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Fundamentals of Quantum Computing
- 1. Fundamentals of Quantum Computing Prof.Amlan Chakrabarti Dean Faculty of Engineering and Technology Director, A.K.Choudhury School of Information Technology University of Calcutta email: acakcs@caluniv.ac.in Quantum Computing Knowledge Workshop, 12th October 2018: Dept. of IT&E, Govt. of W.B.
- 2. Realizations are gettingsmaller (and faster) and reaching a point where “classical” physics is no longer a sufficient model for the laws of physics 2
- 3. 3 MooreMoore’s Law’s Law Courtesy:QuantumComputing and Communications- An Engineering approach: Sandor Imre and Ferenc Balazs Moore’s Law: the amount of information storable on a given amount of silicon has roughly doubled every 18 months. At ~ 10nm. scale quantum effects will upset the classical progression of Moore’s law..
- 4. 4 The Need ofthe ChangeThe Need of the Change • The closer we are to the few-electron transistor disturbingThe closer we are to the few-electron transistor disturbing quantum effects will appear more often and strongerquantum effects will appear more often and stronger • Either we manage to find a new way of miniaturization or weEither we manage to find a new way of miniaturization or we have to learn how to exploit the difficulties and strangeness ofhave to learn how to exploit the difficulties and strangeness of quantum mechanicsquantum mechanics • FeynmanFeynman’s Suggestion:’s Suggestion: – Instead of regarding computers as devices working under theInstead of regarding computers as devices working under the laws of classical physics which is common sense , let uslaws of classical physics which is common sense , let us consider their operation as a special case of a more generalconsider their operation as a special case of a more general theory governed by quantum mechanics.theory governed by quantum mechanics.
- 5. 5 Requirement of NewRequirementof New Computational strategiesComputational strategies • The way becomes open for the development of quantum hardware • We have to study quantum mechanics from a computer science point of view • These software-related efforts are comprehended by quantum computing • Top experts have experimentally validated quantum computing algorithms which overcome the classical competitors • Once we familiarized ourselves with quantum-faced computing why keep communications away from the new chances
- 6. 6 WhyWhy Quantum Computing?QuantumComputing? • It is,It is, apparentlyapparently, exponentially more time-, exponentially more time- efficient thanefficient than any possibleany possible classical computingclassical computing scheme at solvingscheme at solving somesome problems:problems: – Factoring, discrete logarithms, related problemsFactoring, discrete logarithms, related problems – Simulating quantum physical systems accuratelySimulating quantum physical systems accurately • This application was the original motivation for quantumThis application was the original motivation for quantum computing research first suggested by famous physicistcomputing research first suggested by famous physicist Richard Feynman in the early 80Richard Feynman in the early 80’s’s
- 7. Quantum Computation • Acomputation model based on quantum principles of physics • Ability to enter many parallel “states” and use interference to recover important information • Transformations must be unitary
- 8. Why Is ThisHelpful? • Multiple computations simultaneously • Computing power is exponential
- 9. Quantum Algorithms 9 BQP (Bounded-ErrorQuantum Polynomial-Time): Class of problems solvable efficiently by a quantum computer
- 10. Quantum Computing CanPerform Better
- 11. Quantum Computing: ThrustAreas ● Quantum Technology ● Quantum Algorithms ● Quantum Modelling and Simulation ● Quantum Communication and Cryptography 11
- 12. 12 Status of QuantumComputingStatus of Quantum Computing • Computing giants Google and Microsoft recently hired a host of leading lights, and have set challenging goals in QC (Nature news 2017) • D-Wave 2000Q – the firm’s first 2,000 quantum bit (qubit) quantum computer came to existence in 2017 – The quantum 2000Q is capable of outperforming “classical servers” by factors of between 1,000 and 10,000 – IBM established a landmark in computing Friday, announcing a quantum computer that handles 50 quantum bits, or qubits – The company is also making a 20-qubit system available through its cloud computing platform – Scaling up to hundreds or thousands of quantum bits becomes a possibility • Quantum Communication – Teleportation, Entanglement and Zeno Effect has been successfully implemented and tested • Quantum Cryptography – Quantum Key Exchange Protocols, BB84 etc. • Quantum Machine Learning – Quantum Clustering, Quantum Random Walk Algorithms for Heuristic Search
- 13. • Qubit vs.Bit: Bit (Classical) degree of freedom that can take only two possible values. – Qubit • Quantum observable whose spectrum contains two values {0,1}. • Minimal quantum physical system. • The boolean observable of a qbit system is called a sharp observable , as it can have only values 0 and 1. • A qubit can have another observable which has an equal probability of 1 and 0, individual probabilities summed will results to unity. • Qubit in reality: – Electron spin (up or down) – Photon polarization (horizontal/vertical) – Spin of atomic nucleus – Current in a super conducting loop – Presence/absence of a particle 13
- 14. Quantum Phenomenon: Superposition andEntanglement ● Superposition ○ Superposition is the ability of a quantum system to be in multiple states at the same time. ● Entanglement ○ Multiple particles are associated in such a way that measurement of one quantum state of one particle is determined by the measurement of the state of another particle. 14
- 15. Computation with QubitsComputationwith Qubits How does the use of qubits affect computation? Classical Computation Data unit: bit x = 0 x = 1 0 1 0 1 Valid states: x = ‘0’ or ‘1’ |ψ〉 = c1|0〉 + c2|1〉 Quantum Computation Data unit: qubit Valid states: |ψ〉 = |0〉 |ψ〉 = |1〉 |ψ〉 = (|0〉 + |1〉)/√2 =|1〉 =|0〉= ‘1’ = ‘0’ 15
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- 17. 17 Quantum Logic NetworksQuantumLogic Networks • Invented by Deutsch (1989)Invented by Deutsch (1989) – Analogous to classical Boolean logic networksAnalogous to classical Boolean logic networks – Generalization of Fredkin-Toffoli reversible logic circuitsGeneralization of Fredkin-Toffoli reversible logic circuits • System is divided into individual bits, orSystem is divided into individual bits, or qubitsqubits – 2 orthogonal states of each qubit are designated as the2 orthogonal states of each qubit are designated as the computationalcomputational basis statesbasis states,, “0” and “1”“0” and “1” • Quantum logic gates:Quantum logic gates: – Local unitary transforms that operate on only a few state bitsLocal unitary transforms that operate on only a few state bits at a timeat a time • Computation via predetermined sequence of gate applications toComputation via predetermined sequence of gate applications to selected bitsselected bits
- 18. 18 Quantum Gates: NOTQuantumGates: NOT • All classical input-consuming reversible gates can beAll classical input-consuming reversible gates can be represented as unitary transformations!represented as unitary transformations! • E.g.E.g., input-consuming NOT gate (inverter), input-consuming NOT gate (inverter) in out in out in out 0 1 1 0 1 0 10 ≡ 01 10:N 01 10 = = N N 1 01 1 00 ≡ ≡ 1 0: 0 1:
- 19. 19 Controlled-NOTControlled-NOT • A.k.a. CNOT(or input-consuming XOR)A.k.a. CNOT (or input-consuming XOR) A A’ B B’ = A⊕B A A’ B B’ = A⊕B A B A’ B’ 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 11 10 01 00 11100100 ≡ 0100 1000 0010 0001 :X 1110 =X Example: A B A B Jadavpur University November 28, 2016
- 20. 20 Toffoli Gate (CCNOT)ToffoliGate (CCNOT) A B C A’ B’ C’ 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 (XOR) A B C A’=A B’=B C’ = C⊕AB A B’B C A’ C’ 111 110 101 100 011 010 001 000 111110101100011010001000 ≡ 01000000 10000000 00100000 00010000 00001000 00000100 00000010 00000001 :X Now, what happens if the unitary matrix elements are not always 0 or 1?
- 21. 21 The Hadamard TransformTheHadamard Transform • A randomizingA randomizing “square root of identity” gate.“square root of identity” gate. • Used frequently in quantum logic networks.Used frequently in quantum logic networks. 1 0 10 − ≡ 2 1 2 1 2 1 2 1 :H == 10 01 2 2 IH
- 22. 22 • Quantum LogicCircuitsQuantum Logic Circuits – Circuit behavior is governed explicitly by quantumCircuit behavior is governed explicitly by quantum mechanicsmechanics – Signal states are vectors interpreted as a superposition ofSignal states are vectors interpreted as a superposition of binary “binary “qubitqubit”” vectors with complex-number coefficientsvectors with complex-number coefficients – Operations are defined by linear algebra over Hilbert SpaceOperations are defined by linear algebra over Hilbert Space and can be represented by unitary matrices with complexand can be represented by unitary matrices with complex elementselements – Severe restrictions exist on copying and measuring signalsSevere restrictions exist on copying and measuring signals – Many universal gate sets exist but the best types are notMany universal gate sets exist but the best types are not obviousobvious Ψ = ci in −1in−1…i0 i = 0 2n −1 ∑
- 23. 23 • Unitary OperationsUnitaryOperations – Gates and circuits must be reversible (information-Gates and circuits must be reversible (information- lossless)lossless) • Number of output signal lines = Number of input signal linesNumber of output signal lines = Number of input signal lines • The circuit function must be a bijection, implying that outputThe circuit function must be a bijection, implying that output vectors are a permutation of the input vectorsvectors are a permutation of the input vectors – Classical logic behavior can be represented byClassical logic behavior can be represented by permutation matricespermutation matrices – Non-classical logic behavior can be representedNon-classical logic behavior can be represented including state sign (phase) and entanglementincluding state sign (phase) and entanglement Quantum Circuit CharacteristicsQuantum Circuit Characteristics
- 24. 1/√21/√2 00 1/√21/√2 00 11 00 00 00 N CNOT |0〉 + |1〉 |0〉 ExampleCircuit √2 ______ 1/√21/√2 00 1/√21/√2 00 1/√21/√2 00 00 1/√21/√2 00 00 00 11 |0〉 + |1〉 |0〉 √2 ______ ‘0’ ‘0’ or ‘1’ ‘1’ or 50% 50% Separable state: can be written as tensor product |Ψ〉 = |φ〉 ⊗ |χ〉 Entangled state: cannot be written as tensor product |Ψ〉 ≠ |φ〉 ⊗ |χ〉 ? ? 24
- 25. Some Interesting ConsequencesSomeInteresting Consequences No cloning theorem It is impossible to exactly copy an unknown quantum state |ψ〉 |0〉 |ψ〉 |ψ〉 Reversibility Since quantum mechanics is reversible (dynamics are unitary), quantum computation is reversible. |00000000〉 |ψφβπµψ〉 |00000000〉 U.U† = I 25
- 26. Grover’s Search Algorithm Thebest a classical computer can do on average is N/2 queries. 1 Oracle No ... 2 Oracle No 3 Oracle Yes Classical computer Oracle 1+2+3+... No+No+Yes+No+... Quantum computer Using Grover’s algorithm, a quantum computer can find the answer in √N queries! Superposition over all N possible inputs.
- 27. Quantum Cryptography ● Providesfor secure key exchange over physically unprotected channels w. a guarantee of detection of any eavesdropping of the key ● Physically impossible to compromise security (except @ endpoints) barring overthrow of physics! ● Probably secure under known laws Experimentally verified to work perfectly over >48 km distances (so far) (Hughes ‘99) via fiber-optic networks 27
- 28. 28 Typical Implementation MethodTypicalImplementation Method • AnyAny “flying qubit” will do.“flying qubit” will do. – Most common method uses polarized photons.Most common method uses polarized photons. (Bennett & Brassard(Bennett & Brassard ‘84)‘84) θ Arbitrarychoice ofbasis: “0” “1” ↔+= θθ cossin Diffraction grating w. vertical slits θ + π/4 “0” “1” )cos()sin( 44 ππ θθ +++= Diffraction grating w. diagonal slits
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- 30. Simulating Quantum Computers onclassical ones
- 31. 31 Efficient QC SimulationsEfficientQC Simulations • Task: Simulate anTask: Simulate an nn-qubit quantum computer.-qubit quantum computer. • Maximally stupid approach:Maximally stupid approach: – Store a 2Store a 2nn -element vector-element vector – Multiply it by a full 2Multiply it by a full 2nn ××22nn matrix for each gate opmatrix for each gate op • Some obvious optimizations:Some obvious optimizations: – Never store whole matrix (compute dynamically)Never store whole matrix (compute dynamically) – Store only nonzero elements of state vectorStore only nonzero elements of state vector • Especially helpful when qubits are highly correlatedEspecially helpful when qubits are highly correlated – Do only constant work per nonzero vector elementDo only constant work per nonzero vector element • Scatter amplitude from each state to 1 or 2 successorsScatter amplitude from each state to 1 or 2 successors – Drop small-probability-mass sets of statesDrop small-probability-mass sets of states • Linearity of QM implies no chaotic growth of errorsLinearity of QM implies no chaotic growth of errors
- 32. 32 Simulating Quantum Computations •Given:Given: – AnyAny nn-qubit quantum computation, expressed as a-qubit quantum computation, expressed as a sequence of 1-qubit gates and CNOT gates.sequence of 1-qubit gates and CNOT gates. – An initial stateAn initial state ss00 which is just a basis state in thewhich is just a basis state in the classical bitwise basis,classical bitwise basis, e.g.e.g. ||0000000000〉〉.. • Goal:Goal: – Generate a final basis state stochastically with the sameGenerate a final basis state stochastically with the same probability distribution as the quantum computer wouldprobability distribution as the quantum computer would do.do. U1 U3 U4 U2
- 33. 33 Matrix RepresentationMatrix Representation •Consider each gate as rank-2Consider each gate as rank-2nn unitary matrix:unitary matrix: – Each CNOT application is a 0-1 (permutation)Each CNOT application is a 0-1 (permutation) matrix - a classical reversible bit-operation.matrix - a classical reversible bit-operation. – With appropriate row ordering, eachWith appropriate row ordering, each UUii gategate application is block-diagonal, w. each 2×2 blockapplication is block-diagonal, w. each 2×2 block equal toequal to UUii.. – We need never represent these full matrices!We need never represent these full matrices! – The 1 or 2 nonzero entries in a given row can beThe 1 or 2 nonzero entries in a given row can be located & computed immediately given the row idlocated & computed immediately given the row id (bit string) and(bit string) and UUii..
- 34. New Q CircuitSimulation Models 34
- 35. Physical Implementations of QuantumComputing
- 36. QCADQCAD • Motivation – CircuitSynthesis for Quantum Algorithms – Development of Quantum Module library (Technology Independent) – Physical Machine Description (PMD) specific optimization and cost estimation – PMD specific cell library creation (Technology Dependent) – Considering issues of FT-QC and Error coding • Challenges – Integration of Classical (Reversible) and Quantum Modules – Handling larger circuit size in terms of qubits – Appropriate Cost Estimation – Optimization Issues – Functional Verification
- 37. Quantum Algorithm DescriptionQuantumAlgorithm Description using QCLusing QCL • QCL (Quantum Computation Language) is a high level, architecture independent programming language for quantum computers (Omer, http://tph.tuwien.ac.at/~oemer/qcl.html) • Its syntax is similar to C programming language • Both classical and quantum code can be combined in the same program Quantum Algorithm in QCL Input State Output State Quantum Algorithm Simulation
- 38. QCL to QASMQCLto QASM
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- 40. Automated Generation ofQASMAutomated Generation of QASM for Quantum & Reversiblefor Quantum & Reversible ModulesModules • Quantum & Reversible modulesQuantum & Reversible modules considered at presentconsidered at present – QFT/IQFTQFT/IQFT – Bernstein-Vazirani Search (BVS)Bernstein-Vazirani Search (BVS) – GroverGrover’s search’s search – Arithmetic CircuitsArithmetic Circuits • DraperDraper’s Adder’s Adder • CuccaroCuccaro’s adder’s adder • 4 qubit multiplier4 qubit multiplier • modular adder (a+b)%Nmodular adder (a+b)%N • modular subtractormodular subtractor • Constant modular multiplierConstant modular multiplier • Modular exponentiationModular exponentiation Algorithm for generation of QFT Circuit CC Lin, A. Chakrabarti and N.K.Jha, “QLib: Quantum Module Library”, ACM JETC 2014
- 41. PMD specific Synthesis •Each PMD supports a set of primitive quantum operationsEach PMD supports a set of primitive quantum operations • Quantum gate Library :Quantum gate Library : Rx(θ), Ry(θ), Rz(Rx(θ), Ry(θ), Rz(θ),θ), H, CNOT, CZ, ZENO, SWAP, CP(H, CNOT, CZ, ZENO, SWAP, CP(θ),θ), G(G(θ),θ), iSW (iSW (θ),θ), Toffoli,Toffoli, Fredkin, and PeresFredkin, and Peres • Gate implementations are optimized by identity rules, involving both one qubit andGate implementations are optimized by identity rules, involving both one qubit and two-qubit operationstwo-qubit operations PMD One-qubit operations Two-qbit operations QD Rx, Rz, σx, σz, S, T CZ SC Rx, Ry, Rz, iSWAP, CP IT Rxy, Rz, G NA Rxy CZ LP Rx, Ry, Rz, σx, σy, σz, S, T, H CNOT, CZ, SWAP, ZENO NP Asqu, Rx, Ry, Rz, H CNOT
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- 43. Quantum CompilerQuantum Compiler •SKA and STA can only compile one-qubit gatesSKA and STA can only compile one-qubit gates • Conversion of non-FT two-qubit gates to FT two-qubit gates first and then all theConversion of non-FT two-qubit gates to FT two-qubit gates first and then all the non-FT one-qubit gates to FT cascadesnon-FT one-qubit gates to FT cascades Synthesis flow of non-FT one-qubit gates based on the FT table CC Lin, A. Chakrabarti and N.K.Jha, “FTQLS: Fault-Tolerant Quantum Logic Synthesis”, IEEE TVLSI 2013
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- 46. 46 Research work atSchool of I.T., CUResearch work at School of I.T., CU • The present research activities in the area of quantum computing are as follows : – Quantum Machine Learning – Designing of new quantum circuits for quantum algorithms – New circuit optimization techniques • Template based • Heuristic based – Development of CAD tools for quantum circuit design, optimization and simulations – Quantum Cryptography Multi-valued logic and quantum computing • International Collaborations – Department of Computer Engineering, Princeton University, USA – Dept. of Computer Science & Engineering and Department of Physics, New York State University at Buffalo, USA – Iwate Prefecture University, Japan – University of Bremen, Germany – University Linz, Austria – Nanyang Technological University, Singapore
- 47. Our Publications (SelectedList)Our Publications (Selected List) • K. Regan,K. Regan, A. ChakrabartiA. Chakrabarti, C. Guan, “Algebraic and Logical Emulations of Quantum Circuits”, C. Guan, “Algebraic and Logical Emulations of Quantum Circuits” Springer Trans. ComputationalSpringer Trans. Computational ScienceScience 31: 41-76 (2018).31: 41-76 (2018). • M. GhoshM. Ghosh, A. Chakrabarti, A. Chakrabarti, Niraj K. Jha, “Automated Quantum Circuit Synthesis and Cost Estimation for the Binary Welded Tree, Niraj K. Jha, “Automated Quantum Circuit Synthesis and Cost Estimation for the Binary Welded Tree Oracle”Oracle” ACM Journal on Emerging Technologies in Computing Systems (JETC)ACM Journal on Emerging Technologies in Computing Systems (JETC)13(4): 51:1-51:14 (2017).13(4): 51:1-51:14 (2017). • S. Guha Roy andS. Guha Roy and A. ChakrabartiA. Chakrabarti,, “Novel Graph Clustering Algorithm Based On Discrete Time Quantum Random Walk,” Book“Novel Graph Clustering Algorithm Based On Discrete Time Quantum Random Walk,” Book Titled: Quantum Inspired Computational Intelligence: Research and Applications ,Titled: Quantum Inspired Computational Intelligence: Research and Applications , Morgan KaufmannMorgan Kaufmann 2017.2017. • S. Basu, S. B. Mandal,S. Basu, S. B. Mandal, A. ChakrabartiA. Chakrabarti and Susmita Sur-Kolay, "An Efficient Synthesis Method for Ternary Reversible Logic",and Susmita Sur-Kolay, "An Efficient Synthesis Method for Ternary Reversible Logic", Proc. of IEEE International Symposium on Circuits and Systems 2016 (ISCAS 2016)Proc. of IEEE International Symposium on Circuits and Systems 2016 (ISCAS 2016).. • P.Niemann, S. Basu,P.Niemann, S. Basu, A. ChakrabartiA. Chakrabarti, Niraj K. Jha and Robert Wille, "Synthesis of Quantum Circuits for Dedicated Physical, Niraj K. Jha and Robert Wille, "Synthesis of Quantum Circuits for Dedicated Physical Machine Descriptions,Machine Descriptions,"" Proc.of 7th Conference on Reversible ComputationProc.of 7th Conference on Reversible Computation ((RC 2015RC 2015)).. • S.B.Mondal,S.B.Mondal, A.Chakrabarti,A.Chakrabarti, and S.Sur-Kolay,and S.Sur-Kolay, “Quantum Ternary Circuit Synthesis Using Projection Operations,”“Quantum Ternary Circuit Synthesis Using Projection Operations,” Journal ofJournal of Multiple-Valued Logic and Soft ComputingMultiple-Valued Logic and Soft Computing, Vol 21, Issue 1-4, pp. 73-92, January 2015., Vol 21, Issue 1-4, pp. 73-92, January 2015. • C.C. Lin,C.C. Lin, A. ChakrabartiA. Chakrabarti, N. K. Jha,, N. K. Jha, “QLib: Quantum module library,”“QLib: Quantum module library,” ACM Journal on Emerging Technologies in ComputingACM Journal on Emerging Technologies in Computing Systems (JETC)Systems (JETC), V. 11 Issue 1, Article No. 7, September 2014., V. 11 Issue 1, Article No. 7, September 2014. • CC Lin,CC Lin, A. ChakrabartiA. Chakrabarti and N.K.Jha,and N.K.Jha, ““FTQLS: Fault-Tolerant Quantum Logic Synthesis,”““FTQLS: Fault-Tolerant Quantum Logic Synthesis,” IEEE Transactions on Very LargeIEEE Transactions on Very Large Scale Integration (VLSI) SystemsScale Integration (VLSI) Systems, Vol. 22,No.6, pp. 1350-1363, June 2014., Vol. 22,No.6, pp. 1350-1363, June 2014. • S.B.Mondal,S.B.Mondal, A.ChakrabartiA.Chakrabarti and S.Sur-Kolay,and S.Sur-Kolay, “Synthesis of Ternary Grover's Algorithm”“Synthesis of Ternary Grover's Algorithm”, Proc. of IEEE 441st International, Proc. of IEEE 441st International Symposium on Multiple-Valued Logic (ISMVL 2014Symposium on Multiple-Valued Logic (ISMVL 2014)), Bremen Germany, 19-21 May 2014., Bremen Germany, 19-21 May 2014. • C.C. Lin,C.C. Lin, A. ChakrabartiA. Chakrabarti, N. K. Jha,, N. K. Jha, “Optimized Quantum Gate Library for Various Physical Machine Descriptions,”“Optimized Quantum Gate Library for Various Physical Machine Descriptions,” IEEEIEEE Transactions on Very Large Scale Integration (VLSI) SystemsTransactions on Very Large Scale Integration (VLSI) Systems, Vol. 21, No.11, pp. 2055-2068, Nov. 2013., Vol. 21, No.11, pp. 2055-2068, Nov. 2013. 47