1
|
Torlai G, Wood CJ, Acharya A, Carleo G, Carrasquilla J, Aolita L. Quantum process tomography with unsupervised learning and tensor networks. Nat Commun 2023; 14:2858. [PMID: 37208324 DOI: 10.1038/s41467-023-38332-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2021] [Accepted: 04/26/2023] [Indexed: 05/21/2023] Open
Abstract
The impressive pace of advance of quantum technology calls for robust and scalable techniques for the characterization and validation of quantum hardware. Quantum process tomography, the reconstruction of an unknown quantum channel from measurement data, remains the quintessential primitive to completely characterize quantum devices. However, due to the exponential scaling of the required data and classical post-processing, its range of applicability is typically restricted to one- and two-qubit gates. Here, we present a technique for performing quantum process tomography that addresses these issues by combining a tensor network representation of the channel with a data-driven optimization inspired by unsupervised machine learning. We demonstrate our technique through synthetically generated data for ideal one- and two-dimensional random quantum circuits of up to 10 qubits, and a noisy 5-qubit circuit, reaching process fidelities above 0.99 using several orders of magnitude fewer (single-qubit) measurement shots than traditional tomographic techniques. Our results go far beyond state-of-the-art, providing a practical and timely tool for benchmarking quantum circuits in current and near-term quantum computers.
Collapse
Affiliation(s)
- Giacomo Torlai
- AWS Center for Quantum Computing, Pasadena, CA, USA.
- Center for Computational Quantum Physics, Flatiron Institute, New York, NY, 10010, USA.
| | | | - Atithi Acharya
- Center for Computational Quantum Physics, Flatiron Institute, New York, NY, 10010, USA
- Physics and Astronomy Department, Rutgers University, Piscataway, NJ, 08854, USA
| | - Giuseppe Carleo
- Center for Computational Quantum Physics, Flatiron Institute, New York, NY, 10010, USA
- Institute of Physics, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland
| | | | - Leandro Aolita
- Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE
- Instituto de Física, Federal University of Rio de Janeiro, 21941-972, P. O. Box 68528, Rio de Janeiro, Brazil
| |
Collapse
|
2
|
Abstract
Quantum state tomography is the experimental procedure of determining an unknown state. It is not only essential for the verification of resources and processors of quantum information but is also important in its own right with regard to the foundation of quantum mechanics. Standard methods have been elusive for large systems because of the enormous number of observables to be measured and the exponential complexity of data post-processing. Here, we propose a new scheme of quantum state tomography that requires the measurement of only three observables (acting jointly on the system and pointer) regardless of the size of the system. The system is coupled to a "pointer" of single qubit, and the wavefunction of the system is "reaped" onto the pointer upon the measurement of the system. Subsequently, standard two-state tomography on the pointer and classical post-processing are used to reconstruct the quantum state of the system. We also developed an efficient and scalable iterative maximum likelihood algorithm to estimate states from statistically incomplete data.
Collapse
|
3
|
Ahmed S, Sánchez Muñoz C, Nori F, Kockum AF. Quantum State Tomography with Conditional Generative Adversarial Networks. PHYSICAL REVIEW LETTERS 2021; 127:140502. [PMID: 34652197 DOI: 10.1103/physrevlett.127.140502] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2020] [Revised: 05/21/2021] [Accepted: 06/10/2021] [Indexed: 06/13/2023]
Abstract
Quantum state tomography (QST) is a challenging task in intermediate-scale quantum devices. Here, we apply conditional generative adversarial networks (CGANs) to QST. In the CGAN framework, two dueling neural networks, a generator and a discriminator, learn multimodal models from data. We augment a CGAN with custom neural-network layers that enable conversion of output from any standard neural network into a physical density matrix. To reconstruct the density matrix, the generator and discriminator networks train each other on data using standard gradient-based methods. We demonstrate that our QST-CGAN reconstructs optical quantum states with high fidelity, using orders of magnitude fewer iterative steps, and less data, than both accelerated projected-gradient-based and iterative maximum-likelihood estimation. We also show that the QST-CGAN can reconstruct a quantum state in a single evaluation of the generator network if it has been pretrained on similar quantum states.
Collapse
Affiliation(s)
- Shahnawaz Ahmed
- Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
| | - Carlos Sánchez Muñoz
- Departamento de Fisica Teorica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autonoma de Madrid, Madrid 28049, Spain
| | - Franco Nori
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
| | - Anton Frisk Kockum
- Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
| |
Collapse
|
4
|
Zhou Y, Zeng P, Liu Z. Single-Copies Estimation of Entanglement Negativity. PHYSICAL REVIEW LETTERS 2020; 125:200502. [PMID: 33258639 DOI: 10.1103/physrevlett.125.200502] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2020] [Accepted: 09/10/2020] [Indexed: 06/12/2023]
Abstract
Entanglement plays a central role in quantum information processing and quantum physics. However, few effective ways are known to detect the amount of entanglement of an unknown quantum state. Here, we propose a scheme to estimate the entanglement negativity for any bipartition of a composite system. The proposed scheme is based on the random unitary evolution and local measurements on a single-copy quantum state, which is more practical compared to former methods based on collective measurements on many copies of the identical state. Meanwhile, we generalize the scheme to quantify the total correlation. We demonstrate the efficiency of the scheme with statistical analyses and numerical simulations. Our scheme is quite suitable for state-of-the-art quantum platforms, which can serve as a useful benchmarking tool to advance quantum technologies and a probe to study fundamental quantum physics like entanglement dynamics.
Collapse
Affiliation(s)
- You Zhou
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Pei Zeng
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - Zhenhuan Liu
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
- School of Physics, Peking University, Beijing 100871, China
| |
Collapse
|
5
|
Bootstrapping quantum process tomography via a perturbative ansatz. Nat Commun 2020; 11:1084. [PMID: 32107382 PMCID: PMC7046656 DOI: 10.1038/s41467-020-14873-1] [Citation(s) in RCA: 19] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/26/2019] [Accepted: 02/06/2020] [Indexed: 11/23/2022] Open
Abstract
Quantum process tomography has become increasingly critical as the need grows for robust verification and validation of candidate quantum processors, since it plays a key role in both performance assessment and debugging. However, as these processors grow in size, standard process tomography becomes an almost impossible task. Here, we present an approach for efficient quantum process tomography that uses a physically motivated ansatz for an unknown quantum process. Our ansatz bootstraps to an effective description for an unknown process on a multi-qubit processor from pairwise two-qubit tomographic data. Further, our approach can inherit insensitivity to system preparation and measurement error from the two-qubit tomography scheme. We benchmark our approach using numerical simulation of noisy three-qubit gates, and show that it produces highly accurate characterizations of quantum processes. Further, we demonstrate our approach experimentally on a superconducting quantum processor, building three-qubit gate reconstructions from two-qubit tomographic data. Quantum process tomography represents one of the workhorses of quantum information processing, but suffers from exponential resource scaling. Here, the authors propose to efficiently infer general processes by approximating them through a sequence of two-qubit processes, and demonstrate it on a three-qubit case.
Collapse
|
6
|
Stefano QP, Rebón L, Ledesma S, Iemmi C. Set of 4d-3 observables to determine any pure qudit state. OPTICS LETTERS 2019; 44:2558-2561. [PMID: 31090731 DOI: 10.1364/ol.44.002558] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2019] [Accepted: 04/22/2019] [Indexed: 06/09/2023]
Abstract
We present a tomographic method which requires only 4d-3 measurement outcomes to reconstruct any pure quantum state of arbitrary dimension d. Using the proposed scheme, we have experimentally reconstructed a large number of pure states of dimension d=7, obtaining a mean fidelity of 0.94. Moreover, we performed numerical simulations of the reconstruction process, verifying the feasibility of the method for higher dimensions. In addition, the a priori assumption of purity can be certified within the same set of measurements, which represents an improvement with respect to other similar methods and contributes to answering the question of how many observables are needed to uniquely determine any pure state.
Collapse
|
7
|
Carrasquilla J, Torlai G, Melko RG, Aolita L. Reconstructing quantum states with generative models. NAT MACH INTELL 2019. [DOI: 10.1038/s42256-019-0028-1] [Citation(s) in RCA: 121] [Impact Index Per Article: 24.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
8
|
Dhand I, Engelkemeier M, Sansoni L, Barkhofen S, Silberhorn C, Plenio MB. Proposal for Quantum Simulation via All-Optically-Generated Tensor Network States. PHYSICAL REVIEW LETTERS 2018; 120:130501. [PMID: 29694179 DOI: 10.1103/physrevlett.120.130501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/2017] [Revised: 01/16/2018] [Indexed: 06/08/2023]
Abstract
We devise an all-optical scheme for the generation of entangled multimode photonic states encoded in temporal modes of light. The scheme employs a nonlinear down-conversion process in an optical loop to generate one- and higher-dimensional tensor network states of light. We illustrate the principle with the generation of two different classes of entangled tensor network states and report on a variational algorithm to simulate the ground-state physics of many-body systems. We demonstrate that state-of-the-art optical devices are capable of determining the ground-state properties of the spin-1/2 Heisenberg model. Finally, implementations of the scheme are demonstrated to be robust against realistic losses and mode mismatch.
Collapse
Affiliation(s)
- I Dhand
- Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany
- Center for Integrated Quantum Science and Technology (IQST), Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany
| | - M Engelkemeier
- Department of Physics and CeOPP, University of Paderborn, Warburger Strasse 100, D-33098 Paderborn, Germany
| | - L Sansoni
- Department of Physics and CeOPP, University of Paderborn, Warburger Strasse 100, D-33098 Paderborn, Germany
| | - S Barkhofen
- Department of Physics and CeOPP, University of Paderborn, Warburger Strasse 100, D-33098 Paderborn, Germany
| | - C Silberhorn
- Department of Physics and CeOPP, University of Paderborn, Warburger Strasse 100, D-33098 Paderborn, Germany
| | - M B Plenio
- Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany
- Center for Integrated Quantum Science and Technology (IQST), Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany
| |
Collapse
|
9
|
Steffens A, Friesdorf M, Langen T, Rauer B, Schweigler T, Hübener R, Schmiedmayer J, Riofrío C, Eisert J. Towards experimental quantum-field tomography with ultracold atoms. Nat Commun 2015; 6:7663. [PMID: 26138511 PMCID: PMC4506543 DOI: 10.1038/ncomms8663] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2014] [Accepted: 05/22/2015] [Indexed: 11/23/2022] Open
Abstract
The experimental realization of large-scale many-body systems in atomic-optical architectures has seen immense progress in recent years, rendering full tomography tools for state identification inefficient, especially for continuous systems. To work with these emerging physical platforms, new technologies for state identification are required. Here we present first steps towards efficient experimental quantum-field tomography. Our procedure is based on the continuous analogues of matrix-product states, ubiquitous in condensed-matter theory. These states naturally incorporate the locality present in realistic physical settings and are thus prime candidates for describing the physics of locally interacting quantum fields. To experimentally demonstrate the power of our procedure, we quench a one-dimensional Bose gas by a transversal split and use our method for a partial quantum-field reconstruction of the far-from-equilibrium states of this system. We expect our technique to play an important role in future studies of continuous quantum many-body systems.
Collapse
Affiliation(s)
- A. Steffens
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin 14195, Germany
| | - M. Friesdorf
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin 14195, Germany
| | - T. Langen
- Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, Vienna 1020, Austria
| | - B. Rauer
- Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, Vienna 1020, Austria
| | - T. Schweigler
- Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, Vienna 1020, Austria
| | - R. Hübener
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin 14195, Germany
| | - J. Schmiedmayer
- Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, Vienna 1020, Austria
| | - C.A. Riofrío
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin 14195, Germany
| | - J. Eisert
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin 14195, Germany
| |
Collapse
|
10
|
Swingle B, Kim IH. Reconstructing quantum states from local data. PHYSICAL REVIEW LETTERS 2014; 113:260501. [PMID: 25615291 DOI: 10.1103/physrevlett.113.260501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2014] [Indexed: 06/04/2023]
Abstract
We consider the problem of reconstructing global quantum states from local data. Because the reconstruction problem has many solutions in general, we consider the reconstructed state of maximum global entropy consistent with the local data. We show that unique ground states of local Hamiltonians are exactly reconstructed as the maximal entropy state. More generally, we show that if the state in question is a ground state of a local Hamiltonian with a degenerate space of locally indistinguishable ground states, then the maximal entropy state is close to the ground state projector. We also show that local reconstruction is possible for thermal states of local Hamiltonians. Finally, we discuss a procedure to certify that the reconstructed state is close to the true global state. We call the entropy of our reconstructed maximum entropy state the "reconstruction entropy," and we discuss its relation to emergent geometry in the context of holographic duality.
Collapse
Affiliation(s)
- Brian Swingle
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Isaac H Kim
- Perimeter Institute of Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| |
Collapse
|
11
|
Große C, Kabakchiev A, Lutz T, Froidevaux R, Schramm F, Ruben M, Etzkorn M, Schlickum U, Kuhnke K, Kern K. Dynamic control of plasmon generation by an individual quantum system. NANO LETTERS 2014; 14:5693-5697. [PMID: 25181332 DOI: 10.1021/nl502413k] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
Controlling light on the nanoscale in a similar way as electric currents has the potential to revolutionize the exchange and processing of information. Although light can be guided on this scale by coupling it to plasmons, that is, collective electron oscillations in metals, their local electronic control remains a challenge. Here, we demonstrate that an individual quantum system is able to dynamically gate the electrical plasmon generation. Using a single molecule in a double tunnel barrier between two electrodes we show that this gating can be exploited to monitor fast changes of the quantum system itself and to realize a single-molecule plasmon-generating field-effect transistor operable in the gigahertz range. This opens new avenues toward atomic scale quantum interfaces bridging nanoelectronics and nanophotonics.
Collapse
Affiliation(s)
- Christoph Große
- Max-Planck-Institut für Festkörperforschung , Heisenbergstraße 1, 70569 Stuttgart, Germany
| | | | | | | | | | | | | | | | | | | |
Collapse
|
12
|
Schwemmer C, Tóth G, Niggebaum A, Moroder T, Gross D, Gühne O, Weinfurter H. Experimental comparison of efficient tomography schemes for a six-qubit state. PHYSICAL REVIEW LETTERS 2014; 113:040503. [PMID: 25105604 DOI: 10.1103/physrevlett.113.040503] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2014] [Indexed: 06/03/2023]
Abstract
Quantum state tomography suffers from the measurement effort increasing exponentially with the number of qubits. Here, we demonstrate permutationally invariant tomography for which, contrary to conventional tomography, all resources scale polynomially with the number of qubits both in terms of the measurement effort as well as the computational power needed to process and store the recorded data. We demonstrate the benefits of combining permutationally invariant tomography with compressed sensing by studying the influence of the pump power on the noise present in a six-qubit symmetric Dicke state, a case where full tomography is possible only for very high pump powers.
Collapse
Affiliation(s)
- Christian Schwemmer
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany and Department für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| | - Géza Tóth
- Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao, Spain and IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain and Wigner Research Centre for Physics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Alexander Niggebaum
- School of Physics and Astronomy, University of Birmingham, B15 2TT Birmingham, United Kingdom
| | - Tobias Moroder
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany
| | - David Gross
- Physikalisches Institut & FDM, Universität Freiburg, Rheinstraße 10, D-79104 Freiburg, Germany
| | - Otfried Gühne
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany
| | - Harald Weinfurter
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany and Department für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| |
Collapse
|
13
|
Cohen I, Retzker A. Proposal for verification of the haldane phase using trapped ions. PHYSICAL REVIEW LETTERS 2014; 112:040503. [PMID: 24580427 DOI: 10.1103/physrevlett.112.040503] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2013] [Indexed: 06/03/2023]
Abstract
A proposal to use trapped ions to implement spin-one XXZ antiferromagnetic chains as an experimental tool to explore the Haldane phase is presented. We explain how to reach the Haldane phase adiabatically, demonstrate the robustness of the ground states to noise in the magnetic field and Rabi frequencies, and propose a way to detect them using their characteristics: an excitation gap and exponentially decaying correlations, a nonvanishing nonlocal string order, and a double degenerate entanglement spectrum. Scaling up to higher dimensions and more frustrated lattices, we obtain richer phase diagrams, and we can reach spin liquid phase, which can be detected by its entanglement entropy which obeys the boundary law.
Collapse
Affiliation(s)
- I Cohen
- Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, 91904 Givat Ram, Israel
| | - A Retzker
- Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, 91904 Givat Ram, Israel
| |
Collapse
|