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Dr. Andrés Gómez Andres.gomez.tato@cesga.es Jan. 2020 Quantum Computing
Schedule 2  Lecture 1: Introduction to Quantum Computing.  My First Quantum Program.  Lecture 2: Programming Quantum Algorithms  My first Quantum Program with ProjectQ  Lecture 3: Basic Quantum algorithms  Lecture 4: Advanced algorithms
Lecture 1  A brief history of QC and needs. Types of quantum computers.  Basic concepts: qubit, tensors, multiqubit, quantum gates, measurement, amplitudes  My first quantum program.  Quantum Circuits. Width, Depth, Quantum Volume.
Welcome to a Dream! Yuri Manin (1980) and Richard Feynman (1981) proposed independently the concept of Quantum Computer I’m here very “hot”!! -273ºC 4 Source: IBM https://en.wikipedia.org/wiki/Timeline_of_quantum_computing
Welcome to a Dream! Rigetti Intel Google D-Wave Qilimanjaro (Spain) And more in Europe, China, Australia, etc…… 5
Welcome to (my) Nightmare! (*) |𝜱± > = � � ( 𝟎𝟎 >± 𝟏𝟏 >) |𝜳± > = � � 𝟏 � � ( 𝟎𝟏 > ± 𝟏𝟎 >) Bell States Superposition and Entanglement 6 (*) When I was a student long time ago!
Quantum Technologies Ion Trap Transmon NV- Defect Diamond Photons Quantu m Dots And more in the future…. Majorama 7
Quantum Technologies
Quantum Computer  Quantum simulator [1]. Simulate a quantum system using another one, maybe simpler, that can be controlled by the experimenter.  Adiabatic Quantum Computer [2]. Prepares a known and easy Hamiltonian and lets it evolve to solution.  Topological Quantum Computer[4]. Uses topological properties.  Continuous Variable Quantum Computer [5].  Universal Quantum Computer [3]. 1Reviewed in Georgescu, I. M., Ashhab, S., & Nori, F. (2014). Quantum simulation. Reviews of Modern Physics, 86(1), 153– 185. http://doi.org/10.1103/RevModPhys.86.153 arXiv:1308.6253 2 Reviewed in Albash, T., & Lidar, D. A. (2016). Adiabatic Quantum Computing. arxiv:1611.04471 3Proposed in Deutsch, D. (1985). http://doi.org/10.1098/rspa.1985.0070 and Deutsch, D. (1989). http://doi.org/10.1098/rspa.1989.0099 4 Lahtinen V., Pachos J.K.. SciPost Phys. 3, 021 (2017) arXiv:1705.04103 5 Lloyd S. & Braunstein, A.L. Phys.Rev.Lett. 82 (1999) 1784-1787. arXiv:quant-ph/9810082 11
Adiabatic Quantum Computer HB = Initial Hamiltonian, which ground state is easy to find HP = Problem Hamiltonian, whose ground state encodes the solution to the problem H(s) = Combined Hamiltonial to evolve slowly: A(s) decrease smoothly and monotonically B(s) increase smothly and monotonically H(s) = A(s)HB + B(s)HP Li, R. Y., Felice, R. Di, Rohs, R., & Lidar, D. A. (2018). Quantum annealing versus classical machine learning applied to a simplified computational biology problem. Npj Quantum Information 2018 4:1, 4(1), 14. http://doi.org/10.1038/s41534-018-0060-8 12
A real example: Traffic Flow Optimisation Neukart, F., Dollen, D. Von, Compostella, G., Seidel, C., Yarkoni, S., & Parney, B. (2017). Traffic flow optimization using a quantum annealer. arXiv:1708.01625v2 D-Wave Adiabatic Computer Optimisation Classical Computer + QPU Unoptimised Optimised Preprocess Map&GPS Find congestions Find Alternatives Minimisation model Solve model Redistribute cars Classical Computer QPU 13
Xanadu. Continuous Variabl e https:// www.xanadu.ai/ 14  Language : Strawberr y Fields  Cloud service for Research
European Quantum Flagship  AQTION : Trapped Ions  OpenSuperQ : Superconducting  SQUARE: Scalable Rare Earth Ion Quantum Computing Nodes  MicroQC: Microwave driven ion trap quantum computing http://qt.eu 15
Quantum Networks http://english.cas.cn/newsroom/news/201709/t20170928_183577.s html 16
Google Quantum “Supremacy” Arute F, Arya K, Babbush R, Bacon D, Bardin JC, Barends R, et al. Quantum supremacy using a programmable superconducting processor. Nature. 2019;574:505.
Google Quantum “Supremacy”
Lecture 1  A brief history of QC and needs. Types of quantum computers.  Basic concepts: qubit, tensors, multiqubit, quantum gates, measurement, amplitudes  My first quantum program.  Quantum Circuits. Width, Depth, Quantum Volume.
1. A scalable physical system with well characterized qubits. 2. The ability to initialize the state of the qubits to a simple fiducial state, such as |000….000> 3. Long relevant decoherence times, much longer than the gate operation time. 4. A “universal” set of quantum gates. 5. A qubit-specific measurement capability. 20 DiVincenzo’s Criteria D. DiVincenzo (2000). “The Physical Implementation of Quantum Computation“, arXiv:quant-ph/0002077
What do you need (today)? 21 Complex numbers Matrix multiplication Understand TENSOR products Understand measurement and probabilities Imagination
BIT, QUBIT AND SUPERPOSITION BIT: A “classical” physical system with TWO states 0 OR 1 What 0 or 1 means is a convention Information is codified as a list of BITs BIT can be transformed from 0 to 1 and vice versa BITs can be operated with logical gates (OR,XOR,AND…) One BIT can be Classical Computer Business Card 22
BIT, QUBIT AND SUPERPOSITION Quantum Computer Business Card QuBIT: A “Quantum” physical system which yields one of TWO states when is measured 0 OR 1 What 0 or 1 means is a convention* Information is codified in several ways QuBIT can be transformed from 0 to 1 and vice versa QuBITs can be operated with UNITARY gates QuBITs cannot be cloned (no-clone theorem) QuBITs cannot be stored (yet) QuBITs cannot have a long life (yet) Usually, QuBITs are quiet 23
Quantum Technologies Krantz P, Kjaergaard M, Yan F, Orlando TP, Gustavsson S, Oliver WD. A Quantum Engineer’s Guide to Superconducting Qubits. Arxiv: 1904.06560
Our current nightmare! NOISE Noisy Intermediate-Scale Quantum” (NISQ) https://medium.com/@pchojecki/quantum-advantage-b345864 6bd9
Send Circuit “QPU” CPU Return Shots Parametric Quantum Circuit Learnin g
Parametric Quantum Circuit Learnin g
Parametric Quantum Circuit Learnin g
Parametric Quantum Circuit Learnin g SIMULATING WITH NOISE FEW SHOTS
Parametric Quantum Circuit Learnin g N=3, D=3 IBMQ_VIGO Nov. 17th, 2019 Source: IBM© Nov. 24th, 2019
Parametric Quantum Circuit Learnin g
Algorithms with shallow circuit s • QVE: Quantum Variational Eigensolver: https://arxiv.org/abs/1304.3061 • QAOA: Quantum Approximate Optimization Algorithm. http://arxiv.org/abs/1411.4028 • Variational Quantum Factoring: https://arxiv.org/abs/1808.08927 • Quantum Machine Learning: • Quantum Support Vector Machine • Quantum Principal Component Analysis • Quantum Variational Autoencoder, • Etc.
Quantum Variational Eigensolver Figure source: Wang, D., Higgott, O., & Brierley, S. (n.d.). A Generalised Variational Quantum Eigensolver. QPU CPU
www.inl.int “Despite a number of promising results, the theoretical evidence presented in the current literature does not yet allow us to conclude that quantum techniques can obtain an exponential advantage in a realistic learning setting” Ciliberto et.al. “Quantum machine learning: a classical perspective” http://dx.doi.org/10.1098/rspa.2017.0551 34 Quantum Machine Learning ? So: A lot of research to do!!!
www.inl.int Complex Numbers If 𝑖2 = −1, a complex number is defined by: c=𝑎 + 𝑏 ∗ 𝑖, with 𝑎, 𝑏 ∈ ℝ, 𝑐 ∈ ℂ Complex conjugate: 𝑐̅ = 𝑎 − 𝑏 ∗ 𝑖 Modulus: 𝑐 2 = 𝑐𝑐̅ = (𝑎 + 𝑏 ∗ 𝑖)(𝑎 − 𝑏 ∗ 𝑖) = 𝑎2 + 𝑏2 Polar form: c = 𝑐 𝑐𝑜𝑠𝜃 + 𝑐 𝑠𝑖𝑛𝜃 𝑖 = 𝑐 𝑒𝑖𝜃 35
QUBIT |0 > = |1 > = 1 Superposition 0 0 1 |𝜙 > = 𝛼 0 > + 𝛽 1 > Complex numbers 𝛼 2 𝛽 2 + = 1 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝛼 = 𝛼 𝑒𝑖𝜑 𝛼 2 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝜑 Phase 36
QUBIT |𝜙 > = 𝛼 0 > + 𝛽 1 >= � � � � Measurement of |𝜙 > in standard basis (|0>, | 1>) : |0> with probability 𝛼 2. State after measurement | 0> or |1> with probability 𝛽 2. State after measurement |1> < 𝜙 = 𝛼̅ < 0 + 𝛽̅ < 1| = 𝛼̅ 𝛽̅ < 𝜙|𝜙 > = 𝛼 2 + 𝛽 2 |𝜓 > = 𝛾 0 > + 𝛿 1 >= � � � � 37 < 𝜓|𝜙 > =𝛼 𝛾̅ + 𝛽 𝛿
Bloch’s Sphere � � 2 � � 2 |𝜓 > = cos( )|0 > + sin( ) 𝑒 𝑖 𝜑 |1 > 2 𝜋 |0 > −|1 > 𝜃 = , 𝜑 = 𝜋, |− > = 2 𝜃 = � � 2 , 𝜑 = � � 2 , |𝑖 > = |0 > +𝑖|1 > 𝜃 = � � 2 , 𝜑 = 3 𝜋 2 , | − 𝑖 > = 2 |0 > −𝑖|1 > 2 𝜋 𝜃 = 2 , 𝜑 = 0, | +> = |0 > + |1 > 2 y x |𝜓 > � � z |𝟎 > 𝜑 | +> | −> | − 𝒊 > |𝒊 > 38 |𝟏 > Hint: 𝑒𝑖𝜑 = cos(𝜑)+i sin(𝜑)
One-Qubit Transformations |𝜙 > = 𝛼 0 > + 𝛽 1 >= � � � �  Transform vector space in itself |𝜙′ > = U |𝜙 > = 𝑎11 𝑎21 𝑎12 𝑎22 � � � �  Unit lengh vectors must go to unit lenght vectors: < 𝜓 𝑈†𝑈 𝜙 > =< 𝜓|𝜙 >⇒ 𝑈†𝑈=I  Reversible  Geometrically, they are rotations of the complex vector space associated to |𝜙 > 𝑁𝑜𝑡𝑒: 𝑈† = 𝑈𝑇 39
One-Qubit Transformations  Phase shift 𝐾(𝛿) = 𝑒𝑖𝛿𝐼  Rotation, 𝑅(𝛽) = cos(𝛽) −sin(𝛽 ) sin(𝛽 ) 𝑐𝑜𝑠( 𝛽)  Phase rotation, 𝑇(𝛼) = 𝑒𝑖 𝛼 0 0 𝑒−𝑖𝛼  Any other Qubit unitary transformation can be written as: 𝐾 𝛿 𝑇 𝛼 𝑅 𝛽 𝑇 𝛾 = 𝑒𝑖(𝛿+𝛼+𝛾) cos 𝛽 − 𝑒𝑖(𝛿−𝛼+𝛾)sin 𝛽 𝑒𝑖(𝛿+𝛼−𝛾) sin 𝛽 𝑒𝑖(𝛿−𝛼−𝛾) cos 𝛽 40 Source: Eleanor G. Rieffel.Quantum Computing: A Gentle Introduction
One-Qubit Transformations  Phase shift 𝐾(𝛿) = 𝑒𝑖𝛿𝐼 � �  Rotation around 𝑥, 𝑅 𝜃 ≡ 𝑒−𝑖𝜃𝑋 = co s � � 2 −𝑖 sin � � 2 −𝑖 sin 𝜃 2 co s 𝜃 2 = cos 𝜃 2 𝐼 − 𝑖 𝑠𝑖𝑛 𝜃 2 � � � �  Rotation around 𝑦, 𝑅 𝜃 ≡ 𝑒−𝑖𝜃𝑌 = 𝜃 cos( ) 𝜃 −sin( ) 2 𝜃 sin( ) 2 𝜃 2 2 𝑐𝑜𝑠( ) � �  Phase rotation, Rotation around 𝑧, 𝑅 𝜃 ≡ 𝑒−𝑖𝜃𝑍 = 𝑒−𝑖𝜃/2 0 0 𝑒𝑖𝜃/2  Any other QuBit unitary transformation can be written as: 𝑈 = 𝐾(𝛿)𝑅𝑧 𝛾 𝑅𝑦 𝛽 𝑅𝑧 𝛼 42 Source: Nielsen & Chuang, Quantum Computation And Quantum Information
One-Qubit Transformations  Pauli Gates 1 , bit-flip or NOT.  𝑋 =  Y =  Z = 0 1 0 0 −𝑖 𝑖 0 1 0 0 −1  Cliffort group  𝐻𝑎𝑑𝑎𝑚𝑎𝑟𝑑, 𝐻 = 1 2 1 1 1 −1  𝑃ℎ𝑎𝑠𝑒, 𝑆 = 1 0 0 𝑖 � �  , 𝑇 = 8 1 0 0 𝑖 𝜋 𝑒 4 Hint : † 𝑈 = 𝑈 � � 𝑈1 𝜆 = 1 0 0 𝑒𝑖𝜆 𝑈2 𝜙, 𝜆 = 1 1 −𝑒 𝑖 𝜆 2 𝑒𝑖𝜙 𝑒𝑖(𝜙+𝜆) 𝑈3(𝜃, 𝜙, 𝜆) = 𝜃 2 cos( ) 𝜃 2 −sin( ) 𝑒 𝑖 𝜆 𝜃 2 sin( ) 𝑒 𝑖𝜙 𝜃 44 2 cos( ) 𝑒 𝑖(𝜙+𝜆 )  IBM group
Expectation Value of U < 𝑈 >≡< 𝜑|𝑈|𝜑 > Example : < 0 𝑍 0 >= 1 0 1 0 1 0 −1 0 = 1 < 1 𝑍 1 >= 0 1 1 0 0 0 −1 1 = −1 𝜑 > = 𝑎 0 > +𝑏|1 > < 𝜑 𝑍 𝜑 > = 𝑎̅ < 0 +𝑏̅ < 1 𝑍 𝑎 0 > +𝑏 1 > = 𝑎 2 < 0 𝑍 0 > + 𝑏 2 < 1 𝑍 1 >
O P E N Q U I R K . H T M L Exercise with 1 QuBit
Multi-Qubits |𝑎 > = |𝑏 > = 𝑎1 𝑎2 𝑏 1 𝑏 2 TENSOR PRODUCT 𝑎1 𝑎2 ⨂ 𝑏1 𝑏 2 = � � 1 � � 2 𝑏1 𝑏2 𝑏 1 𝑏2 = 𝑎1𝑏 1 𝑎1𝑏 2 𝑎2𝑏 1 𝑎2𝑏 2 47
Multi-Qubits |0 > ⨂|1 >= | 01> = = | 1> |0 > ⨂|0 >= | 00> = 1 0 0 0 = | 0> |1 > ⨂|0 >= | 10> = = | 2> |1 > ⨂|1 >= | 11> = 0 0 1 0 0 1 0 0 0 0 0 1 = | 3> TENSOR PRODUCT |0 > = 1 0 |1 > = 0 1 48
Superposition Multi-Qubits For N QuBits: For 2 QuBits: |𝝍 > = 𝜶 𝟎𝟎 > +𝜷 𝟎𝟏 > +𝜸 𝟏𝟎 > +𝜹 𝟏𝟏 > |𝝍 > = 𝜶 𝟎 > +𝜷 𝟏 > +𝜸 𝟐 > +𝜹 𝟑 > |𝝍 > = 𝟐𝑵−𝟏 ∑ 𝝀𝒊|𝒊 > 𝒊=𝟎 Pay Attention. You can map classical information to: • |𝑖 >, example Shor′s algorithm and/or • 𝜆𝑖, example HHL algorithm 49
Entanglement Multi-Qubits 13/09/201 7 50 When you cannot write a state as a product of single states |𝜱± > = 𝟏 � � 𝟎𝟎 > ± 𝟏𝟏 > ≠ (𝜶 |𝟎 > +𝜷|𝟏 >) ⨂ (𝜸 𝟎 > + 𝜹 𝟏 > ) AND NOW, YOU HAVE WONDERFUL THINGS AS TELEPORTATION!
Multi-Qubit Transformations 51 1 𝐿𝑒𝑡 𝑈 = 𝑈1 1 𝑈2 1 𝑈1 2 𝑈2 1 on qubit 1 2 𝐿𝑒𝑡 𝑉 = 𝑉1 1 𝑉2 1 𝑉1 2 𝑉2 1 on qubit 2 𝑈1 ⊗ 𝑉2 = � � 11 � � 12 𝑈2 1 𝑉1 1 𝑉2 1 𝑉1 1 𝑉2 1 𝑉1 2 𝑉2 1 𝑉1 2 𝑉2 1 𝑈2 2 𝑉1 1 𝑉2 1 𝑉1 1 𝑉2 1 𝑉1 2 𝑉2 1 𝑉1 2 𝑉2 1 𝑈1 ⊗ 𝑉2 = 𝑈11𝑉11 𝑈11𝑉12 𝑈12𝑉11 𝑈12𝑉12 𝑈11𝑉21 𝑈11𝑉22 𝑈12𝑉21 𝑈12𝑉22 𝑈21𝑉1 1 𝑈21𝑉12 𝑈22𝑉11 𝑈22𝑉12 𝑈21𝑉2 1 𝑈21𝑉22 𝑈22𝑉21 𝑈22𝑉22
Multi-Qubit Transformations 13/09/201 7 52 𝐼 ⊗ 𝑋 = 1 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 𝐼 ⊗ 𝑋 = 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 Example: Apply X gate on second qubii. Let first qubit unchanged
Controled Gates 𝐶𝑁𝑂𝑇 = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 Apply one gate on one qubit, depending on the values of other qubits 𝐶𝑁𝑂𝑇 = 0 >< 0 ⊗ 𝐼 + 1 >< 1 ⊗ 𝑋 𝐶𝑁𝑂𝑇|00 > = | 00 > 𝐶𝑁𝑂𝑇|01 > = | 01 > 𝐶𝑁𝑂𝑇|10 > = | 11 > 𝐶𝑁𝑂𝑇|11 > = | 10 > 53
Measurement Classical Bit QuBi t 54
Quantum Circuit Widt h 13/09/201 7 55 Dept h
O P E N Q U I R K . H T M L MY FIRST QUANTUM PROGRAM: Superdense Coding
My First Quantum Program  Using Quirk. Launch quirk.html. QUIRK does not need measurement. Remember to add it in your real circuit.  Apply a Hadamard Gate (H) on the first qubit  Apply a second H to the same qubit. Result?  Remove Second H and apply a CNOT on a second qubit.  Result: an entangled system (Bell’s) 13/09/201 7 57
Superdense Coding 58  Transmit two classical bits with a single qubit A. Bobs generates a Bell’s state B. Bob sends one qubit to Alice. Bob keeps the second. C. Alice applies a single-qubit gate to her qubit to encode 2 bits: • 01 -> X • 10 -> Z • 11 -> Y • 00 -> I D. Alice returns her qubit to Bob. E. Bob uncomputes entanglement (applies the gates in reverse order) F. Bob measures both qubits.
Superdense Coding | 00> 59 | 01> | 10> | 11> q0 q1 q0 q1 q0 q1 q0 q1
Caution!!! 60
C O N N E C T T O : H T T P S : / / Q U A N T U M - C O M P U T I N G . I B M . C O M / Exercise 2: IBM Quantum Experience
Quantum Volume  Width: The number of physical qubits;  Depth: The number of gates that can be applied before errors make the device behave essentially classically;  Topology: The connectivity of the device;  Gate Parallelism: The number of operations that can be run in parallel 62
TOPOLOGY https://medium.com/rigetti/the-rigetti-128-qubit- chip- and-what-it-means-for-quantum-df757d1b71ea https://github.com/Qiskit/ibmq-device- information/blob/master/backends/melbo urne
Quantum Volume  Effective error rate εeff: specifying how well a device can implement arbitrary pairwise interactions between qubits  n is the number of qubits of the Computer  n’ number of qubits used by the algorithm  Depth 𝑑 ≃ 1 𝑛 𝗌𝑒𝑓𝑓  Quantum Volume � � 𝑛′<𝑛 𝑉 = max 𝑚𝑖𝑛𝑛 ′, 1 𝑛′ 𝗌𝑒𝑓𝑓(𝑛 ′) 2 Source: IBM, 2019 64
Classical Resources 65 1 qubit 2 qubits 3 qubits N qubits |0> |00> = |0> |000> = |0> |0….0> = |0> |1> |01> = |1> |001> = |1> |0….1> = |1> |10> = |2> |010> = |2> |11> = |3> |011> = |3> |100> = |4> |101> = |5> |110> = |6> |111> = |7> |1…..1> = |2N-1> 2 4 8 𝟐𝑵 𝜶 |𝟎 > + 𝜷 |𝟏 > 𝟑 ∑ 𝜶𝒏|𝒏 > 𝒏=𝟎 𝟕 ∑ 𝜶𝒏|𝒏 > 𝒏=𝟎 𝟐𝑵 −𝟏 ∑ 𝜶𝒏|𝒏 > 𝒏=𝟎 2 * complex= 2x2x8=32 bytes 4*2*8=64 bytes 8*2*8=128 bytes 𝟐𝑵 ∗ 𝟐𝟒 = 𝟐𝑵+𝟒
How many states? Asimov calculated the number of nucleons+electrons in the Universe as ∼1079 ∼ 10.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.0 00.000.000.000.000 Having a QPU with 270 qubits, one can store in the amplitudes: ∼1081 FPs. Year 2025: ∼170ZB/year ∼ 1023 bytes/year 75 qubits: ∼ 3·1023 FPs = ∼ 24 years!!!!
Classical Resources qubits RAM 1 32 bytes + memory for gates 2 64 bytes + memory for gates 3 128 bytes + memory for gates 4 256 bytes + memory for gates 8 4 kbytes + memory for gates 16 1 Mbytes + memory for gates 32 64 Gbytes + memory for gates 36 1TB + ….. 38 4TB (Limit CESGA FT2 FAT node ….) 45 0,5PB [1] 64 512 ExaBytes!!! THIS IS ONLY TRUE IF YOU NEED ALL POSSIBLE STATES! [1] Häner, T., & Steiger, D. S. (2017). 0.5 Petabyte Simulation of a 45-Qubit Quantum Circuit. Arxiv:1704.01127 67
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quantum computing presentation for professionals

  • 1.
  • 2.
    Schedule 2  Lecture 1:Introduction to Quantum Computing.  My First Quantum Program.  Lecture 2: Programming Quantum Algorithms  My first Quantum Program with ProjectQ  Lecture 3: Basic Quantum algorithms  Lecture 4: Advanced algorithms
  • 3.
    Lecture 1  Abrief history of QC and needs. Types of quantum computers.  Basic concepts: qubit, tensors, multiqubit, quantum gates, measurement, amplitudes  My first quantum program.  Quantum Circuits. Width, Depth, Quantum Volume.
  • 4.
    Welcome to aDream! Yuri Manin (1980) and Richard Feynman (1981) proposed independently the concept of Quantum Computer I’m here very “hot”!! -273ºC 4 Source: IBM https://en.wikipedia.org/wiki/Timeline_of_quantum_computing
  • 5.
    Welcome to aDream! Rigetti Intel Google D-Wave Qilimanjaro (Spain) And more in Europe, China, Australia, etc…… 5
  • 6.
    Welcome to (my)Nightmare! (*) |𝜱± > = � � ( 𝟎𝟎 >± 𝟏𝟏 >) |𝜳± > = � � 𝟏 � � ( 𝟎𝟏 > ± 𝟏𝟎 >) Bell States Superposition and Entanglement 6 (*) When I was a student long time ago!
  • 7.
  • 8.
  • 11.
    Quantum Computer  Quantumsimulator [1]. Simulate a quantum system using another one, maybe simpler, that can be controlled by the experimenter.  Adiabatic Quantum Computer [2]. Prepares a known and easy Hamiltonian and lets it evolve to solution.  Topological Quantum Computer[4]. Uses topological properties.  Continuous Variable Quantum Computer [5].  Universal Quantum Computer [3]. 1Reviewed in Georgescu, I. M., Ashhab, S., & Nori, F. (2014). Quantum simulation. Reviews of Modern Physics, 86(1), 153– 185. http://doi.org/10.1103/RevModPhys.86.153 arXiv:1308.6253 2 Reviewed in Albash, T., & Lidar, D. A. (2016). Adiabatic Quantum Computing. arxiv:1611.04471 3Proposed in Deutsch, D. (1985). http://doi.org/10.1098/rspa.1985.0070 and Deutsch, D. (1989). http://doi.org/10.1098/rspa.1989.0099 4 Lahtinen V., Pachos J.K.. SciPost Phys. 3, 021 (2017) arXiv:1705.04103 5 Lloyd S. & Braunstein, A.L. Phys.Rev.Lett. 82 (1999) 1784-1787. arXiv:quant-ph/9810082 11
  • 12.
    Adiabatic Quantum Computer HB =Initial Hamiltonian, which ground state is easy to find HP = Problem Hamiltonian, whose ground state encodes the solution to the problem H(s) = Combined Hamiltonial to evolve slowly: A(s) decrease smoothly and monotonically B(s) increase smothly and monotonically H(s) = A(s)HB + B(s)HP Li, R. Y., Felice, R. Di, Rohs, R., & Lidar, D. A. (2018). Quantum annealing versus classical machine learning applied to a simplified computational biology problem. Npj Quantum Information 2018 4:1, 4(1), 14. http://doi.org/10.1038/s41534-018-0060-8 12
  • 13.
    A real example:Traffic Flow Optimisation Neukart, F., Dollen, D. Von, Compostella, G., Seidel, C., Yarkoni, S., & Parney, B. (2017). Traffic flow optimization using a quantum annealer. arXiv:1708.01625v2 D-Wave Adiabatic Computer Optimisation Classical Computer + QPU Unoptimised Optimised Preprocess Map&GPS Find congestions Find Alternatives Minimisation model Solve model Redistribute cars Classical Computer QPU 13
  • 14.
  • 15.
    European Quantum Flagship  AQTION: Trapped Ions  OpenSuperQ : Superconducting  SQUARE: Scalable Rare Earth Ion Quantum Computing Nodes  MicroQC: Microwave driven ion trap quantum computing http://qt.eu 15
  • 16.
  • 17.
    Google Quantum “Supremacy” AruteF, Arya K, Babbush R, Bacon D, Bardin JC, Barends R, et al. Quantum supremacy using a programmable superconducting processor. Nature. 2019;574:505.
  • 18.
  • 19.
    Lecture 1  Abrief history of QC and needs. Types of quantum computers.  Basic concepts: qubit, tensors, multiqubit, quantum gates, measurement, amplitudes  My first quantum program.  Quantum Circuits. Width, Depth, Quantum Volume.
  • 20.
    1. A scalablephysical system with well characterized qubits. 2. The ability to initialize the state of the qubits to a simple fiducial state, such as |000….000> 3. Long relevant decoherence times, much longer than the gate operation time. 4. A “universal” set of quantum gates. 5. A qubit-specific measurement capability. 20 DiVincenzo’s Criteria D. DiVincenzo (2000). “The Physical Implementation of Quantum Computation“, arXiv:quant-ph/0002077
  • 21.
    What do youneed (today)? 21 Complex numbers Matrix multiplication Understand TENSOR products Understand measurement and probabilities Imagination
  • 22.
    BIT, QUBIT AND SUPERPOSITION BIT:A “classical” physical system with TWO states 0 OR 1 What 0 or 1 means is a convention Information is codified as a list of BITs BIT can be transformed from 0 to 1 and vice versa BITs can be operated with logical gates (OR,XOR,AND…) One BIT can be Classical Computer Business Card 22
  • 23.
    BIT, QUBIT AND SUPERPOSITION QuantumComputer Business Card QuBIT: A “Quantum” physical system which yields one of TWO states when is measured 0 OR 1 What 0 or 1 means is a convention* Information is codified in several ways QuBIT can be transformed from 0 to 1 and vice versa QuBITs can be operated with UNITARY gates QuBITs cannot be cloned (no-clone theorem) QuBITs cannot be stored (yet) QuBITs cannot have a long life (yet) Usually, QuBITs are quiet 23
  • 24.
    Quantum Technologies Krantz P,Kjaergaard M, Yan F, Orlando TP, Gustavsson S, Oliver WD. A Quantum Engineer’s Guide to Superconducting Qubits. Arxiv: 1904.06560
  • 25.
    Our current nightmare! NOISE NoisyIntermediate-Scale Quantum” (NISQ) https://medium.com/@pchojecki/quantum-advantage-b345864 6bd9
  • 26.
  • 27.
  • 28.
  • 29.
  • 30.
    Parametric Quantum Circuit Learnin g N=3, D=3 IBMQ_VIGO Nov.17th, 2019 Source: IBM© Nov. 24th, 2019
  • 31.
  • 32.
    Algorithms with shallow circuit s • QVE:Quantum Variational Eigensolver: https://arxiv.org/abs/1304.3061 • QAOA: Quantum Approximate Optimization Algorithm. http://arxiv.org/abs/1411.4028 • Variational Quantum Factoring: https://arxiv.org/abs/1808.08927 • Quantum Machine Learning: • Quantum Support Vector Machine • Quantum Principal Component Analysis • Quantum Variational Autoencoder, • Etc.
  • 33.
    Quantum Variational Eigensolver Figure source:Wang, D., Higgott, O., & Brierley, S. (n.d.). A Generalised Variational Quantum Eigensolver. QPU CPU
  • 34.
    www.inl.int “Despite a numberof promising results, the theoretical evidence presented in the current literature does not yet allow us to conclude that quantum techniques can obtain an exponential advantage in a realistic learning setting” Ciliberto et.al. “Quantum machine learning: a classical perspective” http://dx.doi.org/10.1098/rspa.2017.0551 34 Quantum Machine Learning ? So: A lot of research to do!!!
  • 35.
    www.inl.int Complex Numbers If 𝑖2= −1, a complex number is defined by: c=𝑎 + 𝑏 ∗ 𝑖, with 𝑎, 𝑏 ∈ ℝ, 𝑐 ∈ ℂ Complex conjugate: 𝑐̅ = 𝑎 − 𝑏 ∗ 𝑖 Modulus: 𝑐 2 = 𝑐𝑐̅ = (𝑎 + 𝑏 ∗ 𝑖)(𝑎 − 𝑏 ∗ 𝑖) = 𝑎2 + 𝑏2 Polar form: c = 𝑐 𝑐𝑜𝑠𝜃 + 𝑐 𝑠𝑖𝑛𝜃 𝑖 = 𝑐 𝑒𝑖𝜃 35
  • 36.
    QUBIT |0 > = |1 > = 1Superposition 0 0 1 |𝜙 > = 𝛼 0 > + 𝛽 1 > Complex numbers 𝛼 2 𝛽 2 + = 1 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝛼 = 𝛼 𝑒𝑖𝜑 𝛼 2 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝜑 Phase 36
  • 37.
    QUBIT |𝜙 > =𝛼 0 > + 𝛽 1 >= � � � � Measurement of |𝜙 > in standard basis (|0>, | 1>) : |0> with probability 𝛼 2. State after measurement | 0> or |1> with probability 𝛽 2. State after measurement |1> < 𝜙 = 𝛼̅ < 0 + 𝛽̅ < 1| = 𝛼̅ 𝛽̅ < 𝜙|𝜙 > = 𝛼 2 + 𝛽 2 |𝜓 > = 𝛾 0 > + 𝛿 1 >= � � � � 37 < 𝜓|𝜙 > =𝛼 𝛾̅ + 𝛽 𝛿
  • 38.
    Bloch’s Sphere � � 2 � � 2 |𝜓 > =cos( )|0 > + sin( ) 𝑒 𝑖 𝜑 |1 > 2 𝜋 |0 > −|1 > 𝜃 = , 𝜑 = 𝜋, |− > = 2 𝜃 = � � 2 , 𝜑 = � � 2 , |𝑖 > = |0 > +𝑖|1 > 𝜃 = � � 2 , 𝜑 = 3 𝜋 2 , | − 𝑖 > = 2 |0 > −𝑖|1 > 2 𝜋 𝜃 = 2 , 𝜑 = 0, | +> = |0 > + |1 > 2 y x |𝜓 > � � z |𝟎 > 𝜑 | +> | −> | − 𝒊 > |𝒊 > 38 |𝟏 > Hint: 𝑒𝑖𝜑 = cos(𝜑)+i sin(𝜑)
  • 39.
    One-Qubit Transformations |𝜙 >= 𝛼 0 > + 𝛽 1 >= � � � �  Transform vector space in itself |𝜙′ > = U |𝜙 > = 𝑎11 𝑎21 𝑎12 𝑎22 � � � �  Unit lengh vectors must go to unit lenght vectors: < 𝜓 𝑈†𝑈 𝜙 > =< 𝜓|𝜙 >⇒ 𝑈†𝑈=I  Reversible  Geometrically, they are rotations of the complex vector space associated to |𝜙 > 𝑁𝑜𝑡𝑒: 𝑈† = 𝑈𝑇 39
  • 40.
    One-Qubit Transformations  Phaseshift 𝐾(𝛿) = 𝑒𝑖𝛿𝐼  Rotation, 𝑅(𝛽) = cos(𝛽) −sin(𝛽 ) sin(𝛽 ) 𝑐𝑜𝑠( 𝛽)  Phase rotation, 𝑇(𝛼) = 𝑒𝑖 𝛼 0 0 𝑒−𝑖𝛼  Any other Qubit unitary transformation can be written as: 𝐾 𝛿 𝑇 𝛼 𝑅 𝛽 𝑇 𝛾 = 𝑒𝑖(𝛿+𝛼+𝛾) cos 𝛽 − 𝑒𝑖(𝛿−𝛼+𝛾)sin 𝛽 𝑒𝑖(𝛿+𝛼−𝛾) sin 𝛽 𝑒𝑖(𝛿−𝛼−𝛾) cos 𝛽 40 Source: Eleanor G. Rieffel.Quantum Computing: A Gentle Introduction
  • 41.
    One-Qubit Transformations  Phaseshift 𝐾(𝛿) = 𝑒𝑖𝛿𝐼 � �  Rotation around 𝑥, 𝑅 𝜃 ≡ 𝑒−𝑖𝜃𝑋 = co s � � 2 −𝑖 sin � � 2 −𝑖 sin 𝜃 2 co s 𝜃 2 = cos 𝜃 2 𝐼 − 𝑖 𝑠𝑖𝑛 𝜃 2 � � � �  Rotation around 𝑦, 𝑅 𝜃 ≡ 𝑒−𝑖𝜃𝑌 = 𝜃 cos( ) 𝜃 −sin( ) 2 𝜃 sin( ) 2 𝜃 2 2 𝑐𝑜𝑠( ) � �  Phase rotation, Rotation around 𝑧, 𝑅 𝜃 ≡ 𝑒−𝑖𝜃𝑍 = 𝑒−𝑖𝜃/2 0 0 𝑒𝑖𝜃/2  Any other QuBit unitary transformation can be written as: 𝑈 = 𝐾(𝛿)𝑅𝑧 𝛾 𝑅𝑦 𝛽 𝑅𝑧 𝛼 42 Source: Nielsen & Chuang, Quantum Computation And Quantum Information
  • 42.
    One-Qubit Transformations  Pauli Gates1 , bit-flip or NOT.  𝑋 =  Y =  Z = 0 1 0 0 −𝑖 𝑖 0 1 0 0 −1  Cliffort group  𝐻𝑎𝑑𝑎𝑚𝑎𝑟𝑑, 𝐻 = 1 2 1 1 1 −1  𝑃ℎ𝑎𝑠𝑒, 𝑆 = 1 0 0 𝑖 � �  , 𝑇 = 8 1 0 0 𝑖 𝜋 𝑒 4 Hint : † 𝑈 = 𝑈 � � 𝑈1 𝜆 = 1 0 0 𝑒𝑖𝜆 𝑈2 𝜙, 𝜆 = 1 1 −𝑒 𝑖 𝜆 2 𝑒𝑖𝜙 𝑒𝑖(𝜙+𝜆) 𝑈3(𝜃, 𝜙, 𝜆) = 𝜃 2 cos( ) 𝜃 2 −sin( ) 𝑒 𝑖 𝜆 𝜃 2 sin( ) 𝑒 𝑖𝜙 𝜃 44 2 cos( ) 𝑒 𝑖(𝜙+𝜆 )  IBM group
  • 43.
    Expectation Value ofU < 𝑈 >≡< 𝜑|𝑈|𝜑 > Example : < 0 𝑍 0 >= 1 0 1 0 1 0 −1 0 = 1 < 1 𝑍 1 >= 0 1 1 0 0 0 −1 1 = −1 𝜑 > = 𝑎 0 > +𝑏|1 > < 𝜑 𝑍 𝜑 > = 𝑎̅ < 0 +𝑏̅ < 1 𝑍 𝑎 0 > +𝑏 1 > = 𝑎 2 < 0 𝑍 0 > + 𝑏 2 < 1 𝑍 1 >
  • 44.
    O P EN Q U I R K . H T M L Exercise with 1 QuBit
  • 45.
    Multi-Qubits |𝑎 > = |𝑏 > = 𝑎1 𝑎2 𝑏 1 𝑏 2 TENSORPRODUCT 𝑎1 𝑎2 ⨂ 𝑏1 𝑏 2 = � � 1 � � 2 𝑏1 𝑏2 𝑏 1 𝑏2 = 𝑎1𝑏 1 𝑎1𝑏 2 𝑎2𝑏 1 𝑎2𝑏 2 47
  • 46.
    Multi-Qubits |0 > ⨂|1>= | 01> = = | 1> |0 > ⨂|0 >= | 00> = 1 0 0 0 = | 0> |1 > ⨂|0 >= | 10> = = | 2> |1 > ⨂|1 >= | 11> = 0 0 1 0 0 1 0 0 0 0 0 1 = | 3> TENSOR PRODUCT |0 > = 1 0 |1 > = 0 1 48
  • 47.
    Superposition Multi-Qubits For N QuBits: For2 QuBits: |𝝍 > = 𝜶 𝟎𝟎 > +𝜷 𝟎𝟏 > +𝜸 𝟏𝟎 > +𝜹 𝟏𝟏 > |𝝍 > = 𝜶 𝟎 > +𝜷 𝟏 > +𝜸 𝟐 > +𝜹 𝟑 > |𝝍 > = 𝟐𝑵−𝟏 ∑ 𝝀𝒊|𝒊 > 𝒊=𝟎 Pay Attention. You can map classical information to: • |𝑖 >, example Shor′s algorithm and/or • 𝜆𝑖, example HHL algorithm 49
  • 48.
    Entanglement Multi-Qubits 13/09/201 7 50 When youcannot write a state as a product of single states |𝜱± > = 𝟏 � � 𝟎𝟎 > ± 𝟏𝟏 > ≠ (𝜶 |𝟎 > +𝜷|𝟏 >) ⨂ (𝜸 𝟎 > + 𝜹 𝟏 > ) AND NOW, YOU HAVE WONDERFUL THINGS AS TELEPORTATION!
  • 49.
    Multi-Qubit Transformations 51 1 𝐿𝑒𝑡 𝑈 = 𝑈1 1 𝑈2 1 𝑈1 2 𝑈2 1 on qubit 1 2 𝐿𝑒𝑡𝑉 = 𝑉1 1 𝑉2 1 𝑉1 2 𝑉2 1 on qubit 2 𝑈1 ⊗ 𝑉2 = � � 11 � � 12 𝑈2 1 𝑉1 1 𝑉2 1 𝑉1 1 𝑉2 1 𝑉1 2 𝑉2 1 𝑉1 2 𝑉2 1 𝑈2 2 𝑉1 1 𝑉2 1 𝑉1 1 𝑉2 1 𝑉1 2 𝑉2 1 𝑉1 2 𝑉2 1 𝑈1 ⊗ 𝑉2 = 𝑈11𝑉11 𝑈11𝑉12 𝑈12𝑉11 𝑈12𝑉12 𝑈11𝑉21 𝑈11𝑉22 𝑈12𝑉21 𝑈12𝑉22 𝑈21𝑉1 1 𝑈21𝑉12 𝑈22𝑉11 𝑈22𝑉12 𝑈21𝑉2 1 𝑈21𝑉22 𝑈22𝑉21 𝑈22𝑉22
  • 50.
    Multi-Qubit Transformations 13/09/201 7 52 𝐼 ⊗ 𝑋 = 10 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 𝐼 ⊗ 𝑋 = 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 Example: Apply X gate on second qubii. Let first qubit unchanged
  • 51.
    Controled Gates 𝐶𝑁𝑂𝑇 = 1 00 0 0 1 0 0 0 0 0 1 0 0 1 0 Apply one gate on one qubit, depending on the values of other qubits 𝐶𝑁𝑂𝑇 = 0 >< 0 ⊗ 𝐼 + 1 >< 1 ⊗ 𝑋 𝐶𝑁𝑂𝑇|00 > = | 00 > 𝐶𝑁𝑂𝑇|01 > = | 01 > 𝐶𝑁𝑂𝑇|10 > = | 11 > 𝐶𝑁𝑂𝑇|11 > = | 10 > 53
  • 52.
  • 53.
  • 54.
    O P EN Q U I R K . H T M L MY FIRST QUANTUM PROGRAM: Superdense Coding
  • 55.
    My First QuantumProgram  Using Quirk. Launch quirk.html. QUIRK does not need measurement. Remember to add it in your real circuit.  Apply a Hadamard Gate (H) on the first qubit  Apply a second H to the same qubit. Result?  Remove Second H and apply a CNOT on a second qubit.  Result: an entangled system (Bell’s) 13/09/201 7 57
  • 56.
    Superdense Coding 58  Transmittwo classical bits with a single qubit A. Bobs generates a Bell’s state B. Bob sends one qubit to Alice. Bob keeps the second. C. Alice applies a single-qubit gate to her qubit to encode 2 bits: • 01 -> X • 10 -> Z • 11 -> Y • 00 -> I D. Alice returns her qubit to Bob. E. Bob uncomputes entanglement (applies the gates in reverse order) F. Bob measures both qubits.
  • 57.
  • 58.
  • 59.
    C O NN E C T T O : H T T P S : / / Q U A N T U M - C O M P U T I N G . I B M . C O M / Exercise 2: IBM Quantum Experience
  • 60.
    Quantum Volume  Width:The number of physical qubits;  Depth: The number of gates that can be applied before errors make the device behave essentially classically;  Topology: The connectivity of the device;  Gate Parallelism: The number of operations that can be run in parallel 62
  • 61.
  • 62.
    Quantum Volume  Effectiveerror rate εeff: specifying how well a device can implement arbitrary pairwise interactions between qubits  n is the number of qubits of the Computer  n’ number of qubits used by the algorithm  Depth 𝑑 ≃ 1 𝑛 𝗌𝑒𝑓𝑓  Quantum Volume � � 𝑛′<𝑛 𝑉 = max 𝑚𝑖𝑛𝑛 ′, 1 𝑛′ 𝗌𝑒𝑓𝑓(𝑛 ′) 2 Source: IBM, 2019 64
  • 63.
    Classical Resources 65 1 qubit 2qubits 3 qubits N qubits |0> |00> = |0> |000> = |0> |0….0> = |0> |1> |01> = |1> |001> = |1> |0….1> = |1> |10> = |2> |010> = |2> |11> = |3> |011> = |3> |100> = |4> |101> = |5> |110> = |6> |111> = |7> |1…..1> = |2N-1> 2 4 8 𝟐𝑵 𝜶 |𝟎 > + 𝜷 |𝟏 > 𝟑 ∑ 𝜶𝒏|𝒏 > 𝒏=𝟎 𝟕 ∑ 𝜶𝒏|𝒏 > 𝒏=𝟎 𝟐𝑵 −𝟏 ∑ 𝜶𝒏|𝒏 > 𝒏=𝟎 2 * complex= 2x2x8=32 bytes 4*2*8=64 bytes 8*2*8=128 bytes 𝟐𝑵 ∗ 𝟐𝟒 = 𝟐𝑵+𝟒
  • 64.
    How many states? Asimovcalculated the number of nucleons+electrons in the Universe as ∼1079 ∼ 10.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.0 00.000.000.000.000 Having a QPU with 270 qubits, one can store in the amplitudes: ∼1081 FPs. Year 2025: ∼170ZB/year ∼ 1023 bytes/year 75 qubits: ∼ 3·1023 FPs = ∼ 24 years!!!!
  • 65.
    Classical Resources qubits RAM 1 32bytes + memory for gates 2 64 bytes + memory for gates 3 128 bytes + memory for gates 4 256 bytes + memory for gates 8 4 kbytes + memory for gates 16 1 Mbytes + memory for gates 32 64 Gbytes + memory for gates 36 1TB + ….. 38 4TB (Limit CESGA FT2 FAT node ….) 45 0,5PB [1] 64 512 ExaBytes!!! THIS IS ONLY TRUE IF YOU NEED ALL POSSIBLE STATES! [1] Häner, T., & Steiger, D. S. (2017). 0.5 Petabyte Simulation of a 45-Qubit Quantum Circuit. Arxiv:1704.01127 67