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AffiliationAutonomous Non-Profit Organization for Higher Education "Skolkovo Institute of Science and Technology",

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-212 - Quantum data processing methods

KeywordsQuantum adiabaticity, quantum adiabatic theorem, quantum adiabatic conditions, quantum dynamics of many-body systems, adiabatic quantum computer, integrable models

Annotation
Based on the results of the Project-2017, we have identified several promising research directions in the field of quantum many-body adiabaticity. They are as follows. (1) Adiabatic quantum dynamics at a finite temperature. As a rule, the concept of quantum adiabaticity is applied to near-equilibrium quantum dynamics in the vicinity of an eigenstate, i.e. at zero entropy. Within Project-2017, we generalized the concept of many-body quantum adiabaticity to closed quantum systems prepared at a finite temperature, and also proved sufficient conditions for many-body adiabaticity at a finite temperature. Fulfillment of these conditions at a given moment of time ensures that the many-body density matrix of the entire system is close (with a given error) to the quasi-Gibbs density matrix diagonal in the basis of the many-body eigenstates of the instantaneous Hamiltonian. In Project-2020, we intend to conduct a similar study for local adiabaticity, i.e. when instead of a many-body density matrix, a reduced density matrix of a small subsystem is considered. A rigorous conditions for local adiabaticity at a finite temperature will allow, in particular, to determine the theoretical limits on the operation of quantum heat engines (efficiency, power, etc.). In a broader perspective, the study of adiabatic quantum dynamics at finite temperature will allow a more realistic description of the entire range of adiabatic quantum devices, from computers to sensors. (2) Adiabatic quantum algorithms with entangled excited eigenstates. In the framework of Project-2017, a new method was proposed for constructing adiabatic quantum algorithms. Its distinguishing feature is that the excited instantaneous eigenstates remain quantum entangled throughout the calculation, including the end point. This avoids one of the main bottlenecks of the “traditional” adiabatic quantum algorithms - many-body localization, which leads to an exponential slowdown of the adiabatic quantum computation. On the other hand, the disadvantage of the proposed algorithm is the need for at least four-particle interactions between qubits, which are difficult to implement in practice. We plan to modify this algorithm within Project-2020, so that it requires only two-particle interactions. The performance of this algorithm will be tested using simulations on a classical computer. In addition, we will try to run our algorithm on a real quantum device using cloud access (for example, using the platform https://forge.qcware.com/ or its analogues). (3) Quantum speed limits and adiabaticity. Quantum speed limits (QSL) are general bounds on how fast a state vector of a quantum system can move in the Hilbert space. Historically, the first QSL was a rigorous formulation of the time-energy uncertainty relation proved by Mandelstam and Tamm in 1945. Margolus and Levitin in 1998 proved another important QSL. Currently, there are many variations and generalizations of these two fundamental results. QSL underlie fundamental bounds on the speed of quantum information processing, quantum transport and quantum measurements. In the paper by the Project Leader [Phys. Rev. Lett. 119, 200401 (2017)], a quantitative relationship was established between the adiabatic conditions and QSL. Then QSL were used by our group in the framework of the Project-2017 and by foreign colleagues to obtain adiabatic conditions in various situations. Remarkably, as was shown within Project-2017, the relationship between adiabaticity and QSL can work in the opposite direction: the methods developed to describe the adiabatic regime can be generalized to obtain QSL beyond this regime. More specifically, we proved a QSL for systems prepared in thermal states and evolving after a quantum quench. It turned out that for many-body systems this QSL gives much stronger bounds than the Mandelstam-Tamm and Margolus-Levitin QSLs. We plan to develop this approach within the framework of Project-2020. In particular, it is of great fundamental and practical interest to derive a QSL at finite temperature for Hamiltonians that depend on time in an arbitrary way (not necessarily stepwise like in quantum quench). We will try to do this. The resulting new QSLs will be used to analyze the operation of various quantum devices at a finite temperature. (4) Shortcuts to adiabaticity. A shortcut to adiabaticity is a particular path in the parameter space of the Hamiltonian which ensures that a quantum system reaches the same state as if it evolved adiabatically, but in shorter times. The time gain due to a shortcut is limited by quantum speed limits. We plan to exploit particularly stringent QSLs derived by us for thermal initial states in the course of the project to put fundamental limits to the power of quantum heat engines.

Expected results
We expect to obtain the following main results. (1) Stronger conditions for quantum many-body adiabaticity at a finite temperature will be proved. (2) The conditions for local quantum adiabaticity in many-body systems at finite temperature will be proved. (3) A quantum speed limit will be proved for a closed system prepared in an equilibrium thermal state and then evolving under an arbitrary time-dependent Hamiltonian. (4) The results (1) - (3) will be applied to obtain fundamental constraints on the speed, accuracy and / or performance of various quantum devices at finite temperature: quantum sensors, quantum gates, quantum transport devices, quantum thermal machines, etc. (5) An adiabatic quantum algorithm with entangled excited states will be developed, requiring only two-particle interactions between qubits. We will simulate it numerically and identify under what conditions this quantum algorithm outperforms classical and traditional quantum adiabatic algorithms. Results (1)-(3) are of fundamental importance, while results (4)-(5) will have immediate impact on quantum technologies.

Annotation of the results obtained in 2021
The major results of the project in the year 2021-2022 are as follows. (1) The quantum speed limit for an open system interacting with a heat reservoir has been improved. This bound is applicable to an arbitrarily strong and rapidly changing system-reservoir couplings, in contrast to a condition from [A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Quantum speed limits in open system dynamics, Phys. Rev. Lett. 110, 050403 (2013)] based on the Lindblad equation and restricted to the applicability limits of the latter. This general result leads to the upper bounds on rates of qubit initialization, dynamical polarization, quantum heat engines, quantum sensors etc. (2) We have calculated the dynamics of local observables in a one- and two-dimensional quantum Ising models in a time-dependent magnetic field. The problem is reduced to a time-independent one by means of a gauge transformation. In turn, the time-independent problem is tackled by the recursion method. We have created a computer code that automatizes nested commutators in a parallel manner and this way facilitates the application of the method. (3) In the framework of ADS/CFT correspondence, we have calculated a conductance of a quantum point contact that connects two one-dimensional leads.

Publications

1. N. B. Ilyin Quantum adiabatic theorem with energy gap regularization Theoretical and Mathematical Physics, 211(1), 545–557 (year - 2022) https://doi.org/10.1134/S0040577922040080

2. Nikolai Il‘in, Anastasia Aristova, and Oleg Lychkovskiy Adiabatic theorem for closed quantum systems initialized at finite temperature Physical Review A, 104, L030202 (year - 2021) https://doi.org/10.1103/PhysRevA.104.L030202

3. O. Lychkovskiy Entangling Problem Hamiltonian for Adiabatic Quantum Computation Lobachevskii Journal of Mathematics, - (year - 2022)

4. Lychkovskiy O., Gamayun O., Cheianov V. Erratum: Time Scale for Adiabaticity Breakdown in Driven Many-Body Systems and Orthogonality Catastrophe [Phys. Rev. Lett. 119, 200401 (2017)] Physical Review Letters, 129, 119902(E) (year - 2022) https://doi.org/10.1103/PhysRevLett.129.119902

5. - Fundamental Quantum Theorem Now Holds For Finite Temperatures And Not Just Absolute Zero The Science Times, - (year - )

6. - Замораживать до абсолютного нуля не обязательно: фундаментальная квантовая теорема работает и при конечной температуре Пресс-служба РНФ, - (year - )

Annotation of the results obtained in 2020
The main outcomes of the project in the year 2020-2021 are as follows: (1) We have proved new sufficient adiabatic conditions with a free variational parameter. They are applicable for at a finite and zero temperatures. By optimizing the variational parameter one is able to significantly strengthen the condition. (2) Conditions for local adiabaticity are obtained for three systems with periodically driven Hamiltonians, where the Floquet Hamiltonian can be found exactly. Two of them are models of noninteracting fermions, the third one is a spin chain. (3) A quantum sped limit is proved for a closed system prepared in an equilibrium thermal state and then evolving under the action of a time-dependent Hamiltonian. This result generalizes the analogous result for the time-independent Hamiltonians obtained previously within the project. It is shown that our quantum speed limit remains meaningful in the thermodynamic limit, in contrast to other known quantum speed limits. To illustrate our quantum speed limit, we have applied it to two specific models: the spin-boson model and the quantum impurity model. In both cases, we have obtained physically transparent and meaningful results. This study is presented in the paper [Nikolai Il 'in and Oleg Lychkovskiy, Quantum speed limits for thermal states], accepted for publication in Physical Review A. (4) We have numerically compared the performance of adiabatic quantum algorithms with entangled excited states and traditional adiabatic quantum algorithms for systems up to 16 qubits. No significant difference in the performance has been found. (5) A new quantum speed limit for an open system interacting with a heat reservoir has been derived. As a consequence, the power of the modified Carnot and Otto cycles (where the adiabatic strokes are replaced by pseudo-adiabatic ones) has been bounded from above. (6) It has been numerically demonstrated that the operator norm of the adiabatic gauge potential in the one-particle problem on a large lattice saturates with increasing the lattice size.

Publications

1. Nikolai Il'in, Oleg Lychkovskiy Quantum speed limit for thermal states Physical Review A, - (year - 2021)