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Quantum, from Physics to Programming

Quantum, from Physics to Programming. Philippe Jorrand CNRS Leibniz Laboratory, Grenoble, France Philippe.Jorrand@imag.fr. … mechanical,. … or electronic,. … or bio-molecular,. … or quantum, etc. Information may take different physical forms….

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Quantum, from Physics to Programming

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  1. Quantum, from Physics to Programming Philippe Jorrand CNRS Leibniz Laboratory, Grenoble, France Philippe.Jorrand@imag.fr Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  2. … mechanical, … or electronic, … or bio-molecular, … or quantum, etc. Information may take different physical forms… There is no information without a physical carrier, and no computation without a physical process. The laws of physics dictate what computations can be. Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  3. Classical physics Quantum physics At any given time, a physical system can be in one state among a set of possible basis states. But, in general, it is in a state which is a superposition of several basis states. At any given time, a physical system is in one state, and only one state, among a set of possible states of that system. The transformations of the state of a physical system are not, in general, reversible. The transformations of the state of an unobserved physical system are reversible and deterministic. The observation of a physical system in state S does not modify S and it is deterministic: it returns the same information for identical systems in state S. The observation of a physical system in state S irreversibly modifies S and it is probabilistic: it may return different information for identical systems in state S. The state of a physical system A can be copied into another physical system B, if both systems have the same set of possible states. The state of a physical system A cannot, in general, be copied into another physical system B. The state of a physical system composed of n sub-systems is reducible to an n-tuple of the states of these sub-systems. The state of a physical system composed of n sub-systems is not, in general, reducible to an n-tuple of the states of these sub-systems. Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  4. Classical Bit Quantum Bit A classical bit is, at every point in time: - either in state 1: - or in state 0: State of a classical bit: bÎ {0,1} A quantum bit (« qubit ») is, at every point in time: - either in basis state |1Ò: - or in basis state |0Ò: - or in a superposition state, i.e. at the same time |1Òand|0Ò: State of a qubit:|yÒ Î E where E is a 2-dimensional vector space, and {|0Ò, |1Ò} is a basis of E. Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  5. |1Ò |yÒ=a|0Ò+b|1Ò b |0Ò a 1 0 • |y> = a |0> + b |1>can be written as a column vector: |0>= 0 1 |y>= a b |1>= The state of a qubit is a vector|yÒ = a|0Ò + b|1Ò • The state of a qubit is a vector in a 2-dimensional vector space: - either the vector |0Ò, or the vector |1Ò, which form a basis, - or, in general, a vector |yÒ=a|0Ò+b|1Ò, where a and b are complex numbers such that |a|2+|b|2=1 . Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  6. With classical bits With qubits a Ÿ a Ÿb |yÒ b U |yÒ U Computing means transforming the state {0,1} {0,1} More generally: • U is a linear operator, represented by a 2¥2 matrix î deterministic • U unitary (U.U†=Id) î reversible {0,1}n {0,1}m • arbitrary boolean functions • in general not reversible Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  7. |y’ Ò = U |yÒ |yÒ = a |0Ò + b |1Ò U 1 1 1-1 0 1 1 0 Not = |yÒ |y’ Ò = a |1Ò + b |0Ò Not H = 1/√2 |0Ò |1Ò 1/√2 ( |0Ò - |1Ò ) 1/√2 ( |0Ò + |1Ò ) H H Unitary gates on one qubit: examples The Not gate: |0Ò « |1Ò The Hadamard H gate: Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  8. Measuring a qubit which is • in state |0Ò gives the value 0 and • the qubit stays in state |0Ò. • Same for |1Ò. Reading a classical bit which is in state 0 gives the value 0 and the bit stays in state 0. Same for 1. 0 0 Read 1 Read 1 Classical reading Quantum measurement • Measuring a qubit which is in state • |yÒ = a |0Ò + b |1Ò gives either: • • the value 0, with probability |a|2, • and the qubit state becomes |0Ò, • • or the value 1, with probability|b|2, • and the qubit state becomes |1Ò. • a and b: « probability amplitudes » Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  9. 50%: state |0Ò, value 0 1/√2 ( |0Ò + |1Ò ) H Measure probability|a|2: state |0Ò, value 0 50%: state |1Ò, value 1 a|0Ò+b|1Ò 1 1 1-1 probability|b|2: state |1Ò, value 1 Measure H = 1/√2 |0Ò |1Ò 1/√2 ( |0Ò - |1Ò ) 1/√2 ( |0Ò + |1Ò ) H H Measurements on one qubit |0Ò Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  10. probability|a|2: state |0Ò, value 0 Measure |yÒ probability|b|2: state |1Ò, value 1 Probabilistic measurement: state evolution by projection of |yÒ onto a probabilistically chosen basis state (and renormalization), and return of a classical value associated with that choice. Deterministic computation: state evolution by application of a unitaryoperator to |yÒ , i.e. bymultiplying the vector |yÒ by a 2¥2 unitary matrix U. |yÒ U U|yÒ |yÒ U|yÒ U† Two forms of evolutions for|yÒ = a|0Ò + b|1Ò Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  11. 50%: state |0Ò, value 0 1/√2 ( |0Ò + |1Ò ) H Measure 50%: state |1Ò, value 1 50% 50% 50% 50% H Measure H 100%: state |0Ò, value 0 1/√2 ( |0Ò + |1Ò ) 1/√2 ( H(|0Ò) + H(|1Ò) ) = 1/2 ( (|0Ò+ |1Ò) + (|0>-|1Ò) ) = |0Ò Quantum operators: definitely not classical! |0Ò |0Ò Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  12. State of a 2-qubit register • The state of a single qubit is a 2-dimensional vector: • - one of the 2 basis states: |0Ò or |1Ò - or a superposition of the basis states: |yÒ = a |0Ò + b |1Ò, with |a|2+ |b|2 = 1 • The state of a 2 qubit register is a 4 dimensional vector: • - one of the 4 basis states: |00Ò , |01Ò , |10Ò or |11Ò - or a superposition of basis states:|yÒ = a|00Ò + b|01Ò + g|10Ò + d|11Ò, with |a|2+ |b|2+ |g|2+ |d|2 = 1 Examples: |yÒ = 1/2 (|00Ò + |01Ò + |10Ò + |11Ò) = 1/√2 ( |0Ò + |1Ò ) ƒ1/√2 ( |0Ò + |1Ò ) |jÒ = 1/√2 (|00Ò + |11Ò) Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  13. 1/√2 ( |0Ò + |1Ò ) |0Ò H =1/2 ( |00Ò + |01Ò + |10Ò + |11Ò ) 1/√2 ( |0Ò + |1Ò ) |0Ò H H4 1/2 ( |00Ò + |01Ò + |10Ò + |11Ò ) |00Ò 1 1 1 1 1-1 1 -1 1 1 -1 -1 1 -1 -1 1 H = H ƒH = 1/2 H Unitary gates on 2 qubits: example where H4 = HƒH is a 4¥4 matrix: Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  14. |aÒ |aÒ CNot= CNot: |bÒ |a ≈bÒ CNot (|00Ò)  |00Ò CNot(|01Ò)  |01Ò CNot (|10Ò)  |11Ò 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 CNot (|11Ò)  |10Ò The Controlled Not gate a|00Ò + b|01Ò + g|10Ò + d|11Ò a|00Ò + b|01Ò + g|11Ò + d|10Ò Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  15. |0Ò H |0Ò |00Ò 1/√2 (|00Ò+|10Ò) 1/√2 (|00Ò+|11Ò) In general, quantum states are entangled There exist no|y1Ò,|y2Òsuch that 1/√2 (|00Ò+|11Ò) = |y1Òƒ|y2Ò After execution, the two qubits are entangled. They form an “EPR” state: [Einstein, Podolsky, Rosen] « Can quantum mechanical description of physics reality be considered complete? »Physical Review, 1935 Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  16. |yÒ = a|00Ò + b|01Ò + g|10Ò + d|11Ò probability|a|2 : state |00Ò, value 0 |yÒ Measure probability|b|2 : state |01Ò, value 1 1/√2 (|00Ò+|11Ò) probability|g|2 : state |10Ò, value 2 probability|d|2 : state |11Ò, value 3 |0Ò H Measure |0Ò probability50%: state |00Ò, value 0 probability50%: state |11Ò, value 1 Quantum states: definitely not classical! Einstein, 1935: « … spooky action at a distance … » Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  17. No cloning ! Theorem:There exist no unitary transformation U such that for all states |yÒ, U(|yÒ|0Ò) = |yÒ|yÒ. Proof: Suppose that such a U does exist, i.e. for any state |yÒ=a|0Ò+b|1Ò, we have: U(|yÒ|0Ò) = |yÒ|yÒ This also holds for basis states: U(|0Ò|0Ò) = |0Ò|0Ò and U(|1Ò|0Ò) = |1Ò|1Ò Then, by linearity: U(|yÒ|0Ò) = a|0Ò|0Ò + b|1Ò|1Ò ≠|yÒ|yÒ = a2|0Ò|0Ò+ab|0Ò|1Ò + ab|1Ò|0Ò+ b2|1Ò|1Ò Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  18. |yÒ U |yÒ probability|ai|2 |yÒ U Measure U : 2n¥2nmatrix state |iÒ, value i Register of n qubits: state and evolution • The state of a register of n qubits is a 2n-dimensional vector: • - one of the 2n basis states: |00..00Ò (i.e. |0Ò) |00..01Ò (i.e. |1Ò) |00..10Ò (i.e. |2Ò) ... |11..11Ò(i.e. |2n-1Ò) • - or a superposition of basis states:|yÒ = a0|0Ò + a1|1Ò + a2|2Ò + .. + a 2n-1|2n-1Ò, • with |a0|2+ |a1|2+ .. + |a 2n-1|2 = 1 Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  19. 1 0 0eip/4 T= CNot= 1 1 1-1 = = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 -1 11 H Not Not •H 1/√2 1 1 1 1 1-1 1 -1 1 1 -1 -1 1 -1 -1 1 H H= 1/√2 = = H ƒH 1/2 H A universal familly for approximating any 2n¥2nU • Temporal composition: matrix product • Spatial composition: tensor product Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  20. 1 a b |0Ò |1Ò |0Ò |1Ò 1/2 1/2 |2Ò |0Ò |1Ò |2Ò |3Ò |0Ò |1Ò |3Ò -1/2 Diagrams for states with real amplitudes… |0Ò a |0Ò + b |1Ò 1/2 ( |0Ò + |1Ò + |2Ò + |3Ò ) 1/2 ( |0Ò + |1Ò - |2Ò + |3Ò ) Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  21. |0Ò H 1/√2(|0Ò+|1Ò) 1/√2(|0Ò+|1Ò) 1/√2(|0Ò+|1Ò) |0Ò H n 1/√N( |0Ò + |1Ò + |2Ò + .. + |2n-1Ò ) = (where N = 2n) |0Ò H 1 HN 1/√N .. .. .. .. .. |2n-1Ò .. |2n-1Ò |0Ò |1Ò |2Ò .. .. |iÒ .. |0Ò |1Ò |2Ò .. .. |iÒ .. 2n 2n 2n values superposed in a n-qubit register Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  22. aŒ [0,2n-1] f(a)Œ{0,1} bŒ{0,1} |aÒ Register of n+1 qubits in state |a 0Ò Register of n+1 qubits in state |a b≈f(a)Ò |bÒ |00..0Ò HN .. .. .. .. 1/√N |0 0Ò |1 0Ò |2 0Ò .. |0 f(0)Ò |1 f(1) Ò |2 f(2)Ò .. |(2n-1) f(2n-1)Ò |i 0Ò .. |(2n-1) 0Ò |i f(i)Ò .. |0Ò Computing a function: quantum parallelism Uf (N = 2n) Uf Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  23. Vf H |1Ò Vf .. .. |00..0Ò HN .. .. 1/√N .. .. 1/√N - 1/√N Invert a’s amplitudes whenever f(a) = 1 aŒ [0,2n-1] f(a)Œ{0,1} Uf Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  24. .. .. 1/√N -1/√N .. .. .. .. Symmetry with respect to average SN Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  25. aŒ [0,2n-1] f(a)Œ{0,1} f(a)=1 for one and only onea0Œ [0,2n-1] No other information is available about f Problem: find thata0 Find the element, among N=2 n, which satisfies f Classical computing: worst case, N calls to f average, N/2 calls to f Quantum computing: always exactly √N calls to f, out of reach from classical Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  26. aŒ [0,2n-1] f(a)Œ{0,1} f(a)=1 for a singlea=a0Œ [0,2n-1] .. .. |a0Ò .. .. 1/√N .. .. Amplification of |a0Ò GN = SN•Vf |a0Ò Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  27. GN GN 1/√N .. .. |a0Ò GN GN |a0Ò .. .. Quantum algorithm for searching a data base 1-e Measure Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  28. .. .. .. .. Stop at √N times GN ! Geometrical proof q 2q • Goal: get as close as possible to |a0Ò. For that, repeat GN a number k of times such that: p/2 – qbq +k2q b p/2 + q • N is large, so q is small: q» sin q = 1/√N • Hence:k»p/4√N 1/√N |a0Ò |a0Ò 1/√N Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  29. Quantum ingredients for quantum algorithmics • Superposition: up to 2n values are possibly stored at the same time in a register of n qubits, with a probability amplitude for each value. • Unitary operators: linear,deterministic evolution of the state, i.e. of the amplitudes of the values superposed in a register. But no cloning! • Superposition + unitary operators î quantum parallelism: all the values of a function can be computed in one step. • Probabilistic measurement: selects one of the values superposed in a register, all other values are irreversibly lost. Previous evolution of the amplitudes increase or decrease the probabilities of selection. • Entangled states: the contents of two or more registers can be correlated. • Entanglement is a powerful resource: example for f : {0,1}nô {0,1}m • computing f produces 2n pairs |i, f(i)>:i in a register x, f(i) in a register y • measuring y returns one value j of f, and the state of y collapses to |jÒ • measuring y is also measuring the y part of the [x,y] entangled system: the state of the x part collapses to a superposition of all i Œ f -1(j). Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  30. Quantum computation needs classical control • Initialize each quantum register to some basis states, possibly depending upon previous computation: classical computation and control. • Apply unitary operators, in some order, to various parts of quantum registers: classical control. • Measure a quantum register: classical control, and returned value received by classical part. • Because of probabilistic measurement, check the correctness of the received value: classical computation. • If the value is correct, possibly do further classical computation, then terminate or start other quantum part: classical computation and control. • If the value is not correct, start over the quantum part: classical control. • Send qubits (not their states, because of no cloning: move them physically) from a quantum part to another quantum part: classical control. Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  31. Facts and issues in quantum language design • Classical computation and control + Quantum computation under classical control in the same language • Quantum initialization and measurement: no other Classical ´ Quantum exchange of information • Probabilistic quantum measurement î probability distribution over classical and quantum states • No cloning î quantum variables are “single use” î communicating quantum variables from A to B means moving their physical carriers from A to B • Entanglement î when leaving the scope of quantum variables, take care of possible entanglement with variables of larger scopes Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  32. Toward quantum functional programming l-expression: E ô V | lV.E | (E E) b-reduction: ((lx.E) F) == E[x/F] ((lx.(.. x .. x .. x ..) F) == (.. F .. F .. F ..) No-cloningîFis not allowed to be a quantum variable Wanted: typed quantum lambda calculi, linear w.r.t. occurrences of quantum variables: lx:quantum.Ewhere E is linear in x Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  33. Alice broadcasts 0, 1, 2 or 3 on the radio |yÒ = a |0Ò + b |1Ò H Measure Alice |0Ò H |00Ò, |01Ò, |10Ò or |11Ò Bob |0Ò |yÒ 2-1/2(|00Ò+|11Ò) |yÒ0, |yÒ1 , |yÒ2 or |yÒ3 Toward quantum process calculi U0, U1, U2 or U3 Alice == [ x,y: Qubit . [ f: Gate . CNot(x,y) • H(x) • f!Meas(x,y) • end ] ] Bob == [ z: Qubit . [ g: Gate . g?(k: Nat) • [ k=0 ôU0(z) • end, k=1 ôU1(z) • end, k=2 ôU2(z) • end, k=3 ôU3(z) • end ] ] ] Teleport == [ y: Qubit . [ a,b: Qubit . [ h: Gate . BuildEPR[a,b] ; (Alice[y,a][h] ||Bob[b][h]) ] ] ] Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  34. Hot topics in quantum information processing • Algorithms and quantum computational complexity theory • Communication, multi-party communication tasks, quantum communication complexity theory • Quantum cryptography, secret sharing and secure protocols • Distributed and interactive quantum computation • Quantum computation by measurement only • Abstract models, languages, architectures for quantum computing • Understanding and characterizing quantum entanglement • Quantum information theory … and quantum computer hardware… Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  35. YES May be YES May be NO Toward quantum computers 1 2 3 4 5 DiVicenzo (IBM Research) criteria: 1. Ability to initialize qubits 2. Universal family of gates 3. Ability to measure qubits 4. Scalability of physical system 5. Decoherence time >> gate time A B C D E F Six approaches (among others): A. Nuclear Magnetic Resonance B. Trapped Ions C. Neutral atom D. Optical E. Solid State F. Superconducting (Source: A Quantum Information Science and Technology Roadmap, ARDA, April 2004 - http://qist.lanl.gov) Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

  36. (Logo of the Center for Quantum Computation, Oxford - http://www.qubit.org) « Quantum information science is a new field of science and technology. It is an interdisciplinary subject where physicists, mathematicians, computer scientists and engineers are making major contributions. Deep links between the previously unrelated disciplines of quantum physics, computer science and information theory have been forged. On the one hand there have been insights into fundamental issues in physics. On the other hand, totally new methods of computation, communication and information processing have emerged. New technologies have also arisen offering, for example, the potential for immense computing power and secure communications. Quantum information science is one of the most dynamic areas in the physical sciences, computer science and information theory. New ideas and phenomena are appearing at a remarkable rate. There are still very many open questions and fundamental issues to be understood. » (Quantum Information Sciencesemester, Isaac Newton Institute for Mathematical Sciences, Cambridge, August to December 2004 - http://www.newton.cam.ac.uk/programmes/QIS) Quantum: from physics to programming - UPP - Mont Saint Michel - 15-17 Sep. 2004

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