DOCSLIB.ORG

Quantum Computers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Computers Quantum Computers ERIK LIND. PROFESSOR IN NANOELECTRONICS Erik Lind EIT, Lund University 1 Disclaimer Erik Lind EIT, Lund University 2 Quantum Computers Classical Digital Computers Why quantum computers? Basics of QM • Spin-Qubits • Optical qubits • Superconducting Qubits • Topological Qubits Erik Lind EIT, Lund University 3 Classical Computing • We build digital electronics using CMOS • Boolean Logic • Classical Bit – 0/1 (Defined as a voltage level 0/+Vdd) +vDD +vDD +vDD Inverter NAND Logic is built by connecting gates Erik Lind EIT, Lund University 4 Classical Computing • We build digital electronics using CMOS • Boolean Logic • Classical Bit – 0/1 • Very rubust! • Modern CPU – billions of transistors (logic and memory) • We can keep a logical state for years • We can easily copy a bit • Mainly through irreversible computing However – some problems are hard to solve on a classical computer Erik Lind EIT, Lund University 5 Computing Complexity Searching an unsorted list Factoring large numbers Grovers algo. – Shor’s Algorithm BQP • P – easy to solve and check on a 푛 classical computer ~O(N) • NP – easy to check – hard to solve on a classical computer ~O(2N) P P • BQP – easy to solve on a quantum computer. ~O(N) NP • BQP is larger then P • Some problems in P can be more efficiently solved by a quantum computer Simulating quantum systems Note – it is still not known if 푃 = 푁푃. It is very hard to build a quantum computer… Erik Lind EIT, Lund University 6 Basics of Quantum Mechanics A state is a vector (ray) in a complex Hilbert Space Dimension of space – possible eigenstates of the state For quantum computation – two dimensional Hilbert space ۧ↓ Example - Spin ½ particle - ȁ↑ۧ 표푟 Spin can either be ‘up’ or ‘down’ Erik Lind EIT, Lund University 7 Basics of Quantum Mechanics • A state of a system is a vector (ray) in a complex vector (Hilbert) Space. • Denoted 푥 using Dirac notation. (You can think of this as a vector in 3D space.) • Orthonormal basis vectors – various observables (Energy, position, momentum, spin direction…). • A state can be in a superposition of the basis vectors Example – the position of Example – spin of a electron 0 1 ۧ↓ an electron on a molecule ۧ↓ 0 ۧ↑ȁ 1 2 ۧ↑ȁ 5 3 2 4 4 5 3 Erik Lind EIT, Lund University 8 Basics of Quantum Mechanics Depending on the choise of basis – the state of an particle can be described using different basis vectors! 푏 푎 푛 푏1 푛 휓 = 푎1 휓 = 푎 푏1 + 푏 푏2 + 푐 푏3 + ⋯ 푎4 Ex: Difference between position and momentum eigenstates 푏4 Uncertainty relation 푥, 푝 = −푖ℏ 푎1 푏2 푎3 Coefficients 훼, 훽, 훾 are complex numbers 푏3 푎 2 State is normalized: 훼 2 + 훽 2+ 훾 2 + ⋯ = 1 Can be expressed in various bases (observables) • Energy • Position • Momentum • … Erik Lind EIT, Lund University 9 Basics of Quantum Mechanics ۧ↓ For quantum computation – effective 2D dimensional Hilbert space This is called a quantum bit (qubit, qbit) ۧ↑ȁ 휓 = 훼 0 + 훽 1 = 훼 ↑ + 훽 ↓ Example - Spin ½ particle - ȁ↑ۧ ↓ۧ : 2 dimensional Hilbert space • 훼 휓 = 훼 0 + 훽 1 훽 a anb b – two complex numbers State is normalized: 훼2 + 훽2 = 1 Matrix representation of a state Erik Lind EIT, Lund University 10 Quantum Bit – two level system Quantum system with only two distinct (energy) states • Spin ½ particle (spin) • Double quantum dot (position) 0 1 • Atom (orbitals) E2 x E1 Normalized Complex coeff. 휓 = 훼 0 + 훽 1 Bloch Sphere representation 휃 휃 훼 휓 = sin 0 + cos 푒푖휙 1 2 2 훽 Computational basis (with spins) • Spin up in z-direction For one spin ½ particle – the Bloch sphere • Spin down in z-direction represent the spatial orientation of the spin! Spin in x-direction? 1 1 푆푥 = 0 + 1 2 2 Erik Lind EIT, Lund University 11 Many qubits A Qubit A Tensor product 퐻퐴 휓 = 0 퐴 ⊗ 1 퐵 = 01 퐻퐵 B Qubit B Example: Two non-interacting qubits – product state 휓 퐴 = 훼퐴 0 퐴 + 훽퐴 1 퐴 휓 = 휓 퐴 ⊗ 휓 퐵 = 훼퐴훼퐵 00 + 훽퐴훼퐵 10 + 훼퐴훽퐵 01 + 훽퐴훽퐵 11 휓 퐵 = 훼퐵 0 퐵 + 훽퐵 1 퐵 Example: Two correlated qubits – entangled state 1 휓 = ( 10 + 01 ) This can NOT be written 2 as a product state! ۧ↓↓ N Coupled qbits requires 2푁 complex Example: General 2 qubit state numbers to describe 휓 = 훼 00 + 훽 10 +훾 01 +훿 11 ۧ↑↑N=100 : 2100 ≈ 1030 complex numbers ↑↓ۧ ȁ ۧ↑↓ȁ Erik Lind EIT, Lund University 12 Superposition of 2-bit states 1 ۧ↑↓휓 = 0 + 1 one spins – 2 states ȁ 2 1 휓 = 00 + 10 + 01 + 11 Two spins – 4 states 4 1 휓 = 000 + 001 + 010 + 100 + 011 + 110 + 101 + 111 Three spins – 8 states 8 휓 = ⋯ . 100 spin ½ : 2100 = 1030 dimensional space • There are 1080 atoms in the observable 휓 = ⋯ . 300 spin ½ : 2300 = 1090 dimensional space universe • Hilbert spaces are big! A quantum state can represent a huge number of 0’s and 1’ simultaneously! Erik Lind EIT, Lund University 13 Quantum System Dynamics A quantum system evolves in time set by the 휓(푡) = 훼(푡) 푎1 + 훽(푡) 푎2 + 훾(푡) 푎3 + ⋯ Schrödinger Equation, where H is the Hamilton operator. 푎푛 푑 푖ℏ 휓(푡) = 퐻෡(푡) 휓(푡) 푎4 푑푡 푝Ƹ 2 퐻෡ = + 푉(푥ො) Electron in potential 2푚 푎1 퐻෡ = −훾퐵푧휎푧 Spin in magnetic field 푎3 푎 • This give a set of coupled differential equations for 2 the coefficients a b g… • By applying an Hamiltonian to a quantum state – move the state in the Hilbert space over time. We can in this way control a quantum state. • This is deterministic and time reversible! Erik Lind EIT, Lund University 14 Measurement in Quantum Mechanics Measurement – apparatus which determines an 휓 = 훼 푎1 + 훽 푎2 + 훾 푎3 + ⋯ observable (Energy, position, Spin direction…) 푎푛 1 1 푎 휓 = ↑ + ↓ 4 2 2 푎1 50% ↑ ↓ ? 푎3 푎2 During a measurement of an observable a State collapses to an eigenstate with probability 50% 2 푃푛 = 푎푛 휓 푑 • Not deterministic and NOT time reversible! 푖ℏ 휓(푡) = 퐻෡(푡) 휓(푡) ? 푑푡 Erik Lind EIT, Lund University 15 Measurement of states 푁 = 100 ↔ 1030 푠푡푎푡푒푠 1 휓 = 000 … + 001 … + 010 … + 100 … + 011 … + 110 … + ⋯ 2푁 Will randomly select ONE of the states with equal probability! Gives 100 bits of information or out 1030 Direct application of quantum parallelism is not an easy task! Entangeled state 1) Separate two entangled qubits (by light years (!)) 2) Measure qubit A (will show 0 or 1 with 50%) 3) If A shows 0 퐴 - we now know that qubit B is 1 퐵 1 휓 = ( 10 + 01 ) 4) Spooky action at a distance! 2 Erik Lind EIT, Lund University 16 Single quantum Gate – Bloch Sphere rotation 휓 휓′ Quantum gates are U reversible! Let H(t) act on 휓 for some time t0. These can in general be build by applying time varying B/E field to a spin/position qbit in x,y and z direction for 휃 휃 a fixed time. 휓 = sin 0 + cos 푒푖휙 1 2 2 0 1 NOT gate p rotation around x axis 0 → 1 Matrix representation 1 0 of a operator 훼 휓 = 훼 0 + 훽 1 훽 1 1 1 1 Matrix representation Hadamard gate: 0 → 0 + 1 of a operator Matrix representation of a state 2 2 1 −1 These also work on superpositions: 휓 = 훼 0 + 훽 1 Erik Lind EIT, Lund University 17 Quantum Gate – Controlled Gate Let H(t) act on x and y for some time t0. 휓′ 푥 U 푥 푦 푈 푦 or 푦 If x=0 do nothing to qubit y If x=1 apply rotation U to qubit y CNOT Gate 휃 휃 휓 = sin 0 + cos 푒푖휙 1 2 2 If x = 0 then y=y If x = 1 then apply NOT to y • All controlled gates will be unitary • Controlled Gates also work on super positions – quantum parallelism! Erik Lind EIT, Lund University 18 Universial set of quantum gates These set of gates are ‘universal’ Can implement all logic functions NAND, NOR etc. 1 qbit gate – controlled rotation of Bloch Sphere Controlled Gates – almost anyone will do, controlled NOT (CNOT) or controlled phase CPHASE most common. 1 1 1 1 1 1 1 (퐻ȁ0ۧ ∶ = 퐻ȁ0ۧ = (ȁ0ۧ + ȁ1ۧ 2 1 −1 0 2 1 2 휓 = 훼 00 + 훽 10 +훾 01 +훿 11 훼 훽 훾 훿 Erik Lind EIT, Lund University 19 Example - Deutsch’s Algorithm 푓 푥 =? x ?f ?f ?f ?f 푥 푓 0 ≠ 푓 1 “balanced” Uf 푥 0 0 0 1 1 푦 푦⨁푓(푥) 푓 0 = 푓 1 “constant” 1 0 1 0 1 c b b c 푓(푥) takes a single bit {0,1} into a single output bit {0,1}. If we don’t know if f(x) is “balanced” or “constant” – but would like to know Classical Computer – calculate f(0) and f(1) and compare – need to evaluate f twice! The quantum circuit below can achieve this by evaluating f(x) only once! 0 퐴 H Uf H Measure qubit A 0 푓 0 = 푓 1 1 푓 0 ≠ 푓 1 1 퐵 H Erik Lind EIT, Lund University 20 General Quantum Computing x0 x0 x1 x1 x2 … x2 … … 0 f0 푥 0 → 푥 푓(푥) 0 f1 0 f2 … … 2푁−1 1 푁 1 Input register 0 + 1 = ෍ 푥 This can be done with N Hadamard gates 2 2푁/2 0 By just computing the f(x) once, f(x) is 2푁−1 1 “calculated” for all 2N values! 푥 0 → ෍ 푥 푓(푥) 30 2푁/2 N=100 – 10 speedup! 0 2푁−1 1 However, a direct measurement of x ෍ 푥 푓(푥) → 푥0 푓(푥0) 2푁/2 gives only ONE random value of f(x0)! 0 Measurement of 푥 Need clever algorithms similar to Deutsch Problem to try and extract more information from the correlation between x and f(x) Erik Lind EIT, Lund University 21 Decoherence – random Bloch sphere rotation A single qbit will couple to the environment (example for a single ½ spin) • Nuclear spins • Stray B-fields Couples through spin- • Electrical fluctuations orbit interactions • Interactions with photons • Electrical Noise This will induce random 퐻(푡) which randomly shifts around the quantum state.
Load more

Recommended publications
  • Optimization of Transmon Design for Longer Coherence Time
    Optimization of transmon design for longer coherence time Simon Burkhard Semester thesis Spring semester 2012 Supervisor: Dr. Arkady Fedorov ETH Zurich Quantum Device Lab Prof. Dr. A. Wallraff Abstract Using electrostatic simulations, a transmon qubit with new shape and a large geometrical size was designed. The coherence time of the qubits was increased by a factor of 3-4 to about 4 µs. Additionally, a new formula for calculating the coupling strength of the qubit to a transmission line resonator was obtained, allowing reasonably accurate tuning of qubit parameters prior to production using electrostatic simulations. 2 Contents 1 Introduction 4 2 Theory 5 2.1 Coplanar waveguide resonator . 5 2.1.1 Quantum mechanical treatment . 5 2.2 Superconducting qubits . 5 2.2.1 Cooper pair box . 6 2.2.2 Transmon . 7 2.3 Resonator-qubit coupling (Jaynes-Cummings model) . 9 2.4 Effective transmon network . 9 2.4.1 Calculation of CΣ ............................... 10 2.4.2 Calculation of β ............................... 10 2.4.3 Qubits coupled to 2 resonators . 11 2.4.4 Charge line and flux line couplings . 11 2.5 Transmon relaxation time (T1) ........................... 11 3 Transmon simulation 12 3.1 Ansoft Maxwell . 12 3.1.1 Convergence of simulations . 13 3.1.2 Field integral . 13 3.2 Optimization of transmon parameters . 14 3.3 Intermediate transmon design . 15 3.4 Final transmon design . 15 3.5 Comparison of the surface electric field . 17 3.6 Simple estimate of T1 due to coupling to the charge line . 17 3.7 Simulation of the magnetic flux through the split junction .
    [Show full text]
  • Superconducting Transmon Machines
    Quantum Computing Hardware Platforms: Superconducting Transmon Machines • CSC591/ECE592 – Quantum Computing • Fall 2020 • Professor Patrick Dreher • Research Professor, Department of Computer Science • Chief Scientist, NC State IBM Q Hub 3 November 2020 Superconducting Transmon Quantum Computers 1 5 November 2020 Patrick Dreher Outline • Introduction – Digital versus Quantum Computation • Conceptual design for superconducting transmon QC hardware platforms • Construction of qubits with electronic circuits • Superconductivity • Josephson junction and nonlinear LC circuits • Transmon design with example IBM chip layout • Working with a superconducting transmon hardware platform • Quantum Mechanics of Two State Systems - Rabi oscillations • Performance characteristics of superconducting transmon hardware • Construct the basic CNOT gate • Entangled transmons and controlled gates • Appendices • References • Quantum mechanical solution of the 2 level system 3 November 2020 Superconducting Transmon Quantum Computers 2 5 November 2020 Patrick Dreher Digital Computation Hardware Platforms Based on a Base2 Mathematics • Design principle for a digital computer is built on base2 mathematics • Identify voltage changes in electrical circuits that map base2 math into a “zero” or “one” corresponding to an on/off state • Build electrical circuits from simple on/off states that apply these basic rules to construct digital computers 3 November 2020 Superconducting Transmon Quantum Computers 3 5 November 2020 Patrick Dreher From Previous Lectures It was
    [Show full text]
  • Measurement of Superconducting Qubits and Causality Alexander Korotkov University of California, Riverside
    Michigan State Univ., 09/21/15 Measurement of superconducting qubits and causality Alexander Korotkov University of California, Riverside Outline: Introduction (causality, POVM measurement) Circuit QED setup for measurement of superconducting qubits (transmons, Xmons) Qubit evolution due to measurement in circuit QED (simple and better theories) Experiments Alexander Korotkov University of California, Riverside Causality principle in quantum mechanics objects a and b observers A and B (and C) observers have “free will”; A B they can choose an action A choice made by observer A can affect light a b evolution of object b “back in time” cones time However, this retroactive control cannot pass space “useful” information to B (no signaling) C Randomness saves causality (even C Our focus: continuous cannot predict result of A measurement) |0 collapse Ensemble-averaged evolution of object b cannot depend on actions of observer A |1 Alexander Korotkov University of California, Riverside Werner Heisenberg Books: Physics and Philosophy: The Revolution in Modern Science Philosophical Problems of Quantum Physics The Physicist's Conception of Nature Across the Frontiers Niels Bohr Immanuel Kant (1724-1804), German philosopher Critique of pure reason (materialism, but not naive materialism) Nature - “Thing-in-itself” (noumenon, not phenomenon) Humans use “concepts (categories) of understanding”; make sense of phenomena, but never know noumena directly A priori: space, time, causality A naïve philosophy should not be a roadblock for good physics,
    [Show full text]
  • Introduction and Motivation the Transmon Qubits the Combined
    Noise Insensitive Superconducting Cavity Qubtis Pan Zheng, Thomas Alexander, Ivar Taminiau, Han Le, Michal Kononenko, Hamid Mohebbi, Vadiraj Ananthapadmanabha Rao, Karthik Sampath Kumar, Thomas McConkey, Evan Peters, Joseph Emerson, Matteo Mariantoni, Christopher Wilson, David G. Cory - - When driven in the single-photon level, the entire system is in Introduction and Motivation The Transmon Qubits the superposition of the states where the photon exists in either - Circuit quantum electrodynamics (QED) studies the - When the detuning between a qubit and a cavity is much larger of the cavities. The symmetric and antisymmetric superposition interactions between artificial atoms (superconducting qubits) than the coupling strength, the second order dispersive coupling of this collective mode can be used to encode a two level system, and electromagnetic fields in the microwave regime at low causes the cavity frequency depending on the state of the qubit realizing a logical qubit: temperatures [1]. It provides a fundamental architecture for [1]. The dispersive coupling provides possible links between realizing quantum computation. isolated cavities. - When a qubit is placed in a 3D superconducting cavity, the - A transmon qubit is a superconducting charge qubit insensitive cavity acts as both a resonator with low loss and a package to charge noise [5]. In the project they will be fabricated from providing an isolated environment [2]. Coherence times of evaporation deposited aluminum on a sapphire chip. transmon qubits up to ~100 μs have been observed [3]. - Advantages of the logical qubits: - Quantum memories made of 3D cavities have achieved 1) A logical qubit of this form will be immune to the collective coherence times as long as 1 ms [4].
    [Show full text]
  • Virtualized Logical Qubits: a 2.5D Architecture for Error-Corrected Quantum Computing
    2020 53rd Annual IEEE/ACM International Symposium on Microarchitecture (MICRO) Virtualized Logical Qubits: A 2.5D Architecture for Error-Corrected Quantum Computing Casey Duckering, Jonathan M. Baker, David I. Schuster, and Frederic T. Chong University of Chicago {cduck, jmbaker, david.schuster, ftchong}@uchicago.edu Abstract—Current, near-term quantum devices have shown execution. These efforts are directed at enabling NISQ (Noisy great progress in the last several years culminating recently with Intermediate-Scale Quantum) [2] algorithms to demonstrate a demonstration of quantum supremacy. In the medium-term, the power of quantum computing. Machines in this era are however, quantum machines will need to transition to greater reliability through error correction, likely through promising expected to run some important programs and have recently techniques like surface codes which are well suited for near-term been used to by Google to demonstrate “quantum supremacy” devices with limited qubit connectivity. We discover quantum [3]. memory, particularly resonant cavities with transmon qubits Despite this, these machines will be too small for error arranged in a 2.5D architecture, can efficiently implement surface correction and unable to run large-scale programs due to codes with substantial hardware savings and performance/fidelity gains. Specifically, we virtualize logical qubits by storing them in unreliable qubits. The ultimate goal is to construct fault- layers of qubit memories connected to each transmon. tolerant machines capable of executing thousands of gates Surprisingly, distributing each logical qubit across many and in the long-term to execute large-scale algorithms such memories has a minimal impact on fault tolerance and results as Shor’s [4] and Grover’s [5] with speedups over classical in substantially more efficient operations.
    [Show full text]
  • Quantum Information Processing and Quantum Optics with Circuit Quantum Electrodynamics
    FOCUS | REVIEW ARTICLE FOCUS | REVIEWhttps://doi.org/10.1038/s41567-020-0806-z ARTICLE Quantum information processing and quantum optics with circuit quantum electrodynamics Alexandre Blais 1,2,3 ✉ , Steven M. Girvin 4,5 and William D. Oliver6,7,8 Since the first observation of coherent quantum behaviour in a superconducting qubit, now more than 20 years ago, there have been substantial developments in the field of superconducting quantum circuits. One such advance is the introduction of the concepts of cavity quantum electrodynamics (QED) to superconducting circuits, to yield what is now known as circuit QED. This approach realizes in a single architecture the essential requirements for quantum computation, and has already been used to run simple quantum algorithms and to operate tens of superconducting qubits simultaneously. For these reasons, circuit QED is one of the leading architectures for quantum computation. In parallel to these advances towards quantum information process- ing, circuit QED offers new opportunities for the exploration of the rich physics of quantum optics in novel parameter regimes in which strongly nonlinear effects are readily visible at the level of individual microwave photons. We review circuit QED in the context of quantum information processing and quantum optics, and discuss some of the challenges on the road towards scalable quantum computation. avity quantum electrodynamics (QED) studies the interac- introduce the basic concepts of circuit QED and how the new tion of light and matter at its most fundamental level: the parameter regimes that can be obtained in circuit QED, compared Ccoherent interaction of single atoms with single photons. with cavity QED, lead to new possibilities for quantum optics.
    [Show full text]
  • Supercomputer Simulations of Transmon Quantum Computers Quantum Simulations of Transmon Supercomputer
    IAS Series IAS Supercomputer simulations of transmon quantum computers quantum simulations of transmon Supercomputer 45 Supercomputer simulations of transmon quantum computers Dennis Willsch IAS Series IAS Series Band / Volume 45 Band / Volume 45 ISBN 978-3-95806-505-5 ISBN 978-3-95806-505-5 Dennis Willsch Schriften des Forschungszentrums Jülich IAS Series Band / Volume 45 Forschungszentrum Jülich GmbH Institute for Advanced Simulation (IAS) Jülich Supercomputing Centre (JSC) Supercomputer simulations of transmon quantum computers Dennis Willsch Schriften des Forschungszentrums Jülich IAS Series Band / Volume 45 ISSN 1868-8489 ISBN 978-3-95806-505-5 Bibliografsche Information der Deutschen Nationalbibliothek. Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografe; detaillierte Bibliografsche Daten sind im Internet über http://dnb.d-nb.de abrufbar. Herausgeber Forschungszentrum Jülich GmbH und Vertrieb: Zentralbibliothek, Verlag 52425 Jülich Tel.: +49 2461 61-5368 Fax: +49 2461 61-6103 [email protected] www.fz-juelich.de/zb Umschlaggestaltung: Grafsche Medien, Forschungszentrum Jülich GmbH Titelbild: Quantum Flagship/H.Ritsch Druck: Grafsche Medien, Forschungszentrum Jülich GmbH Copyright: Forschungszentrum Jülich 2020 Schriften des Forschungszentrums Jülich IAS Series, Band / Volume 45 D 82 (Diss. RWTH Aachen University, 2020) ISSN 1868-8489 ISBN 978-3-95806-505-5 Vollständig frei verfügbar über das Publikationsportal des Forschungszentrums Jülich (JuSER) unter www.fz-juelich.de/zb/openaccess. This is an Open Access publication distributed under the terms of the Creative Commons Attribution License 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We develop a simulator for quantum computers composed of superconducting transmon qubits.
    [Show full text]
  • Quantum Information Processing with Superconducting Circuits: a Review
    Quantum Information Processing with Superconducting Circuits: a Review G. Wendin Department of Microtechnology and Nanoscience - MC2, Chalmers University of Technology, SE-41296 Gothenburg, Sweden Abstract. During the last ten years, superconducting circuits have passed from being interesting physical devices to becoming contenders for near-future useful and scalable quantum information processing (QIP). Advanced quantum simulation experiments have been shown with up to nine qubits, while a demonstration of Quantum Supremacy with fifty qubits is anticipated in just a few years. Quantum Supremacy means that the quantum system can no longer be simulated by the most powerful classical supercomputers. Integrated classical-quantum computing systems are already emerging that can be used for software development and experimentation, even via web interfaces. Therefore, the time is ripe for describing some of the recent development of super- conducting devices, systems and applications. As such, the discussion of superconduct- ing qubits and circuits is limited to devices that are proven useful for current or near future applications. Consequently, the centre of interest is the practical applications of QIP, such as computation and simulation in Physics and Chemistry. Keywords: superconducting circuits, microwave resonators, Josephson junctions, qubits, quantum computing, simulation, quantum control, quantum error correction, superposition, entanglement arXiv:1610.02208v2 [quant-ph] 8 Oct 2017 Contents 1 Introduction 6 2 Easy and hard problems 8 2.1 Computational complexity . .9 2.2 Hard problems . .9 2.3 Quantum speedup . 10 2.4 Quantum Supremacy . 11 3 Superconducting circuits and systems 12 3.1 The DiVincenzo criteria (DV1-DV7) . 12 3.2 Josephson quantum circuits . 12 3.3 Qubits (DV1) .
    [Show full text]
  • Superconducting Qubits: Dephasing and Quantum Chemistry
    UNIVERSITY of CALIFORNIA Santa Barbara Superconducting Qubits: Dephasing and Quantum Chemistry A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics by Peter James Joyce O'Malley Committee in charge: Professor John Martinis, Chair Professor David Weld Professor Chetan Nayak June 2016 The dissertation of Peter James Joyce O'Malley is approved: Professor David Weld Professor Chetan Nayak Professor John Martinis, Chair June 2016 Copyright c 2016 by Peter James Joyce O'Malley v vi Any work that aims to further human knowledge is inherently dedicated to future generations. There is one particular member of the next generation to which I dedicate this particular work. vii viii Acknowledgements It is a truth universally acknowledged that a dissertation is not the work of a single person. Without John Martinis, of course, this work would not exist in any form. I will be eter- nally indebted to him for ideas, guidance, resources, and|perhaps most importantly| assembling a truly great group of people to surround myself with. To these people I must extend my gratitude, insufficient though it may be; thank you for helping me as I ventured away from superconducting qubits and welcoming me back as I returned. While the nature of a university research group is to always be in flux, this group is lucky enough to have the possibility to continue to work together to build something great, and perhaps an order of magnitude luckier that we should wish to remain so. It has been an honor. Also indispensable on this journey have been all the members of the physics depart- ment who have provided the support I needed (and PCS, I apologize for repeatedly ending up, somehow, on your naughty list).
    [Show full text]
  • AQIS16 Preface Content
    The Theory of Statistical Comparison with Applications in Quantum Information Science……………………………………………………………………….….……...………..… Introduction to measurement-based quantum computation…………………………. Engineering the quantum: probing atoms with light and light with atoms in a transmon circuit QED system…………………………………………………………….……….3 The device-independent outlook on quantum physics……………………..…...……. Superconducting qubit systems: recent experimental progress towards fault-tolerant quantum computing at IBM……………………………………...… Observation of frequency-domain Hong-Ou-Mandel interference……. Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair…………………………………………………………………………………….….... One-way and reference-frame independent EPR-steering………...… Are Incoherent Operations Physically Consistent? — A Physical Appraisal of Incoherent Operations and an Overview of Coherence Measures………............................. Relating the Resource Theories of Entanglement and Quantum Coherence An infinite dimensional Birkhoff’s Theorem and LOCC- convertibility………..… How local is the information in MPS/PEPS tensor networks……………......….. Information-theoretical analysis of topological entanglement entropy and multipartite correlations ……................................................................................................ Phase-like transitions in low-number quantum dots Bayesian magnetometry... Separation between quantum Lovász number and entanglement-assisted zero-error classical capacity ………..……..............................................................………...
    [Show full text]
  • The University of Chicago Quantum Computing with Distributed Modules a Dissertation Submitted to the Faculty of the Division Of
    THE UNIVERSITY OF CHICAGO QUANTUM COMPUTING WITH DISTRIBUTED MODULES A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS BY NELSON LEUNG CHICAGO, ILLINOIS MARCH 2019 Copyright c 2019 by Nelson Leung All Rights Reserved TABLE OF CONTENTS LIST OF FIGURES . vi LIST OF TABLES . viii ACKNOWLEDGMENTS . ix ABSTRACT . x PREFACE............................................ 1 1 QUANTUM COMPUTING . 2 1.1 Introduction to Quantum Computing . .2 1.2 High-Level Description of Some Quantum Algorithms . .3 1.2.1 Grover’s algorithm . .3 1.2.2 Shor’s algorithm . .3 1.2.3 Variational Quantum Eigensolver (VQE) . .4 1.3 Quantum Architectures . .4 1.3.1 Trapped Ions . .4 1.3.2 Silicon . .5 1.3.3 Superconducting qubits . .5 1.4 Thesis Overview . .6 2 CIRCUIT QUANTUM ELECTRODYNAMICS . 7 2.1 Quantization of Quantum Harmonic Oscillator . .7 2.2 Circuit Quantum Harmonic Oscillator . 10 2.3 Josephson Junction . 12 2.4 Quantization of Transmon Qubit . 13 2.5 Quantization of Split Junction Transmon Qubit with External Flux . 16 2.6 Coupling between Resonators . 18 2.7 Dispersive Shift between Transmon and Resonator . 21 3 CONTROLLING SUPERCONDUCTING QUBITS . 23 3.1 Charge Control . 23 3.2 Flux Control . 25 3.3 Transmon State Measurement . 25 3.4 Cryogenic Environment . 26 3.5 Microwave Control Electronics . 26 3.5.1 Oscillator . 26 3.5.2 Arbitrary waveform generator . 27 3.5.3 IQ mixer . 27 3.6 Other Electronics . 28 3.7 Microwave Components . 28 3.7.1 Attenuator .
    [Show full text]
  • The Transmon Qubit
    The Transmon Qubit Sam Bader December 14, 2013 Contents 1 Introduction 2 2 Qubit Architecture 4 2.1 Cooper Pair Box . .4 2.1.1 Classical Hamiltonian . .4 2.1.2 Quantized Hamiltonian . .5 2.2 Capacitively-shunted CPB . .7 2.2.1 The ratio EJ =Ec .....................................7 3 Circuit QED 9 3.1 Vocabulary of Cavity QED . .9 3.1.1 Jaynes-Cummings Hamiltonian . 10 3.1.2 Effects of the coupling: resonant and dispersive limits . 10 3.1.3 Purcell Effect . 11 3.2 Translating into Circuit QED . 12 3.2.1 Why Circuit QED is easier than Cavity QED . 12 3.2.2 Circuit QED Hamiltonian . 13 4 Control and Readout Protocol 15 4.1 Measurement . 15 4.2 Single qubit gates . 16 4.2.1 Modeling drives . 16 4.2.2 Applying gates . 16 4.3 Multi-qubit gates and entanglement . 17 5 Conclusion 19 A Derivation of Classical Hamiltonians for Qubit Systems 20 A.1 Cooper Pair Box . 20 A.2 Transmon with transmission line . 21 B Quantum Circuits 23 B.1 Charge basis . 23 B.2 Phase basis . 24 C Perturbation Theory for the Transmon 25 C.1 Periodic Potentials . 25 C.2 Transforming away the offset charge . 26 C.3 Duffing Oscillator . 26 C.3.1 Relative Anharmonicity . 26 C.3.2 Number operator matrix elements . 27 1 Chapter 1 Introduction Quantum information processing is one of the most thrilling prospects to emerge from the interaction of physics and computer science. In recent decades, scientists have transitioned from merely observing microscopic systems to actually controlling those same systems on the scale of individual quanta, and the future of information processing based on these techniques will revolutionize the computing industry.
    [Show full text]