1
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Brand D, Sinayskiy I, Petruccione F. Markovian noise modelling and parameter extraction framework for quantum devices. Sci Rep 2024; 14:4769. [PMID: 38413630 PMCID: PMC10899264 DOI: 10.1038/s41598-024-54598-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2023] [Accepted: 02/14/2024] [Indexed: 02/29/2024] Open
Abstract
In recent years, Noisy Intermediate Scale Quantum (NISQ) computers have been widely used as a test bed for quantum dynamics. This work provides a new hardware-agnostic framework for modelling the Markovian noise and dynamics of quantum systems in benchmark procedures used to evaluate device performance. As an accessible example, the application and performance of this framework is demonstrated on IBM Quantum computers. This framework serves to extract multiple calibration parameters simultaneously through a simplified process which is more reliable than previously studied calibration experiments and tomographic procedures. Additionally, this method allows for real-time calibration of several hardware parameters of a quantum computer within a comprehensive procedure, providing quantitative insight into the performance of each device to be accounted for in future quantum circuits. The framework proposed here has the additional benefit of highlighting the consistency among qubit pairs when extracting parameters, which leads to a less computationally expensive calibration process than evaluating the entire device at once.
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Affiliation(s)
- Dean Brand
- Department of Physics, School of Data Science and Computational Thinking, Stellenbosch University, Stellenbosch, 7604, South Africa.
| | - Ilya Sinayskiy
- School of Chemistry and Physics, University of KwaZulu-Natal, Durban, 4001, South Africa.
- National Institute for Theoretical and Computational Sciences (NITheCS), Stellenbosch, 7604, South Africa.
| | - Francesco Petruccione
- Department of Physics, School of Data Science and Computational Thinking, Stellenbosch University, Stellenbosch, 7604, South Africa
- National Institute for Theoretical and Computational Sciences (NITheCS), Stellenbosch, 7604, South Africa
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2
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Butler EP, Fux GE, Ortega-Taberner C, Lovett BW, Keeling J, Eastham PR. Optimizing Performance of Quantum Operations with Non-Markovian Decoherence: The Tortoise or the Hare? PHYSICAL REVIEW LETTERS 2024; 132:060401. [PMID: 38394576 DOI: 10.1103/physrevlett.132.060401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2023] [Accepted: 12/19/2023] [Indexed: 02/25/2024]
Abstract
The interaction between a quantum system and its environment limits our ability to control it and perform quantum operations on it. We present an efficient method to find optimal controls for quantum systems coupled to non-Markovian environments, by using the process tensor to compute the gradient of an objective function. We consider state transfer for a driven two-level system coupled to a bosonic environment, and characterize performance in terms of speed and fidelity. This allows us to determine the best achievable fidelity as a function of process duration. We show there can be a trade-off between speed and fidelity, and that slower processes can have higher fidelity by exploiting non-Markovian effects.
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Affiliation(s)
- Eoin P Butler
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
- Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, Ireland
| | - Gerald E Fux
- SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
- Abdus Salam International Center for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
| | - Carlos Ortega-Taberner
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
- Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, Ireland
| | - Brendon W Lovett
- SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
| | - Jonathan Keeling
- SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
| | - Paul R Eastham
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
- Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, Ireland
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3
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White GAL, Modi K, Hill CD. Filtering Crosstalk from Bath Non-Markovianity via Spacetime Classical Shadows. PHYSICAL REVIEW LETTERS 2023; 130:160401. [PMID: 37154634 DOI: 10.1103/physrevlett.130.160401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2022] [Accepted: 03/02/2023] [Indexed: 05/10/2023]
Abstract
From an open system perspective non-Markovian effects due to a nearby bath or neighboring qubits are dynamically equivalent. However, there is a conceptual distinction to account for: neighboring qubits may be controlled. We combine recent advances in non-Markovian quantum process tomography with the framework of classical shadows to characterize spatiotemporal quantum correlations. Observables here constitute operations applied to the system, where the free operation is the maximally depolarizing channel. Using this as a causal break, we systematically erase causal pathways to narrow down the progenitors of temporal correlations. We show that one application of this is to filter out the effects of crosstalk and probe only non-Markovianity from an inaccessible bath. It also provides a lens on spatiotemporally spreading correlated noise throughout a lattice from common environments. We demonstrate both examples on synthetic data. Owing to the scaling of classical shadows, we can erase arbitrarily many neighboring qubits at no extra cost. Our procedure is thus efficient and amenable to systems even with all-to-all interactions.
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Affiliation(s)
- G A L White
- School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - K Modi
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
- Centre for Quantum Technology, Transport for New South Wales, Sydney, New South Wales 2000, Australia
| | - C D Hill
- School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia
- School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, 3010, Australia
- Silicon Quantum Computing, The University of New South Wales, Sydney, New South Wales 2052, Australia
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4
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Chruściński D, Hesabi S, Lonigro D. On Markovianity and classicality in multilevel spin-boson models. Sci Rep 2023; 13:1518. [PMID: 36707631 PMCID: PMC9883298 DOI: 10.1038/s41598-023-28606-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/11/2022] [Accepted: 01/20/2023] [Indexed: 01/28/2023] Open
Abstract
We provide a detailed discussion about the unitary and reduced evolution induced by family of Hamiltonian models describing a multilevel system, with a ground state and a possibly multilevel excited sector, coupled to a multimode boson field via a rotating-wave interaction. We prove explicitly that the system, in the limit in which the coupling is flat with respect to the boson frequencies, is Markovian under sharp measurements in arbitrary bases; we also find necessary and sufficient conditions under which the process is classical, i.e. its family of multitime joint probability distributions satisfies the Kolmogorov consistency condition, and may thus be equivalently obtained by a classical stochastic process.
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Affiliation(s)
- Dariusz Chruściński
- grid.5374.50000 0001 0943 6490Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń, Poland
| | - Samaneh Hesabi
- grid.5374.50000 0001 0943 6490Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń, Poland
| | - Davide Lonigro
- grid.7644.10000 0001 0120 3326Dipartimento di Fisica and MECENAS, Università di Bari, 70126 Bari, Italy ,grid.470190.bINFN, Sezione di Bari, 70126 Bari, Italy
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5
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Sayer T, Montoya-Castillo A. Compact and complete description of non-Markovian dynamics. J Chem Phys 2023; 158:014105. [PMID: 36610963 DOI: 10.1063/5.0132614] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023] Open
Abstract
Generalized master equations provide a theoretically rigorous framework to capture the dynamics of processes ranging from energy harvesting in plants and photovoltaic devices to qubit decoherence in quantum technologies and even protein folding. At their center is the concept of memory. The explicit time-nonlocal description of memory is both protracted and elaborate. When physical intuition is at a premium, one would desire a more compact, yet complete, description. Here, we demonstrate how and when the time-convolutionless formalism constitutes such a description. In particular, by focusing on the dissipative dynamics of the spin-boson and Frenkel exciton models, we show how to: easily construct the time-local generator from reference reduced dynamics, elucidate the dependence of its existence on the system parameters and the choice of reduced observables, identify the physical origin of its apparent divergences, and offer analysis tools to diagnose their severity and circumvent their deleterious effects. We demonstrate that, when applicable, the time-local approach requires as little information as the more commonly used time-nonlocal scheme, with the important advantages of providing a more compact description, greater algorithmic simplicity, and physical interpretability. We conclude by introducing the discrete-time analog and a straightforward protocol to employ it in cases where the reference dynamics have limited resolution. The insights we present here offer the potential for extending the reach of dynamical methods, reducing both their cost and conceptual complexity.
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Affiliation(s)
- Thomas Sayer
- Department of Chemistry, University of Colorado Boulder, Boulder, Colorado 80309, USA
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6
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Chakraborty S, Das A, Chruściński D. Strongly coupled quantum Otto cycle with single qubit bath. Phys Rev E 2022; 106:064133. [PMID: 36671160 DOI: 10.1103/physreve.106.064133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2022] [Accepted: 11/15/2022] [Indexed: 12/27/2022]
Abstract
We discuss a model of a closed quantum evolution of two qubits where the joint Hamiltonian is so chosen such that one of the qubits acts as a bath and thermalizes the other qubit which is acting as the system. The corresponding exact master equation for the system is derived. Interestingly, for a specific choice of parameters the master equation takes the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form, with constant coefficients representing pumping and damping of a single qubit system. Based on this model we construct an Otto cycle connected to a single qubit bath and study its thermodynamic properties. Our analysis goes beyond the conventional weak coupling scenario and illustrates the effects of finite baths, including non-Markovianity. We find closed form expressions for efficiency (coefficient of performance), power (cooling power) for the heat engine regime (refrigerator regime), and for different modifications of the joint Hamiltonian.
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Affiliation(s)
- Sagnik Chakraborty
- Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziądzka 5/7, 87-100 Toruń, Poland
| | - Arpan Das
- Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziądzka 5/7, 87-100 Toruń, Poland
| | - Dariusz Chruściński
- Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziądzka 5/7, 87-100 Toruń, Poland
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7
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Budini AA. Quantum Non-Markovian Environment-to-System Backflows of Information: Nonoperational vs. Operational Approaches. ENTROPY 2022; 24:e24050649. [PMID: 35626534 PMCID: PMC9140462 DOI: 10.3390/e24050649] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/07/2022] [Revised: 04/27/2022] [Accepted: 04/29/2022] [Indexed: 12/04/2022]
Abstract
Quantum memory effects can be qualitatively understood as a consequence of an environment-to-system backflow of information. Here, we analyze and compare how this concept is interpreted and implemented in different approaches to quantum non-Markovianity. We study a nonoperational approach, defined by the distinguishability between two system states characterized by different initial conditions, and an operational approach, which is defined by the correlation between different outcomes associated to successive measurement processes performed over the system of interest. The differences, limitations, and vantages of each approach are characterized in detail by considering diverse system–environment models and dynamics. As a specific example, we study a non-Markovian depolarizing map induced by the interaction of the system of interest with an environment characterized by incoherent and coherent self-dynamics.
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Affiliation(s)
- Adrián A Budini
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Centro Atómico Bariloche, Avenida E. Bustillo Km 9.5, Bariloche 8400, Argentina
- Universidad Tecnológica Nacional (UTN-FRBA), Fanny Newbery 111, Bariloche 8400, Argentina
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8
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Zonnios M, Levinsen J, Parish MM, Pollock FA, Modi K. Signatures of Quantum Chaos in an Out-of-Time-Order Tensor. PHYSICAL REVIEW LETTERS 2022; 128:150601. [PMID: 35499886 DOI: 10.1103/physrevlett.128.150601] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2021] [Revised: 02/26/2022] [Accepted: 03/02/2022] [Indexed: 06/14/2023]
Abstract
Motivated by the famous ink-drop experiment, where ink droplets are used to determine the chaoticity of a fluid, we propose an experimentally implementable method for measuring the scrambling capacity of quantum processes. Here, a system of interest interacts with a small quantum probe whose dynamical properties identify the chaoticity of the system. Specifically, we propose a fully quantum version of the out-of-time-order correlator-which we term the out-of-time-order tensor-whose correlations offer clear information theoretic meanings about the chaoticity of a process. We illustrate the utility of the out-of-time-order tensor as a signature of chaos using random unitary processes as well as in the quantum kicked rotor, where the chaoticity is tunable.
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Affiliation(s)
- Magdalini Zonnios
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Jesper Levinsen
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Meera M Parish
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Felix A Pollock
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Kavan Modi
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
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9
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Multipartite Correlations in Quantum Collision Models. ENTROPY 2022; 24:e24040508. [PMID: 35455171 PMCID: PMC9032730 DOI: 10.3390/e24040508] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/10/2022] [Revised: 04/01/2022] [Accepted: 04/03/2022] [Indexed: 12/10/2022]
Abstract
Quantum collision models have proved to be useful for a clear and concise description of many physical phenomena in the field of open quantum systems: thermalization, decoherence, homogenization, nonequilibrium steady state, entanglement generation, simulation of many-body dynamics, and quantum thermometry. A challenge in the standard collision model, where the system and many ancillas are all initially uncorrelated, is how to describe quantum correlations among ancillas induced by successive system-ancilla interactions. Another challenge is how to deal with initially correlated ancillas. Here we develop a tensor network formalism to address both challenges. We show that the induced correlations in the standard collision model are well captured by a matrix product state (a matrix product density operator) if the colliding particles are in pure (mixed) states. In the case of the initially correlated ancillas, we construct a general tensor diagram for the system dynamics and derive a memory-kernel master equation. Analyzing the perturbation series for the memory kernel, we go beyond the recent results concerning the leading role of two-point correlations and consider multipoint correlations (Waldenfelds cumulants) that become relevant in the higher-order stroboscopic limits. These results open an avenue for the further analysis of memory effects in collisional quantum dynamics.
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10
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Megier N, Smirne A, Campbell S, Vacchini B. Correlations, Information Backflow, and Objectivity in a Class of Pure Dephasing Models. ENTROPY (BASEL, SWITZERLAND) 2022; 24:304. [PMID: 35205599 PMCID: PMC8871357 DOI: 10.3390/e24020304] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/23/2022] [Revised: 02/11/2022] [Accepted: 02/14/2022] [Indexed: 02/04/2023]
Abstract
We critically examine the role that correlations established between a system and fragments of its environment play in characterising the ensuing dynamics. We employ a dephasing model with different initial conditions, where the state of the initial environment represents a tunable degree of freedom that qualitatively and quantitatively affects the correlation profiles, but nevertheless results in the same reduced dynamics for the system. We apply recently developed tools for the characterisation of non-Markovianity to carefully assess the role that correlations, as quantified by the (quantum) Jensen-Shannon divergence and relative entropy, as well as changes in the environmental state, play in whether the conditions for classical objectivity within the quantum Darwinism paradigm are met. We demonstrate that for precisely the same non-Markovian reduced dynamics of the system arising from different microscopic models, some exhibit quantum Darwinistic features, while others show that no meaningful notion of classical objectivity is present. Furthermore, our results highlight that the non-Markovian nature of an environment does not a priori prevent a system from redundantly proliferating relevant information, but rather it is the system's ability to establish the requisite correlations that is the crucial factor in the manifestation of classical objectivity.
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Affiliation(s)
- Nina Megier
- Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, 20133 Milan, Italy; (A.S.); (B.V.)
- Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milan, Italy
- International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gdansk, Poland
| | - Andrea Smirne
- Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, 20133 Milan, Italy; (A.S.); (B.V.)
- Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milan, Italy
| | - Steve Campbell
- School of Physics, University College Dublin, Belfield, D04 Dublin, Ireland
- Centre for Quantum Engineering, Science, and Technology, University College Dublin, Belfield, D04 Dublin, Ireland
| | - Bassano Vacchini
- Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, 20133 Milan, Italy; (A.S.); (B.V.)
- Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milan, Italy
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11
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Guo Y, Taranto P, Liu BH, Hu XM, Huang YF, Li CF, Guo GC. Experimental Demonstration of Instrument-Specific Quantum Memory Effects and Non-Markovian Process Recovery for Common-Cause Processes. PHYSICAL REVIEW LETTERS 2021; 126:230401. [PMID: 34170148 DOI: 10.1103/physrevlett.126.230401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Accepted: 04/12/2021] [Indexed: 06/13/2023]
Abstract
The duration, strength, and structure of memory effects are crucial properties of physical evolution. Because of the invasive nature of quantum measurement, such properties must be defined with respect to the probing instruments employed. Here, using a photonic platform, we experimentally demonstrate this necessity via two paradigmatic processes: future-history correlations in the first process can be erased by an intermediate quantum measurement; for the second process, a noisy classical measurement blocks the effect of history. We then apply memory truncation techniques to recover an efficient description that approximates expectation values for multitime observables. Our proof-of-principle analysis paves the way for experiments concerning more general non-Markovian quantum processes and highlights where standard open systems techniques break down.
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Affiliation(s)
- Yu Guo
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Philip Taranto
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
- Vienna Center for Quantum Science and Technology, Atominstitut, Technische Universität Wien, 1020 Vienna, Austria
| | - Bi-Heng Liu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Xiao-Min Hu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Yun-Feng Huang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Chuan-Feng Li
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Guang-Can Guo
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
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12
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Fux GE, Butler EP, Eastham PR, Lovett BW, Keeling J. Efficient Exploration of Hamiltonian Parameter Space for Optimal Control of Non-Markovian Open Quantum Systems. PHYSICAL REVIEW LETTERS 2021; 126:200401. [PMID: 34110219 DOI: 10.1103/physrevlett.126.200401] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2021] [Accepted: 04/14/2021] [Indexed: 06/12/2023]
Abstract
We present a general method to efficiently design optimal control sequences for non-Markovian open quantum systems, and illustrate it by optimizing the shape of a laser pulse to prepare a quantum dot in a specific state. The optimization of control procedures for quantum systems with strong coupling to structured environments-where time-local descriptions fail-is a computationally challenging task. We modify the numerically exact time evolving matrix product operator (TEMPO) method, such that it allows the repeated computation of the time evolution of the reduced system density matrix for various sets of control parameters at very low computational cost. This method is potentially useful for studying numerous optimal control problems, in particular in solid state quantum devices where the coupling to vibrational modes is typically strong.
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Affiliation(s)
- Gerald E Fux
- SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
| | - Eoin P Butler
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
| | - Paul R Eastham
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
| | - Brendon W Lovett
- SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
| | - Jonathan Keeling
- SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
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13
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Demonstration of non-Markovian process characterisation and control on a quantum processor. Nat Commun 2020; 11:6301. [PMID: 33298929 PMCID: PMC7725842 DOI: 10.1038/s41467-020-20113-3] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2020] [Accepted: 11/10/2020] [Indexed: 11/10/2022] Open
Abstract
In the scale-up of quantum computers, the framework underpinning fault-tolerance generally relies on the strong assumption that environmental noise affecting qubit logic is uncorrelated (Markovian). However, as physical devices progress well into the complex multi-qubit regime, attention is turning to understanding the appearance and mitigation of correlated — or non-Markovian — noise, which poses a serious challenge to the progression of quantum technology. This error type has previously remained elusive to characterisation techniques. Here, we develop a framework for characterising non-Markovian dynamics in quantum systems and experimentally test it on multi-qubit superconducting quantum devices. Where noisy processes cannot be accounted for using standard Markovian techniques, our reconstruction predicts the behaviour of the devices with an infidelity of 10−3. Our results show this characterisation technique leads to superior quantum control and extension of coherence time by effective decoupling from the non-Markovian environment. This framework, validated by our results, is applicable to any controlled quantum device and offers a significant step towards optimal device operation and noise reduction. As quantum computing devices become more complex, they enter the realm of correlated noise, which is difficult to characterise and mitigate. Here, the authors demonstrate, over a range of superconducting devices, a method for non-Markovian dynamics characterisation based on the process tensor framework.
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14
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Figueroa-Romero P, Modi K, Pollock FA. Equilibration on average in quantum processes with finite temporal resolution. Phys Rev E 2020; 102:032144. [PMID: 33075897 DOI: 10.1103/physreve.102.032144] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2020] [Accepted: 09/15/2020] [Indexed: 11/07/2022]
Abstract
We characterize the conditions under which a multitime quantum process with a finite temporal resolution can be approximately described by an equilibrium one. By providing a generalization of the notion of equilibration on average, where a system remains closed to a fixed equilibrium for most times, to one which can be operationally assessed at multiple times, we place an upper-bound on a new observable distinguishability measure comparing a multitime process with a finite temporal resolution against a fixed equilibrium one. While the same conditions on single-time equilibration, such as a large occupation of energy levels in the initial state remain necessary, we obtain genuine multitime contributions depending on the temporal resolution of the process and the amount of disturbance of the observer's operations on it.
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Affiliation(s)
| | - Kavan Modi
- School of Physics & Astronomy, Monash University, Victoria 3800, Australia
| | - Felix A Pollock
- School of Physics & Astronomy, Monash University, Victoria 3800, Australia
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15
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Han L, Zou J, Li H, Shao B. Non-Markovianity of a Central Spin Interacting with a Lipkin-Meshkov-Glick Bath via a Conditional Past-Future Correlation. ENTROPY 2020; 22:e22080895. [PMID: 33286664 PMCID: PMC7517521 DOI: 10.3390/e22080895] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/28/2020] [Revised: 08/10/2020] [Accepted: 08/12/2020] [Indexed: 11/24/2022]
Abstract
Based on conditional past–future (CPF) correlations, we study the non-Markovianity of a central spin coupled to an isotropic Lipkin–Meshkov–Glick (LMG) bath. Although the dynamics of a system is always non-Markovian, it is found that some measurement time intervals considering a specific process, with respect to a particular set of CPF measurement operators, can be zero, which means that in this case the non-Markovianity of the system could not be detected. Furthermore, the initial system–bath correlations only slightly influence the non-Markovianity of the system in our model. Significantly, it is also found that the dynamics of the system for LMG baths, initially in the ground states corresponding to the symmetric phase and symmetry broken phase, exhibit different properties, and the maximal value of the CPF at the critical point is the smallest, independent of the measurement operator, which means that the criticality can manifest itself by the CPF. Moreover, the effect of bath temperature on the quantum criticality of the CPF depends on the measurement operator.
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Affiliation(s)
- Liping Han
- School of Physics, Beijing Institute of Technology, Beijing 100081, China; (L.H.); (B.S.)
- School of Science, Tianjin University of Technology, Tianjin 300384, China
| | - Jian Zou
- School of Physics, Beijing Institute of Technology, Beijing 100081, China; (L.H.); (B.S.)
- Correspondence:
| | - Hai Li
- School of Information and Electronic Engineering, Shandong Technology and Business University, Yantai 264005, China;
| | - Bin Shao
- School of Physics, Beijing Institute of Technology, Beijing 100081, China; (L.H.); (B.S.)
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16
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Luchnikov IA, Vintskevich SV, Grigoriev DA, Filippov SN. Machine Learning Non-Markovian Quantum Dynamics. PHYSICAL REVIEW LETTERS 2020; 124:140502. [PMID: 32338970 DOI: 10.1103/physrevlett.124.140502] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2019] [Revised: 01/28/2020] [Accepted: 03/16/2020] [Indexed: 06/11/2023]
Abstract
Machine learning methods have proved to be useful for the recognition of patterns in statistical data. The measurement outcomes are intrinsically random in quantum physics, however, they do have a pattern when the measurements are performed successively on an open quantum system. This pattern is due to the system-environment interaction and contains information about the relaxation rates as well as non-Markovian memory effects. Here we develop a method to extract the information about the unknown environment from a series of projective single-shot measurements on the system (without resorting to the process tomography). The method is based on embedding the non-Markovian system dynamics into a Markovian dynamics of the system and the effective reservoir of finite dimension. The generator of Markovian embedding is learned by the maximum likelihood estimation. We verify the method by comparing its prediction with an exactly solvable non-Markovian dynamics. The developed algorithm to learn unknown quantum environments enables one to efficiently control and manipulate quantum systems.
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Affiliation(s)
- I A Luchnikov
- Moscow Institute of Physics and Technology, Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia
- Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, Moscow Region 121205, Russia
| | - S V Vintskevich
- Moscow Institute of Physics and Technology, Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia
| | - D A Grigoriev
- Moscow Institute of Physics and Technology, Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia
| | - S N Filippov
- Moscow Institute of Physics and Technology, Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia
- Valiev Institute of Physics and Technology of Russian Academy of Sciences, Nakhimovskii Prospekt 34, Moscow 117218, Russia
- Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina Street 8, Moscow 119991, Russia
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17
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Quantum Memristors in Frequency-Entangled Optical Fields. MATERIALS 2020; 13:ma13040864. [PMID: 32074986 PMCID: PMC7079656 DOI: 10.3390/ma13040864] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/24/2019] [Revised: 01/17/2020] [Accepted: 02/05/2020] [Indexed: 11/17/2022]
Abstract
A quantum memristor is a passive resistive circuit element with memory, engineered in a given quantum platform. It can be represented by a quantum system coupled to a dissipative environment, in which a system-bath coupling is mediated through a weak measurement scheme and classical feedback on the system. In quantum photonics, such a device can be designed from a beam splitter with tunable reflectivity, which is modified depending on the results of measurements in one of the outgoing beams. Here, we show that a similar implementation can be achieved with frequency-entangled optical fields and a frequency mixer that, working similarly to a beam splitter, produces state superpositions. We show that the characteristic hysteretic behavior of memristors can be reproduced when analyzing the response of the system with respect to the control, for different experimentally attainable states. Since memory effects in memristors can be exploited for classical and neuromorphic computation, the results presented in this work could be a building block for constructing quantum neural networks in quantum photonics, when scaling up.
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18
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Jørgensen MR, Pollock FA. Exploiting the Causal Tensor Network Structure of Quantum Processes to Efficiently Simulate Non-Markovian Path Integrals. PHYSICAL REVIEW LETTERS 2019; 123:240602. [PMID: 31922869 DOI: 10.1103/physrevlett.123.240602] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/24/2019] [Revised: 09/21/2019] [Indexed: 06/10/2023]
Abstract
In the path integral formulation of the evolution of an open quantum system coupled to a Gaussian, noninteracting environment, the dynamical contribution of the latter is encoded in an object called the influence functional. Here, we relate the influence functional to the process tensor-a more general representation of a quantum stochastic process-describing the evolution. Then, we use this connection to motivate a tensor network algorithm for the simulation of multitime correlations in open systems, building on recent work where the influence functional is represented in terms of time evolving matrix product operators. By exploiting the symmetries of the influence functional, we are able to use our algorithm to achieve orders-of-magnitude improvement in the efficiency of the resulting numerical simulation. Our improved algorithm is then applied to compute exact phonon emission spectra for the spin-boson model with strong coupling, demonstrating a significant divergence from spectra derived under commonly used assumptions of memorylessness.
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Affiliation(s)
- Mathias R Jørgensen
- Department of Physics, Technical University of Denmark, 2800 Kongens Lyngby, Denmark
| | - Felix A Pollock
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
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19
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Strasberg P. Repeated Interactions and Quantum Stochastic Thermodynamics at Strong Coupling. PHYSICAL REVIEW LETTERS 2019; 123:180604. [PMID: 31763881 DOI: 10.1103/physrevlett.123.180604] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2019] [Indexed: 06/10/2023]
Abstract
The thermodynamic framework of repeated interactions is generalized to an arbitrary open quantum system in contact with a heat bath. Based on these findings, the theory is then extended to arbitrary measurements performed on the system. This constitutes a direct experimentally testable framework in strong coupling quantum thermodynamics. By construction, it provides many quantum stochastic processes and quantum causal models with a consistent thermodynamic interpretation. The setting can be further used, for instance, to rigorously investigate the interplay between non-Markovianity and nonequilibrium thermodynamics.
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Affiliation(s)
- Philipp Strasberg
- Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
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20
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Strasberg P. Operational approach to quantum stochastic thermodynamics. Phys Rev E 2019; 100:022127. [PMID: 31574666 DOI: 10.1103/physreve.100.022127] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/08/2018] [Indexed: 06/10/2023]
Abstract
We set up a framework for quantum stochastic thermodynamics based solely on experimentally controllable but otherwise arbitrary interventions at discrete times. Using standard assumptions about the system-bath dynamics and insights from the repeated interaction framework, we define internal energy, heat, work, and entropy at the trajectory level. The validity of the first law (at the trajectory level) and the second law (on average) is established. The theory naturally allows one to treat incomplete information and it is able to smoothly interpolate between a trajectory-based and an ensemble level description. We use our theory to compute the thermodynamic efficiency of recent experiments reporting on the stabilization of photon number states using real-time quantum feedback control. Special attention is paid to limiting cases of our general theory, where we recover or contrast it with previous results. We point out various interesting problems, which the theory is able to address rigorously, such as the detection of quantum effects in thermodynamics.
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Affiliation(s)
- Philipp Strasberg
- Physics and Materials Science Research Unit, University of Luxembourg, 1511 Luxembourg, Luxembourg and Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
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21
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Strasberg P, Winter A. Stochastic thermodynamics with arbitrary interventions. Phys Rev E 2019; 100:022135. [PMID: 31574732 DOI: 10.1103/physreve.100.022135] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2019] [Indexed: 06/10/2023]
Abstract
We extend the theory of stochastic thermodynamics in three directions: (i) instead of a continuously monitored system we consider measurements only at an arbitrary set of discrete times, (ii) we allow for imperfect measurements and incomplete information in the description, and (iii) we treat arbitrary manipulations (e.g., feedback control operations) which are allowed to depend on the entire measurement record. For this purpose we define for a driven system in contact with a single heat bath the four key thermodynamic quantities-internal energy, heat, work, and entropy-along a single "trajectory" for a causal model. The first law at the trajectory level and the second law on average is verified. We highlight the special case of Bayesian or "bare" measurements (incomplete information, but no average disturbance) which allows us to compare our theory with the literature and to derive a general inequality for the estimated free energy difference in Jarzynski-type experiments. An analysis of a recent Maxwell demon experiment using real-time feedback control is also given. As a mathematical tool, we prove a classical version of Stinespring's dilation theorem, which might be of independent interest.
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Affiliation(s)
- Philipp Strasberg
- Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Andreas Winter
- Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
- ICREA - Institució Catalana de Recerca i Estudis Avançats, Passeig Lluis Companys 23, 08010 Barcelona, Spain
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22
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Milz S, Kim MS, Pollock FA, Modi K. Completely Positive Divisibility Does Not Mean Markovianity. PHYSICAL REVIEW LETTERS 2019; 123:040401. [PMID: 31491272 DOI: 10.1103/physrevlett.123.040401] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2019] [Indexed: 06/10/2023]
Abstract
In the classical domain, it is well known that divisibility does not imply that a stochastic process is Markovian. However, for quantum processes, divisibility is often considered to be synonymous with Markovianity. We show that completely positive divisible quantum processes can still involve non-Markovian temporal correlations, that we then fully classify using the recently developed process tensor formalism, which generalizes the theory of stochastic processes to the quantum domain.
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Affiliation(s)
- Simon Milz
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - M S Kim
- QOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
- Korea Institute for Advanced Study, 02455 Seoul, Korea
| | - Felix A Pollock
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Kavan Modi
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
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23
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Luchnikov IA, Vintskevich SV, Ouerdane H, Filippov SN. Simulation Complexity of Open Quantum Dynamics: Connection with Tensor Networks. PHYSICAL REVIEW LETTERS 2019; 122:160401. [PMID: 31075029 DOI: 10.1103/physrevlett.122.160401] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2018] [Indexed: 06/09/2023]
Abstract
The difficulty to simulate the dynamics of open quantum systems resides in their coupling to many-body reservoirs with exponentially large Hilbert space. Applying a tensor network approach in the time domain, we demonstrate that effective small reservoirs can be defined and used for modeling open quantum dynamics. The key element of our technique is the timeline reservoir network (TRN), which contains all the information on the reservoir's characteristics, in particular, the memory effects timescale. The TRN has a one-dimensional tensor network structure, which can be effectively approximated in full analogy with the matrix product approximation of spin-chain states. We derive the sufficient bond dimension in the approximated TRN with a reduced set of physical parameters: coupling strength, reservoir correlation time, minimal timescale, and the system's number of degrees of freedom interacting with the environment. The bond dimension can be viewed as a measure of the open dynamics complexity. Simulation is based on the semigroup dynamics of the system and effective reservoir of finite dimension. We provide an illustrative example showing the scope for new numerical and machine learning-based methods for open quantum systems.
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Affiliation(s)
- I A Luchnikov
- Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, Moscow Region 121205, Russia
- Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia
| | - S V Vintskevich
- Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia
- A. M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov St. 38, Moscow 119991, Russia
| | - H Ouerdane
- Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, Moscow Region 121205, Russia
| | - S N Filippov
- Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia
- Valiev Institute of Physics and Technology of Russian Academy of Sciences, Nakhimovskii Pr. 34, Moscow 117218, Russia
- Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina St. 8, Moscow 119991, Russia
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24
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Taranto P, Pollock FA, Milz S, Tomamichel M, Modi K. Quantum Markov Order. PHYSICAL REVIEW LETTERS 2019; 122:140401. [PMID: 31050460 DOI: 10.1103/physrevlett.122.140401] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/09/2018] [Indexed: 05/23/2023]
Abstract
We formally extend the notion of Markov order to open quantum processes by accounting for the instruments used to probe the system of interest at different times. Our description recovers the classical property in the appropriate limit: when the stochastic process is classical and the instruments are noninvasive, i.e., restricted to orthogonal, projective measurements. We then prove that there do not exist non-Markovian quantum processes that have finite Markov order with respect to all possible instruments; the same process exhibits distinct memory effects when probed by different instruments. This naturally leads to a relaxed definition of quantum Markov order with respect to specified instrument sequences. The memory effects captured by different choices of instruments vary dramatically, providing a rich landscape for future exploration.
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Affiliation(s)
- Philip Taranto
- School of Physics & Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Felix A Pollock
- School of Physics & Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Simon Milz
- School of Physics & Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Marco Tomamichel
- Centre for Quantum Software and Information, School of Software, University of Technology Sydney, Sydney NSW 2007, Australia
| | - Kavan Modi
- School of Physics & Astronomy, Monash University, Clayton, Victoria 3800, Australia
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25
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Obeid AK, Bruza PD, Wittek P. Evaluating probabilistic programming languages for simulating quantum correlations. PLoS One 2019; 14:e0208555. [PMID: 30608937 PMCID: PMC6319741 DOI: 10.1371/journal.pone.0208555] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2018] [Accepted: 11/17/2018] [Indexed: 12/01/2022] Open
Abstract
This article explores how probabilistic programming can be used to simulate quantum correlations in an EPR experimental setting. Probabilistic programs are based on standard probability which cannot produce quantum correlations. In order to address this limitation, a hypergraph formalism was programmed which both expresses the measurement contexts of the EPR experimental design as well as associated constraints. Four contemporary open source probabilistic programming frameworks were used to simulate an EPR experiment in order to shed light on their relative effectiveness from both qualitative and quantitative dimensions. We found that all four probabilistic languages successfully simulated quantum correlations. Detailed analysis revealed that no language was clearly superior across all dimensions, however, the comparison does highlight aspects that can be considered when using probabilistic programs to simulate experiments in quantum physics.
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Affiliation(s)
- Abdul Karim Obeid
- School of Information Systems, Queensland University of Technology, Brisbane, Queensland, Australia
| | - Peter D. Bruza
- School of Information Systems, Queensland University of Technology, Brisbane, Queensland, Australia
| | - Peter Wittek
- Rotman School of Management, University of Toronto, Toronto, Ontario, Canada
- Creative Destruction Lab, Toronto, Ontario, Canada
- Vector Institute for Artificial Intelligence, Toronto, Ontario, Canada
- Perimeter Institute for Theoretical Physics, Toronto, Ontario, Canada
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26
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Budini AA. Quantum Non-Markovian Processes Break Conditional Past-Future Independence. PHYSICAL REVIEW LETTERS 2018; 121:240401. [PMID: 30608721 DOI: 10.1103/physrevlett.121.240401] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2018] [Indexed: 06/09/2023]
Abstract
For classical Markovian stochastic systems, past and future events become statistically independent when conditioned to a given state at the present time. Memory non-Markovian effects break this condition, inducing a nonvanishing conditional past-future correlation. Here, this classical memory indicator is extended to a quantum regime, which provides an operational definition of quantum non-Markovianity based on a minimal set of three time-ordered quantum system measurements and postselection. The detection of memory effects through the measurement scheme is univocally related to departures from Born-Markov and white noise approximations in quantum and classical environments, respectively.
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Affiliation(s)
- Adrián A Budini
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Centro Atómico Bariloche, Avenida E. Bustillo Km 9.5, (8400) Bariloche, Argentina, and Universidad Tecnológica Nacional (UTN-FRBA), Fanny Newbery 111, (8400) Bariloche, Argentina
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27
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Strasberg P, Esposito M. Response Functions as Quantifiers of Non-Markovianity. PHYSICAL REVIEW LETTERS 2018; 121:040601. [PMID: 30095950 DOI: 10.1103/physrevlett.121.040601] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2017] [Indexed: 06/08/2023]
Abstract
Quantum non-Markovianity is crucially related to the study of dynamical maps, which are usually derived for initially factorized system-bath states. We demonstrate that linear response theory also provides a way to derive dynamical maps but for initially correlated (and, in general, entangled) states. Importantly, these maps are always time-translational invariant and allow for a much simpler quantification of non-Markovianity compared to previous approaches. We apply our theory to the Caldeira-Leggett model, for which our quantifier is valid beyond linear response and can be expressed analytically. We find that a classical Brownian particle coupled to an Ohmic bath can already exhibit non-Markovian behavior, a phenomenon related to the initial state preparation procedure. Furthermore, for a peaked spectral density, we show that there is no monotonic relation between our quantifier and the system-bath coupling strength, the sharpness of the peak or the resonance frequency in the bath.
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Affiliation(s)
- Philipp Strasberg
- Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Massimiliano Esposito
- Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
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