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Zwolak JP, Taylor JM. Colloquium: Advances in automation of quantum dot devices control. REVIEWS OF MODERN PHYSICS 2023; 95:10.1103/revmodphys.95.011006. [PMID: 37051403 PMCID: PMC10088060 DOI: 10.1103/revmodphys.95.011006] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/19/2023]
Abstract
Arrays of quantum dots (QDs) are a promising candidate system to realize scalable, coupled qubit systems and serve as a fundamental building block for quantum computers. In such semiconductor quantum systems, devices now have tens of individual electrostatic and dynamical voltages that must be carefully set to localize the system into the single-electron regime and to realize good qubit operational performance. The mapping of requisite QD locations and charges to gate voltages presents a challenging classical control problem. With an increasing number of QD qubits, the relevant parameter space grows sufficiently to make heuristic control unfeasible. In recent years, there has been considerable effort to automate device control that combines script-based algorithms with machine learning (ML) techniques. In this Colloquium, a comprehensive overview of the recent progress in the automation of QD device control is presented, with a particular emphasis on silicon- and GaAs-based QDs formed in two-dimensional electron gases. Combining physics-based modeling with modern numerical optimization and ML has proven effective in yielding efficient, scalable control. Further integration of theoretical, computational, and experimental efforts with computer science and ML holds vast potential in advancing semiconductor and other platforms for quantum computing.
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Affiliation(s)
| | - Jacob M. Taylor
- Joint Quantum Institute, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
- Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland 20742, USA
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2
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Yannouleas C, Landman U. Wigner molecules and hybrid qubits. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:21LT01. [PMID: 35379767 DOI: 10.1088/1361-648x/ac5c28] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2021] [Accepted: 03/09/2022] [Indexed: 06/14/2023]
Abstract
It is demonstrated that exact diagonalization of the microscopic many-body Hamiltonian via systematic full configuration-interaction (FCI) calculations is able to predict the spectra as a function of detuning of three-electron hybrid qubits based on GaAs asymmetric double quantum dots (QDs). It is further shown that, as a result of strong inter-electron correlations, these spectroscopic patterns, including avoided crossings between states associated with different electron occupancies of the left and right wells, are inextricably related to the formation of Wigner molecules (WMs). These physical entities cannot be captured by the previously employed independent-particle or Hubbard-type theoretical modeling of the hybrid qubit. We report remarkable agreement with recent experimental results. Moreover, the present FCI methodology for multi-well QDs can be straightforwardly extended to treat Si/SiGe hybrid qubits, where the central role of WMs was recently experimentally confirmed as well.
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Affiliation(s)
| | - Uzi Landman
- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430
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Ha W, Ha SD, Choi MD, Tang Y, Schmitz AE, Levendorf MP, Lee K, Chappell JM, Adams TS, Hulbert DR, Acuna E, Noah RS, Matten JW, Jura MP, Wright JA, Rakher MT, Borselli MG. A Flexible Design Platform for Si/SiGe Exchange-Only Qubits with Low Disorder. NANO LETTERS 2022; 22:1443-1448. [PMID: 34806894 DOI: 10.1021/acs.nanolett.1c03026] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Spin-based silicon quantum dots are an attractive qubit technology for quantum information processing with respect to coherence time, control, and engineering. Here we present an exchange-only Si qubit device platform that combines the throughput of CMOS-like wafer processing with the versatility of direct-write lithography. The technology, which we coin "SLEDGE", features dot-shaped gates that are patterned simultaneously on one topographical plane and subsequently connected by vias to interconnect metal lines. The process design enables nontrivial layouts as well as flexibility in gate dimensions, material selection, and additional device features such as for rf qubit control. We show that the SLEDGE process has reduced electrostatic disorder with respect to traditional overlapping gate devices with lift-off metallization, and we present spin coherent exchange oscillations and single qubit blind randomized benchmarking data.
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Affiliation(s)
- Wonill Ha
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Sieu D Ha
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Maxwell D Choi
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Yan Tang
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Adele E Schmitz
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Mark P Levendorf
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Kangmu Lee
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - James M Chappell
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Tower S Adams
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Daniel R Hulbert
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Edwin Acuna
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Ramsey S Noah
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Justine W Matten
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Michael P Jura
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Jeffrey A Wright
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Matthew T Rakher
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
| | - Matthew G Borselli
- HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, United States
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Alfieri A, Anantharaman SB, Zhang H, Jariwala D. Nanomaterials for Quantum Information Science and Engineering. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2022:e2109621. [PMID: 35139247 DOI: 10.1002/adma.202109621] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2021] [Revised: 02/04/2022] [Indexed: 06/14/2023]
Abstract
Quantum information science and engineering (QISE)-which entails the use of quantum mechanical states for information processing, communications, and sensing-and the area of nanoscience and nanotechnology have dominated condensed matter physics and materials science research in the 21st century. Solid-state devices for QISE have, to this point, predominantly been designed with bulk materials as their constituents. This review considers how nanomaterials (i.e., materials with intrinsic quantum confinement) may offer inherent advantages over conventional materials for QISE. The materials challenges for specific types of qubits, along with how emerging nanomaterials may overcome these challenges, are identified. Challenges for and progress toward nanomaterials-based quantum devices are condidered. The overall aim of the review is to help close the gap between the nanotechnology and quantum information communities and inspire research that will lead to next-generation quantum devices for scalable and practical quantum applications.
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Affiliation(s)
- Adam Alfieri
- Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA
| | - Surendra B Anantharaman
- Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA
| | - Huiqin Zhang
- Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA
| | - Deep Jariwala
- Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA
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Jia Z, Fu Y, Cao Z, Cheng W, Zhao Y, Dou M, Duan P, Kong W, Cao G, Li H, Guo G. Superconducting and Silicon-Based Semiconductor Quantum Computers: A Review. IEEE NANOTECHNOLOGY MAGAZINE 2022. [DOI: 10.1109/mnano.2022.3175394] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Affiliation(s)
- Zhilong Jia
- University of Science and Technology of China
| | - Yaobin Fu
- Hefei Origin Quantum Computing Technology
| | - Zhen Cao
- Hefei Origin Quantum Computing Technology
| | | | | | | | - Peng Duan
- University of Science and Technology of China
| | | | - Gang Cao
- University of Science and Technology of China
| | - Haiou Li
- University of Science and Technology of China
| | - Guoping Guo
- University of Science and Technology of China
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Nikandish R, Blokhina E, Leipold D, Staszewski RB. Semiconductor Quantum Computing: Toward a CMOS quantum computer on chip. IEEE NANOTECHNOLOGY MAGAZINE 2021. [DOI: 10.1109/mnano.2021.3113216] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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Arumona AE, Amiri IS, Punthawanunt S, Ray K, Singh G, Bharti GK, Yupapin P. 3D‐quantum interferometer using silicon microring‐embedded gold grating circuit. Microsc Res Tech 2020; 83:1217-1224. [DOI: 10.1002/jemt.23513] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2020] [Revised: 05/01/2020] [Accepted: 05/02/2020] [Indexed: 11/09/2022]
Affiliation(s)
- Arumona Edward Arumona
- Computational Optics Research GroupAdvanced Institute of Materials Science, Ton Duc Thang University District 7, Ho Chi Minh City Vietnam
- Faculty of Applied SciencesTon Duc Thang University District 7, Ho Chi Minh City Vietnam
- Division of Computational PhysicsInstitute for Computational Science, Ton Duc Thang University Ho Chi Minh City Vietnam
| | - Iraj Sandegh Amiri
- Computational Optics Research GroupAdvanced Institute of Materials Science, Ton Duc Thang University District 7, Ho Chi Minh City Vietnam
| | | | - Kanad Ray
- Amity School of Applied SciencesAmity University Rajasthan Jaipur Rajasthan India
| | - Ghanshyam Singh
- Department of ECEMalaviya National Institute of Technology Jaipur (MNIT) Jaipur Rajasthan India
| | - Gaurav Kumar Bharti
- Department of Electronics and Communication EngineeringTechno Engineering College Banipur West Bengal India
| | - Preecha Yupapin
- Computational Optics Research GroupAdvanced Institute of Materials Science, Ton Duc Thang University District 7, Ho Chi Minh City Vietnam
- Faculty of Applied SciencesTon Duc Thang University District 7, Ho Chi Minh City Vietnam
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Russ M, Péterfalvi CG, Burkard G. Theory of valley-resolved spectroscopy of a Si triple quantum dot coupled to a microwave resonator. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:165301. [PMID: 31829981 DOI: 10.1088/1361-648x/ab613f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
We theoretically study a silicon triple quantum dot (TQD) system coupled to a superconducting microwave resonator. The response signal of an injected probe signal can be used to extract information about the level structure by measuring the transmission and phase shift of the output field. This information can further be used to gain knowledge about the valley splittings and valley phases in the individual dots. Since relevant valley states are typically split by several [Formula: see text], a finite temperature or an applied external bias voltage is required to populate energetically excited states. The theoretical methods in this paper include a capacitor model to fit experimental charging energies, an extended Hubbard model to describe the tunneling dynamics, a rate equation model to find the occupation probabilities, and an input-output model to determine the response signal of the resonator.
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Abstract
Spin qubits and superconducting qubits are among the promising candidates for realizing a solid state quantum computer. For the implementation of a hybrid architecture which can profit from the advantages of either approach, a coherent link is necessary that integrates and controllably couples both qubit types on the same chip over a distance that is several orders of magnitude longer than the physical size of the spin qubit. We realize such a link with a frequency-tunable high impedance SQUID array resonator. The spin qubit is a resonant exchange qubit hosted in a GaAs triple quantum dot. It can be operated at zero magnetic field, allowing it to coexist with superconducting qubits on the same chip. We spectroscopically observe coherent interaction between the resonant exchange qubit and a transmon qubit in both resonant and dispersive regimes, where the interaction is mediated either by real or virtual resonator photons. Different qubit platforms each have their own advantages and disadvantages. By engineering couplings between them it may be possible to create a more capable hybrid device. Here the authors demonstrate coherent coupling between a semiconductor spin qubit and a superconducting transmon.
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Andrews RW, Jones C, Reed MD, Jones AM, Ha SD, Jura MP, Kerckhoff J, Levendorf M, Meenehan S, Merkel ST, Smith A, Sun B, Weinstein AJ, Rakher MT, Ladd TD, Borselli MG. Quantifying error and leakage in an encoded Si/SiGe triple-dot qubit. NATURE NANOTECHNOLOGY 2019; 14:747-750. [PMID: 31308497 DOI: 10.1038/s41565-019-0500-4] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2018] [Accepted: 06/06/2019] [Indexed: 06/10/2023]
Abstract
Quantum computation requires qubits that satisfy often-conflicting criteria, which include long-lasting coherence and scalable control1. One approach to creating a suitable qubit is to operate in an encoded subspace of several physical qubits. Although such encoded qubits may be particularly susceptible to leakage out of their computational subspace, they can be insensitive to certain noise processes2,3 and can also allow logical control with a single type of entangling interaction4 while maintaining favourable features of the underlying physical system. Here we demonstrate high-fidelity operation of an exchange-only qubit encoded in a subsystem of three coupled electron spins5 confined in gated, isotopically enhanced silicon quantum dots6. This encoding requires neither high-frequency electric nor magnetic fields for control, and instead relies exclusively on the exchange interaction4,5, which is highly local and can be modulated with a large on-off ratio using only fast voltage pulses. It is also compatible with very low and gradient-free magnetic field environments, which simplifies integration with superconducting materials. We developed and employed a modified blind randomized benchmarking protocol that determines both computational and leakage errors7,8, and found that unitary operations have an average total error of 0.35%, with half of that, 0.17%, coming from leakage driven by interactions with substrate nuclear spins. The combination of this proven performance with complete control via gate voltages makes the exchange-only qubit especially attractive for use in many-qubit systems.
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Affiliation(s)
| | | | | | | | - Sieu D Ha
- HRL Laboratories, LLC, Malibu, CA, USA
| | | | | | | | | | | | | | - Bo Sun
- HRL Laboratories, LLC, Malibu, CA, USA
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van Meter JR, Knill E. Approximate exchange-only entangling gates for the three-spin-1/2 decoherence-free subsystem. PHYSICAL REVIEW. A 2019; 99:10.1103/physreva.99.042331. [PMID: 34136734 PMCID: PMC8204502 DOI: 10.1103/physreva.99.042331] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The three-spin-1/2 decoherence-free subsystem defines a logical qubit protected from collective noise and supports exchange-only universal gates. Such logical qubits are well suited for implementation with electrically defined quantum dots. Exact exchange-only entangling logical gates exist but are challenging to construct and understand. We use a decoupling strategy to obtain straightforward approximate entangling gates. A benefit of the strategy is that, if the physical spins are aligned, then it can implement evolution under entangling Hamiltonians. Hamiltonians expressible as linear combinations of logical Pauli products not involving σ y can be implemented directly. Self-inverse gates that are constructible from these Hamiltonians, such as the controlled-not (cnot) gate, can be implemented without the assumption on the physical spins. We compare the control complexity of implementing cnot to previous methods and find that the complexity for fault-tolerant fidelities is competitive.
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Affiliation(s)
- James R van Meter
- Department of Mathematics, University of Colorado, Boulder, Colorado 80309, USA
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
| | - Emanuel Knill
- National Institute of Standards and Technology, Boulder, Colorado 80305, USA
- Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA
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Russ M, Petta JR, Burkard G. Quadrupolar Exchange-Only Spin Qubit. PHYSICAL REVIEW LETTERS 2018; 121:177701. [PMID: 30411952 DOI: 10.1103/physrevlett.121.177701] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2018] [Indexed: 06/08/2023]
Abstract
We propose a quadrupolar exchange-only spin qubit that is highly robust against charge noise and nuclear spin dephasing, the dominant decoherence mechanisms in quantum dots. The qubit consists of four electrons trapped in three quantum dots, and operates in a decoherence-free subspace to mitigate dephasing due to nuclear spins. To reduce sensitivity to charge noise, the qubit can be completely operated at an extended charge noise sweet spot that is first-order insensitive to electrical fluctuations. Because of on-site exchange mediated by the Coulomb interaction, the qubit energy splitting is electrically controllable and can amount to several GHz even in the "off" configuration, making it compatible with conventional microwave cavities.
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Affiliation(s)
- Maximilian Russ
- Department of Physics, University of Konstanz, D-78457 Konstanz, Germany
| | - J R Petta
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Guido Burkard
- Department of Physics, University of Konstanz, D-78457 Konstanz, Germany
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Landig AJ, Koski JV, Scarlino P, Mendes UC, Blais A, Reichl C, Wegscheider W, Wallraff A, Ensslin K, Ihn T. Coherent spin-photon coupling using a resonant exchange qubit. Nature 2018; 560:179-184. [PMID: 30046114 DOI: 10.1038/s41586-018-0365-y] [Citation(s) in RCA: 40] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2018] [Accepted: 06/19/2018] [Indexed: 11/09/2022]
Abstract
Electron spins hold great promise for quantum computation because of their long coherence times. Long-distance coherent coupling of spins is a crucial step towards quantum information processing with spin qubits. One approach to realizing interactions between distant spin qubits is to use photons as carriers of quantum information. Here we demonstrate strong coupling between single microwave photons in a niobium titanium nitride high-impedance resonator and a three-electron spin qubit (also known as a resonant exchange qubit) in a gallium arsenide device consisting of three quantum dots. We observe the vacuum Rabi mode splitting of the resonance of the resonator, which is a signature of strong coupling; specifically, we observe a coherent coupling strength of about 31 megahertz and a qubit decoherence rate of about 20 megahertz. We can tune the decoherence electrostatically to obtain a minimal decoherence rate of around 10 megahertz for a coupling strength of around 23 megahertz. We directly measure the dependence of the qubit-photon coupling strength on the tunable electric dipole moment of the qubit using the 'AC Stark' effect. Our demonstration of strong qubit-photon coupling for a three-electron spin qubit is an important step towards coherent long-distance coupling of spin qubits.
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Affiliation(s)
- A J Landig
- Department of Physics, ETH Zürich, Zurich, Switzerland.
| | - J V Koski
- Department of Physics, ETH Zürich, Zurich, Switzerland
| | - P Scarlino
- Department of Physics, ETH Zürich, Zurich, Switzerland
| | - U C Mendes
- Institut quantique and Départment de Physique, Université de Sherbrooke, Sherbrooke, Quebec, Canada
| | - A Blais
- Institut quantique and Départment de Physique, Université de Sherbrooke, Sherbrooke, Quebec, Canada.,Canadian Institute for Advanced Research, Toronto, Ontario, Canada
| | - C Reichl
- Department of Physics, ETH Zürich, Zurich, Switzerland
| | - W Wegscheider
- Department of Physics, ETH Zürich, Zurich, Switzerland
| | - A Wallraff
- Department of Physics, ETH Zürich, Zurich, Switzerland
| | - K Ensslin
- Department of Physics, ETH Zürich, Zurich, Switzerland
| | - T Ihn
- Department of Physics, ETH Zürich, Zurich, Switzerland
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Łuczak J, Bułka BR. Two-qubit logical operations in three quantum dots system. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:225601. [PMID: 29658887 DOI: 10.1088/1361-648x/aabe50] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We consider a model of two interacting always-on, exchange-only qubits for which controlled phase (CPHASE), controlled NOT (CNOT), quantum Fourier transform (QFT) and SWAP operations can be implemented only in a few electrical pulses in a nanosecond time scale. Each qubit is built of three quantum dots (TQD) in a triangular geometry with three electron spins which are always kept coupled by exchange interactions only. The qubit states are encoded in a doublet subspace and are fully electrically controlled by a voltage applied to gate electrodes. The two qubit quantum gates are realized by short electrical pulses which change the triangular symmetry of TQD and switch on exchange interaction between the qubits. We found an optimal configuration to implement the CPHASE gate by a single pulse of the order 2.3 ns. Using this gate, in combination with single qubit operations, we searched for optimal conditions to perform the other gates: CNOT, QFT and SWAP. Our studies take into account environment effects and leakage processes as well. The results suggest that the system can be implemented for fault tolerant quantum computations.
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Affiliation(s)
- Jakub Łuczak
- Institute of Molecular Physics, Polish Academy of Sciences, ul. M. Smoluchowskiego 17, 60-179 Poznań, Poland
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Milivojević M. Symmetric spin-orbit interaction in triple quantum dot and minimisation of spin-orbit leakage in CNOT gate. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:085302. [PMID: 29328053 DOI: 10.1088/1361-648x/aaa736] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We analyse spin-orbit interaction in triple quantum dots and show that a symmetric spin-orbit Hamiltonian does not follow the standard form used in double quantum dots, as a consequence of the presence of the third dot in the setup. Furthermore, CNOT implementation schemes based on the exchange interaction were studied. It was shown that an antisymmetric Dzyaloshinsky-Moriya term is the dominant source of spin-orbit leakage from the computational space. We present a simple scheme for the minimisation of leakage that can be implemented in cases where interacting spins enclose parallelogram or equilateral triangle loops.
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Affiliation(s)
- Marko Milivojević
- Department of Physics, University of Belgrade, Studentski trg 12-16, 11000 Belgrade, Serbia
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