This talk sketches how quantum computers are build and the current state of the art. In a second part we discuss possibilities and limitations of quantum computers

Epping, Michael

Schuhmacher, Peter

Institute of Transport Research:Publications

www.DLR.de  Chart 1 > 2021-11-03 > DLR-SC > 2nd WAW IntroSecond WAW Quantum ComputingIntroductory TalkMichael EppingPeter Ken Schuhmacher– based on talks of the HPC Seminar –Quantum Computing GroupHigh-Performance ComputingInstitute for Software TechnologyNovember 3rd, 2021www.DLR.de  Chart 2 > 2021-11-03 > DLR-SC > 2nd WAW IntroOutlineHow to build a quantum computerState of DevelopmentWhat Quantum Computers can doAnd what they can’twww.DLR.de  Chart 3 > 2021-11-03 > DLR-SC > 2nd WAW IntroOriginal Idea Challenge: the Hilbert space dimension of a quantumsystem scales exponentially with the number of particles Simulation of quantum systems is hard for classical computingmachines Solution: Use the exponentially large Hilbert space ofquantum systems as a resource for computing“Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make itquantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.”Richard Feynman, “Simulating Physics with Computers”, 1st conf. on Phys. and Comp., MIT (1981)www.DLR.de  Chart 4 > 2021-11-03 > DLR-SC > 2nd WAW IntroSuperposition Single qubit basis: j0i, j1i Composite system via tensor product.Basis: j0i  j0i, j0i  j1i, j1i  j0i, ... Superposition-principle introduces “intermediate” states, e.g.j i = 1p2(j0i  j0i+ j1i  j1i) j i 6= ji  ji entanglementj0i  j0ij1i  j1ij iwww.DLR.de  Chart 5 > 2021-11-03 > DLR-SC > 2nd WAW IntroDiVincenzo’s CriteriaRequirements for quantum computers1:1. scalable2. initializable3. low noise4. a universal set of gates5. readout1D.P. DiVincenzo, The Physical Implementation of Quantum Computation, Fortschr. Phys. 48: 771-783 (2000)www.DLR.de  Chart 6 > 2021-11-03 > DLR-SC > 2nd WAW IntroQuantum Error Correction (QEC) Quantum information is prone to noise. Any noise can be decomposed into:1. bit-flips j0i 7! j1i, j1i 7! j0i2. phase-flips j0i 7! j0i, j1i 7!   j1i It’s not possible to copy quantum states (No-Cloning-Theorem)! Create redundancy via entangled multi-qubit states, e.g. j0i+  j1i  !  j000i+  j111i We need five qubits to correct both error types.www.DLR.de  Chart 7 > 2021-11-03 > DLR-SC > 2nd WAW IntroThreshold Theorem Error correction is implemented vianoisy gates. Increasing the code size improves thecapabilities, but requires more gates. There is a threshold gate fidelity forwhere more error correction helps. High fidelities (e.g. 99.99%) withmany qubits (e.g. 106) enable arbitrarylong computation.Image: Google (Martinis group)www.DLR.de  Chart 8 > 2021-11-03 > DLR-SC > 2nd WAW IntroHow to build a quantum computerState of DevelopmentWhat Quantum Computers can doAnd what they can’twww.DLR.de  Chart 9 > 2021-11-03 > DLR-SC > 2nd WAW IntroDevelopment Levels1 Level A: Basic functionality 2 qubits, gate operations, readout Level B: QualityGate error rates below error correction threshold Level C: Quantum Error CorrectionQuantum error correction helps Level D: Fault-tolerant operationsFully error corrected universal gates Level E: AlgorithmsComplex algorithms have been executed1 F.K. Wilhelm et. al., Entwicklungsstand Quantencomputer, BSI Studie 283 (2020)www.DLR.de  Chart 10 > 2021-11-03 > DLR-SC > 2nd WAW IntroHardware Platforms: State of DevelopmentA Basic functionality B Quality C Error Correction D/ETopological PlatformsMolecular SpinsGaAs-Quantum-DotsSilicon DonorsNuclear Magnetic ResonancePhotonsColor CenterSiGe-Quantum-Dots3D-TransmonsRydberg-AtomsFlux qubits2D-TransmonsIon TrapsState of Development11 F.K. Wilhelm et. al., Entwicklungsstand Quantencomputer, BSI Studie 283 (2020)www.DLR.de  Chart 11 > 2021-11-03 > DLR-SC > 2nd WAW IntroPhotons Qubits based on non-classical states of light:Single photons, Multi-photon squeezed states, ... Circuits are waveguides integrated intomicro-chips. Gates are formed by Beam splitters, phaseshifters, interferometers and more. Pro: No cooling, easily integrates with networks Con: Difficult interaction, photon loss Image: Xanaduwww.DLR.de  Chart 12 > 2021-11-03 > DLR-SC > 2nd WAW IntroNitrogen-Vacancy-Centers The NV-center is a charged defect in diamond. The electron spin forms the qubit. The spin is read via photoluminescence. Pro: Long lifetime. Con: Each center has different characteristics.Image: Wikipediawww.DLR.de  Chart 13 > 2021-11-03 > DLR-SC > 2nd WAW IntroRydberg-Atoms Rydberg-Atoms are highly excited Atoms, withlarger diameter. They can be trapped in optical lattices. Multi-qubit gates via strong interaction withneighbours. Pro: Many qubits Con: Lower gate fidelities, atom lossImage: University of Stuttgartwww.DLR.de  Chart 14 > 2021-11-03 > DLR-SC > 2nd WAW IntroSuperconducting Qubits Macroscopic (m) circuits, e.g. rings or junctions Cooled to low temperatures (. 100mK) Commercial front-runner (IBM, Google, ...) Pro: Established manufacturing, scalable Con: Cryogenic cooling, noise, different qubitsImage: Fraunhoferwww.DLR.de  Chart 15 > 2021-11-03 > DLR-SC > 2nd WAW IntroIon Traps Charged atoms are trapped in a time-dependentelectromagnetic field. The qubit states are atomic levels. Qubits interact via motion. Pro: Low noise, multi-qubit interactions Con: Difficult scaling, ion lossImage: IonQwww.DLR.de  Chart 16 > 2021-11-03 > DLR-SC > 2nd WAW IntroHow to build a quantum computerState of DevelopmentWhat Quantum Computers can doAnd what they can’twww.DLR.de  Chart 17 > 2021-11-03 > DLR-SC > 2nd WAW IntroThe input-output-problem State preparation can set all amplitudes in the state may require exponentially many gates State tomography unveils the 2n complex amplitudes of the state takes an exponential amount of measurements(and copies of the state!) good problems for quantum computers1. are difficult2. are specified by few parameters3. have a short answer (yes, no, few integers)www.DLR.de  Chart 18 > 2021-11-03 > DLR-SC > 2nd WAW IntroGrover search Every quantum operation acts linearly on all states in a superposition. For example:G j i = 1p2(G j00i+ G j11i) An oracle is a black box which marks solutions, e.g. G j00i = j00i, G j11i =   j11i. Grover (roughly): Start with uniform superposition, iteratively suppress unmarked states. For problems without structure:Classically try all N = 2n possible solutions O(N)Grover O(pN) quadratic speed-up. Known to be optimal.www.DLR.de  Chart 19 > 2021-11-03 > DLR-SC > 2nd WAW IntroQuantum Fourier transform (QFT) The QFT performs the discrete fourier transform on quantum amplitudesjji 7!2n 1Σk=0e2ijk=2n jki It uses O(n2) gates, while Fast Fourier Transform (FFT) requires O(n2n) gates Preparing and accessing amplitudes is difficult! QFT is a key component of many quantum algorithms: Phase estimation Period finding Discrete logarithm and factorization ...www.DLR.de  Chart 20 > 2021-11-03 > DLR-SC > 2nd WAW IntroHow to build a quantum computerState of DevelopmentWhat Quantum Computers can doAnd what they can’twww.DLR.de  Chart 21 > 2021-11-03 > DLR-SC > 2nd WAW IntroNo-Signalling Despite entanglement we can’t communicate faster thanlight. Every quantum operation on Alice’s subsystem only has nomeasurable effect on Bob’s system. Useful: Any idea that violates no-signalling is unphysical.www.DLR.de  Chart 22 > 2021-11-03 > DLR-SC > 2nd WAW IntroNo-Cloning It is impossible to copy an unknown quantum state:j'i 6 ! j'i j'i Cloning would violate the no-signalling principle:The choice of measurement basis on one half of anentangled pair could be determined from the second half. No-cloning makes quantum error correction difficult. It enables quantum cryptography.www.DLR.de  Chart 23 > 2021-11-03 > DLR-SC > 2nd WAW IntroThank you! 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Second WAW Quantum Computing: Introductory Talk

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Institute of Transport Research:Publications