Quantum noise can enhance algorithmic cooling Zahra Farahmand,1 Reyhaneh Aghaei Saem,1 and Sadegh Raeisi1, ∗ 1Department of Physics, Sharif University of Technology, Tehran, Iran 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 performance and improve the cooling limit of Heat-Bath Algorithmic Cooling techniques. We numerically simulate the noisy algorithmic cooling for the two optimal strategies, the Partner Pairing, and the Two-sort algorithms. We find that for both of them, in the presence of the generalized amplitude damping noise, the process converges and the asymptotic purity can be higher than the noiseless process. This opens up new avenues for increasing the purity beyond the heat-bath algorithmic cooling. Most quantum applications require pure qubits. ments of the density matrix for compression. They However, it is not always easy to prepare pure quan- showed that sort is the optimal compression oper- tum states. For instance, for spin qubits, the states ation for HBAC. They also proved that even this are often close to maximally mixed states. optimal technique cannot always converge to a com- Heat-Bath Algorithmic Cooling techniques pletely pure state. This indicated that HBAC tech- (HBAC) provide implementation-agnostic methods niques are physically limited. Later, Raeisi and for the purification of qubits. These techniques Mosca established the asymptotic state and found exploit auxiliary qubits to increase the purity of the cooling limit of HBAC techniques [4]. Recently target elements. the physical roots of the limit have been further in- Algorithmic cooling techniques work based on vestigated by Raeisi and it was shown that the limit quantum compression algorithms. They compress is due to the unitarity of compression operations the entropy of the system and transfer that away from [5]. Specifically, a no-go theorem was proved which the target elements and into the auxiliary elements shows that for two qubits, it is not possible to in- in the system [1]. crease the purity beyond the purity of the individual The initial algorithmic cooling techniques were qubits. limited by Shannon’s bound for compression. In 2002, Boykin et al. proposed to use a heat-bath to One of the main challenges of PPA is that it is reset the auxiliary qubits, and repeat the cooling pro- state-dependent and in each iteration, the complete cess [2]. This turned the original cooling algorithms information of the state of the system is required to iterative processes that could go beyond the Shan- to determine the sort operation. This also means non’s bound for compression. In each iteration, first, that the compression operation of PPA is constantly the compression operation would transfer the entropy changing through the HBAC process. These make away from the target qubit to the auxiliary ones, and the implementation of PPA challenging and imprac- then through the interaction with the heat-bath, the tical. In [6], Raeisi et al. introduced a new optimal accumulated entropy would be transferred out of the HBAC technique. The new algorithm is called the system and to the heat-bath. “Two-Sort Algorithmic Cooling” (TSAC) technique arXiv:2107.07321v1 [quant-ph] 15 Jul 2021 In [3], Shulman et al. introduced the Partner Pair- and resolves these issues. They proved that this new ing Algorithm (PPA) and showed that it is the opti- method converges to the HBAC limit with a fixed mal HBAC technique. PPA sorts the diagonal ele- compression operation. It means that TSAC can cool down the target qubits to the HBAC limit without knowing the state of the system. This was a signif- icant step towards bringing HBAC to the realm of ∗ [email protected] feasibility [7]. 2 0 00 One remaining caveat that has limited practical ρi ρi = C(ρi) ρi = ρi+1 applications of HBAC is the assumption of ideal experimental conditions. Namely, most of the stud- N ies in this field investigate HBAC under noiseless conditions and do not take into account realistic im- N perfections. This problem is interesting from both U fundamental and practical points of view. Fundamen- tally, it is interesting to know if a noisy implemen- N tation of HBAC would converge to a limit and also if the limit changes from the noiseless limit. Prac- R tically, it is unclear if HBAC can be helpful under realistic noisy implementation. Figure 1. The schematic picture of one iteration of HBAC. Here we investigate noisy HBAC and answer these ρi is a density matrix of n + 1 qubits which the first one is questions. We find that for the generalized amplitude the target qubit and the last one is the reset qubit. In a real- damping channel, which is one of the most relevant istic implementation, there is also noise. We consider the experimental noise models, noisy HBAC converges uncorrelated noise as an extra operation in each iteration and interestingly, it also exceeds the HBAC limit. which is demonstrated by N. During the reset process, We also study the effect of other relevant noise mod- the noise channels are applied to the computational qubits. 0 els such as the depolarizing channel on the cooling ρi = C(ρi) represents the state after effect of the quan- algorithms. We find that only the generalized am- tum noise. Then the reset and compression steps make one y 0 plitude damping channel could enhance the HBAC HBAC iteration and ρi+1 = U (TrR(ρi) ⊗ ρR)U. The limit. reset step includes the partial trace over the reset qubit and replacement of the reset qubit with ρR which is in the ther- From a practical point of view, our results provide mal state of the reset element in contact with the heat-bath. a more realistic assessment of HBAC. These find- U is the compression operation. For TSAC, this is given ings show how HBAC is expected to perform under by unitary operator UTS in Eq. (4). For PPA, U represents imperfect experimental conditions. In particular, for the sort operator that is changing from one iteration to the a specific experimental setting, the noise model can other. be characterized and our findings can help determine if and by how much HBAC is expected to help in increasing the purity. The computation qubits (or a subset of them) are the We start by introducing our notation, the noise target elements that are to be cooled or purified. model, and our assumptions. We investigate the ef- Each iteration of HBAC involves two steps. First, fect of quantum noise for both TSAC and PPA meth- there is the reset which refreshes the reset qubit to ods and identify the regimes where the noisy HBAC its initial state, a.k.a the reset state. Often the reset goes beyond the cooling limit of HBAC. We intro- step is the thermal equilibration of the reset qubit in duce some metrics to characterize the enhancement contact with the heat-bath. This transfers the accu- due to the noise. We find that the noise enhances mulated entropy during the previous iteration from PPA more effectively compared to TSAC, in the the reset qubit to the heat-bath. Mathematically, this sense that, first, for identical noise parameters, the process is described by the following quantum chan- limit of noisy PPA is greater than the limit of noisy nel TSAC, and second, PPA is enhanced for a wider R(ρ) = Tr (ρ) ⊗ ρ ; (1) range of noise parameters compared to TSAC. R R The setting for HBAC is as follows. We consider where ρ is the density matrix of the n + 1 qubits and an ensemble of n + 1 qubits, the first n of them are TrR is the partial trace over the reset element. ρR is referred to as the computation qubits and the last one the reset state, which can be expressed as is referred to as the reset qubit. Our results can be generalized to multi-qubit reset. The system is de- 1 e"0 0 ρR = −"0 ; (2) scribed by a density matrix in HC⊗HR Hilbert space. e"0 + e−"0 0 e 3 where "0 is referred to as the polarization of the reset means that the reset element can be reset faster than qubit. This is inspired by the thermal distribution the computation element. given by the Boltzmann distribution where "0 de- Typically, it is also assumed that the state of the pends linearly on the energy gap of the qubit and computation qubits remains unchanged while the inversely on the temperature. reset qubits are reset. This is the key assumption For an arbitrary qubit state, the polarization can that we are revisiting and relaxing in this work. In be characterized in terms of the eigenvalues of the particular, we take into account that during the reset density matrix. We define Λ(ρ) to denote the largest process, the computation qubits could undergo some eigenvalue of the qubit state ρ. The polarization can noise. be expressed as Note that there are two main assumptions in HBAC that need to be revisited. The first one is that 1 Λ(ρ) " = log( ): (3) the compression operation is noiseless. The second 2 1 − Λ(ρ) one is that while the reset takes place, the compu- tation qubits remain unchanged.