having correlation in the qubits-bath interaction.
We report here the experimental realization of multi-step cooling of a quantum system via heat-bath algorithmic cooling. The experiment was carried out using nuclear magnetic resonance (NMR) of a solid-state ensemble three-qubit system.
In this paper we prove that a fundamental constraint on the cooling dynamic implies that it is impossible to cool, via a unitary system-bath quantum evolution, a system that is embedded in a thermal environment down to its ground state, if the initial state is a factorized product of system and bath states. The latter is a crucial but artificial assumption often included in many descriptions of system-bath dynamics. The analogous conclusion holds for 'cooling' to any pure state of the system.
We suggest alternative quantum Otto engines, using heat bath algorithmic cooling with partner pairing algorithm instead of isochoric cooling. Liquid state nuclear magnetic resonance systems in one entropy sink are considered as working fluids. Then, the extractable work and thermal efficiency are analyzed in detail for four-stroke and two-stroke type of quantum Otto engines. The role of heat bath algorithmic cooling in these cycles is to use a single entropy sink instead of two. Also, this cooling algorithm increases the power of engines reducing the time required for one cycle.
Controlled preparation of highly pure quantum states is at the core of practical applications of quantum information science, from the state initialization of most quantum algorithms to a reliable supply of ancilla qubits that satisfy the fault-tolerance threshold for quantum error correction. Heat-bath algorithmic cooling has been shown to purify qubits by controlled redistribution of entropy and multiple contact with a bath, not only for ensemble implementations but also for technologies with strong but imperfect measurements. However, an implicit restriction about the interaction with the b…
Quantum technologies require pure states, which are often generated by extreme refrigeration. Heat-bath algorithmic cooling is the theoretically optimal refrigeration technique: it shuttles entropy from a multiparticle system to a thermal bath, thereby generating a quantum state with a high degree of purity. Here, we show how to surpass this hitherto-optimal technique by taking advantage of a single binary-outcome measurement. Our protocols can create arbitrary numbers of pure quantum states without any residual mixedness by using a recently discovered device known as a quantum switch to put t…
The microscopic basis of Newton's law of cooling and its modification when the difference in temperature between the system and the surroundings is very large is discussed. When the system of interest is interacting with a small bath, the effect of the dynamical evolution of the bath variables is important to find out its dynamical feedback on the system. As in the usual system-bath approach, however, the bath is finally considered to be in thermal equilibrium and thereby provides an effective generalization of the Born-Markov master equation. It is shown that the cooling at early time is fast…
Heat-Bath Algorithmic Cooling techniques (HBAC) are techniques that are used to purify a target element in a quantum system. These methods compress and transfer entropy away from the target element into auxiliary elements of the system. The performance of Algorithmic Cooling has been investigated under ideal noiseless conditions. However, realistic implementations are imperfect and for practical purposes, noise should be taken into account. Here we analyze Heat-Bath Algorithmic Cooling techniques under realistic noise models. Surprisingly, we find that noise can in some cases enhance the perfo…
Algorithmic cooling is a novel technique to generate ensembles of highly polarized spins, which could significantly improve the signal strength in Nuclear Magnetic Resonance (NMR) spectroscopy. It combines reversible (entropy-preserving) manipulations and irreversible controlled interactions with the environment, using simple quantum computing techniques to increase spin polarization far beyond the Shannon entropy-conservation bound. Notably, thermalization is beneficially employed as an integral part of the cooling scheme, contrary to its ordinary destructive implications. We report the first…
The ability to perform quantum error correction is a significant hurdle for scalable quantum information processing. A key requirement for multiple-round quantum error correction is the ability to dynamically extract entropy from ancilla qubits. Heat-bath algorithmic cooling is a method that uses quantum logic operations to move entropy from one subsystem to another, and permits cooling of a spin qubit below the closed system (Shannon) bound. Gamma-irradiated, $^{13}$C-labeled malonic acid provides up to 5 spin qubits: 1 spin-half electron and 4 spin-half nuclei. The nuclei are strongly hyperf…
Pure quantum states play a central role in applications of quantum information, both as initial states for many 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 set of qubits coupled to a bath. We investigated the achievable polarization by analyzing the state when no more entropy can be extracted. In particular we give an …
We propose a method for increasing purity of interacting quantum systems that takes advantage of correlations present due to the internal interaction. In particular we show that by using the system's quantum correlations one can achieve cooling beyond established limits of previous conventional algorithmic cooling proposals which assume no interaction.
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 $^{13}$C$_2$-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 …
Algorithmic Cooling (AC) of Spins is potentially the first near-future application of quantum computing devices. Straightforward quantum algorithms combined with novel entropy manipulations can result in a method to improve the identification of molecules. We introduce here several new exhaustive cooling algorithms, such as the Tribonacci and k-bonacci algorithms. In particular, we present the ``all-bonacci'' algorithm, which appears to reach the maximal degree of cooling obtainable by the optimal AC approach.
A minimal model of a quantum refrigerator (QR), i.e. a periodically phase-flipped two-level system permanently coupled to a finite-capacity bath (cold bath) and an infinite heat dump (hot bath), is introduced and used to investigate the cooling of the cold bath towards the absolute zero (T=0). Remarkably, the temperature scaling of the cold-bath cooling rate reveals that it does not vanish as T->0 for certain realistic quantized baths, e.g. phonons in strongly disordered media (fractons) or quantized spin-waves in ferromagnets (magnons). This result challenges Nernst's third-law formulation kn…
Optomechanical systems show tremendous promise for high sensitivity sensing of forces and modification of mechanical properties via light. For example, similar to neutral atoms and trapped ions, laser cooling of mechanical motion by radiation pressure can take single mechanical modes to their ground state. Conventional optomechanical cooling is able to introduce additional damping channel to mechanical motion, while keeping its thermal noise at the same level, and as a consequence, the effective temperature of the mechanical mode is lowered. However, the ratio of temperature to quality factor …
A suitable sequence of sharp pulses applied to a spin coupled to a bosonic bath can cool its state, i.e., increase its polarization or ground state occupation probability. Starting from an unpolarized state of the spin in equilibrium with the bath, one can reach very low temperatures or sizeable polarizations within a time shorter than the decoherence time. Both the bath and external fields are necessary for the effect which comes from the backreaction of the spin on the bath. This method can be applied to cool at once a disordered ensemble of spins. Since the bath is crucial for this mechanis…
Algorithmic cooling (AC) is a method to purify quantum systems, such as ensembles of nuclear spins, or cold atoms in an optical lattice. When applied to spins, AC produces ensembles of highly polarized spins, which enhance the signal strength in nuclear magnetic resonance (NMR). According to this cooling approach, spin-half nuclei in a constant magnetic field are considered as bits, or more precisely, quantum bits, in a known probability distribution. Algorithmic steps on these bits are then translated into specially designed NMR pulse sequences using common NMR quantum computation tools. The …
Application of multiple rounds of Quantum Error Correction (QEC) is an essential milestone towards the construction of scalable quantum information processing devices. However, experimental realizations of it are still in their infancy. The requirements for multiple round QEC are high control fidelity and the ability to extract entropy from ancilla qubits. Nuclear Magnetic Resonance (NMR) based quantum devices have demonstrated high control fidelity with up to 12 qubits. On the other hand, the major challenge in the NMR QEC experiment is to efficiently supply ancilla qubits in highly pure stat…
Heat-bath algorithmic cooling (HBAC) provides algorithmic ways to improve the purity of quantum states. These techniques are complex iterative processes that change from each iteration to the next and this poses a significant challenge to implementing these algorithms. Here, we introduce a new technique that on a fundamental level, shows that it is possible to do algorithmic cooling and even reach the cooling limit without any knowledge of the state and using only a single fixed operation, and on a practical level, presents a more feasible and robust alternative for implementing HBAC. We also …
Heat-Bath Algorithmic Cooling is a set of techniques for producing highly pure quantum systems by utilizing a surrounding heat-bath and unitary interactions. These techniques originally used the thermal environment only to fully thermalize ancillas at the environment temperature. Here we extend HBAC protocols by optimizing over the thermalization strategy. We find, for any $d$-dimensional system in an arbitrary initial state, provably optimal cooling protocols with surprisingly simple structure and exponential convergence to the ground state. Compared to the standard ones, these schemes can us…
We propose a scheme to cool down a mechanical resonator to its quantum ground-state, which is interacting with a working fluid via an optomechanical-like coupling. As opposed to standard laser cooling schemes where coherence renders the motion of a resonator to its ground-state, we consider an incoherent thermal source to achieve the same aim. We show that simultaneous cooling of two degenerate or near-degenerate mechanical resonators is possible, which is otherwise a challenging goal to achieve. The generalization of this method to the simultaneous cooling of multiple resonators is straightfo…
We show experimental results demonstrating multiple rounds of heat-bath algorithmic cooling in a 3 qubit solid-state nuclear magnetic resonance quantum information processor. By dynamically pumping entropy out of the system of interest and into the heat-bath, we are able show purification of a single qubit to a polarization 1.69 times that of the heat-bath and thus go beyond the Shannon bound for closed system cooling. The cooling algorithm implemented requires both high fidelity coherent control and a deliberate controlled interaction with the environment. We discuss the improvements in contr…
We present here algorithmic cooling (via polarization-heat-bath)- a powerful method for obtaining a large number of highly polarized spins in liquid nuclear-spin systems at finite temperature. Given that spin-half states represent (quantum) bits, algorithmic cooling cleans dirty bits beyond the Shannon's bound on data compression, by employing a set of rapidly thermal-relaxing bits. Such auxiliary bits could be implemented using spins that rapidly get into thermal equilibrium with the environment, e.g., electron spins. Cooling spins to a very low temperature without cooling the environment cou…
In this work, we study the thermodynamics of a two-level system (qubit) embedded in a finite-temperature structured-bath under periodical measurements. The system under measurements will reach a quasi-steady state, whose effective temperature can be maintained lower than that of the surrounding environment. To study the influence of the environmental oscillators from different regimes of frequency on the qubit, the spectrum of the bath consisting of a large number of bosonic harmonic oscillators can be approximately divided into three parts according to their effects of cooling or heating. Due…
State preparation and measurement errors are commonly regarded as indistinguishable. The problem of distinguishing state preparation (SPAM) errors from measurement errors is important to the field of characterizing quantum processors. In this work, we propose a method to separately characterize SPAM errors using a novel type of algorithmic cooling protocol called measurement-based algorithmic cooling (MBAC). MBAC assumes the ability to perform (potentially imperfect) projective measurements on individual qubits, which is available on many modern quantum processors. We demonstrate that MBAC can…
Algorithmic Cooling is a method that uses novel data compression techniques and simplecquantum computing devices to improve NMR spectroscopy, and to offer scalable NMR quantum computers. The algorithm recursively employs two steps. A reversible entropy compression of the computation quantum-bits (qubits) of the system and an irreversible heat transfer from the system to the environment through a set of reset qubits that reach thermal relaxation rapidly. Is it possible to experimentally demonstrate algorithmic cooling using existing technology? To allow experimental algorithmic cooling, the the…
Heat-Bath Algorithmic cooling (HBAC) techniques provide ways to selectively enhance the polarization of target quantum subsystems. However, the cooling in these techniques are bounded. Here we report the first experimental observation of the HBAC cooling bound. We use HBAC to hyperpolarize nuclear spins in diamond. Using two carbon nuclear spins as the source of polarization (reset) and the 14N nuclear spin as the computation bit, we demonstrate that repeating a single cooling step increases the polarization beyond the initial reset polarization and reaches the cooling limit of HBAC. We benchm…
Heat-bath cooling is a component of practicable algorithmic cooling of spins, an approach which might be useful for in vivo 13C spectroscopy, in particular for prolonged metabolic processes where substrates that are hyperpolarized ex-vivo are not effective. We applied heat-bath cooling to 1,2-13C2-amino acids, using the alpha protons to shift entropy from selected carbons to the environment. For glutamate and glycine, both carbons were cooled by about 2.5-fold, and in other experiments the polarization of C1 nearly doubled while all other spins had equilibrium polarization, indicating reductio…
We introduce a scheme to perform the cooling algorithm, first presented by Oscar 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.
An efficient technique to generate ensembles of spins that are highly polarized by external magnetic fields is the Holy Grail in Nuclear Magnetic Resonance (NMR) spectroscopy. Since spin-half nuclei have steady-state polarization biases that increase inversely with temperature, spins exhibiting high polarization biases are considered cool, even when their environment is warm. Existing spin-cooling techniques are highly limited in their efficiency and usefulness. Algorithmic cooling is a promising new spin-cooling approach that employs data compression methods in open systems. It reduces the en…
Heat-bath algorithmic cooling (AC) of spins is a theoretically powerful effective cooling approach, that (ideally) cools spins with low polarization exponentially better than cooling by reversible entropy manipulations alone. Here, we investigate the limitations and prospects of AC. For non-ideal and semioptimal AC, we study the impact of finite relaxation times of reset and computation spins on the achievable effective cooling. We derive, via simulations, the attainable cooling levels for given ratios of relaxation times using two semioptimal practicable algorithms. We expect this analysis to…
The Shannon's bound for compression is one of the key restrictions for the compression of quantum information. Here we show that the unitarity of the compression operation imposes new bounds on the compression that are more limiting than Shannon's compression bound. This translates to a no-go theorem for the purification of quantum states. For a specific case of a two-qubit system, our results indicate that it is not possible to distill purity beyond the maximum of the individual purities. We show that this restriction results in the cooling limit of the heat-bath algorithmic cooling technique…
Controlled quantum mechanical devices provide a means of simulating more complex quantum systems exponentially faster than classical computers. Such "quantum simulators" rely heavily upon being able to prepare the ground state of Hamiltonians, whose properties can be used to calculate correlation functions or even the solution to certain classical computations. While adiabatic preparation remains the primary means of producing such ground states, here we provide a different avenue of preparation: cooling to the ground state via simulated dissipation. This is in direct analogy to contemporary e…
This study introduces a novel minimalistic variational quantum ansatz inspired by algorithmic cooling principles. The proposed Heat Exchange algorithmic cooling ansatz (HE ansatz) facilitates efficient population redistribution without requiring bath resets, simplifying implementation on noisy intermediate-scale quantum (NISQ) devices. The HE ansatz achieves superior approximation ratios with the complete network \textsc{Maxcut} optimization problem compared to the conventional Hardware efficient and QAOA ansatz. We also proposed a new variational algorithm that utilize HE ansatz to compute th…
We show that quantum absorption refrigerators, which have traditionally been studied as of three qubits, each of which is connected to a thermal reservoir, can also be constructed by using three qubits and two thermal baths, where two of the qubits, including the qubit to be locally cooled, are connected to a common bath. With a careful choice of the system, bath, and qubit-bath interaction parameters within the Born-Markov and rotating-wave approximations, one of the qubits attached to the common bath achieves a cooling in the steady state. We observe that the proposed refrigerator may also o…
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Improving Quantum Machine Learning via Heat-Bath Algorithmic Cooling
Atominstitut, Technische Universität Wien（维也纳技术大学原子研究所）；Miller Institute for Basic Research in Science, University of California Berkeley（加州大学伯克利分校米勒基础科学研究所）；Yonsei University（延世大学）
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