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Quantum Computer

Quantum Computer. Qin Cai Gang Zhai Tianli Zhang. Topics/Contents. Introduction to Quantum Computer(QC) Quantum bits - Qubit Quantum Memory Register Quantum Parallelism Obstacle of QC Milestones of QC References. Introduction. Conventional Computers: Use bits

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Quantum Computer

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  1. Quantum Computer Qin Cai Gang Zhai Tianli Zhang

  2. Topics/Contents • Introduction to Quantum Computer(QC) • Quantum bits-Qubit • Quantum Memory Register • Quantum Parallelism • Obstacle of QC • Milestones of QC • References

  3. Introduction • Conventional Computers: Use bits • Quantum Computers: Use Quantum Bits (Qubits)

  4. Introduction - Linear Ion Traps • This linear ion trap uses 4 rod electrodes which are fed an AC voltage of about 1 kilovolt at a frequency of a few megahertz. • This confines the ions along a central line. The end-rings are electrically charged to prevent the ions from escaping at the ends.

  5. Nuclear Magnetic Resonance (NMR) • The nuclei of some atoms have a magnetic moment, (behave in some ways like tiny magnets). • Laws of quantum mechanics require direction of each nuclear magnet to be aligned either parallel or antiparallel to an external magnetic field. • The nuclear magnets are used as qubits. Orange arrows: direction of nuclear magnet. Green arrow: External magnetic field.

  6. Quantum Dots • Each dot can be as small as 30nm across and confine an electron in discrete energy levels. • The quantum dot can be used as a qubit. • Quantum dots can be accessed by focused laser beams that flip the electron between two discrete energy levels or place it into a superposition of the two levels. • The required interaction between qubits occurs through externally applied electric and optical fields.

  7. Other Technological Candidates • Cavity QED • In cavity Quantum ElectroDynamics, atoms interact with photons trapped between reflective mirrors. • Linear optics • Linear optical elements such as mirrors, beamsplitters and single photon sources, can be used to perform quantum algorithms.

  8. So What is a qubit?

  9. Beams of protons Stern-Gerlach Apparatus

  10. Stern-Gerlach Apparatus (Cntd) • Classical physics: • All orientations of the magnetic moment are possible. There should be a continuous smear of positions on the screen. • Also, a magnetic dipole moment of a proton should be able to point in any direction in space.

  11. Stern-Gerlach Apparatus (Cntd) • Actually, this is not what happened… • The beam didn’t smear • It also didn’t point in many directions

  12. Stern-Gerlach Apparatus Results • A beam of protons sent through a static, non-uniform magnetic field splits into two separate beams.

  13. Quantum State Stern-Gerlach Apparatus Results • States |-) and |+) are stable, meaning that if initially placed in |+) state, the proton will remain in that state for a while. • However, it may eventually emit a photon and switch to the lower energy state |-). (spontaneous emission)

  14. Stern-Gerlach Apparatus Applet • Applet: • http://www.ba.infn.it/~zito/museo/frame177.html

  15. Qubits True • In quantum computing these quantum states are called computational basis states • These states allow us to store one bit of information on a single proton, or “qubit” meaning quantum bit. False

  16. Quantum Memory and Register

  17. Quantum Memory • The memory of a quantum computer is usually a combination of 2-state subsystems, referred to as quantum bits (qubits). • Quantum memory holds the current machine state. • Machine State • The state of a classical computer can be given as the distinct states of all bits in memory and processor registers. • The “memory content” of a quantum computer is the combined state of all qubits. This state is referred to as the (quantum) machine state.

  18. Machine State The machine state of a n-qubit quantum computer is the current state of a combined system of n identical qubit subsystems. Generally, the machine state of a n-qubit quantum computer is given by: The combined Hilbert space is thus a composition of n 1-qubit-Hilbert spaces , i.e.

  19. The machine state of a n-qubit quantum computer is a vector in the above Hilbert space , due to the destructive nature of measurement, it cannot be directly observed. • The machine state of a quantum computer is different from the program state which is the current state of the controlling (classic) algorithm (e.g. contents of variable, execution stack, etc.).

  20. Quantum registers are an interface between the machine state and the controlling algorithm. A quantum register is a pointer to a sequence of (mutually different) qubits. • A m-qubit quantum register s is a sequence of mutually different zero-based qubit positions of some machine state with .

  21. An n qubit quantum computer allows for n!/(n-m)! different m qubit registers, any unitary or measurement operation on a m qubit register can result in n!/(n-m)! different operations on the machine state. • In general, a "qubit" can be 0 and 1 simultaneously!- A k-bit quantum register can store 2^k values simultaneously!

  22. Simulation of quantum registers in QCL • General Register • A general quantum Register with n=expr qubits can be declared with • For example: qcl> qureg q[1]; // allocate a qubit register • Constant Register • Registers can be declared constant, by using the register type quconst. A quantum constant has to be invariant to all applied operators. • For example: qcl> quconst c[1];

  23. Register References • To conveniently address subregisters or combined registers, quantum expressions can be named by declaring a register reference: • For example qcl> qureg q[8]; qcl> qureg oddbits=q[1]&q[3]&q[5]&q[7]; qcl> qureg lowbits=q[0:3];

  24. Empty Registers • A quantum register s is considered empty if • Subregisters • Subregisters can be addressed with the subscript operator […]. • For example qcl> qureg q[8]; qcl> print q[3],q[3:4],q[3\4]; • Combined Registers • Registers can be combined with the concatenation operator &. If the registers overlap, an error is triggered. • For example qcl> print q[4:7]&q[0:3];

  25. Memory Management • In QCL, the relation between registers and qubits is handled transparently by allocation and deallocation from qubits of the quantum heap, which allows the use of local quantum variables. All free (i.e. unallocated) quantum memory has to be empty. • At startup or after the reset command, the whole machine state is empty, thus • When temporary scratch registers ( quscratch) are allocated, memory management has to keep track of all applied operators until the local register is deallocated again.

  26. Quantum memory a possibility • Using ultrafast lasers and a beam of cesium atoms, U-M physicists have created a database that stores and retrieves data in atomic quantum phase, instead of the bits and bytes used by today’s computers. • The U-M experiment is the first to test a theoretical approach to using quantum phase for data storage and retrieval, which was proposed by L.K. Grover in a 1997 paper published in Physics Review Letters.

  27. Quantum Parallelism

  28. 1. Quantum State Function for a single particle. • |>=a1|> + a2|> • |a1|2 + |a2|2 = 1

  29. 2. Quantum State Function for a multi-particle system. • 1 2 3 4 … N • |φ1> |> |> |> |> |> • |φ2> |> |> |> |> |> . • |Ψm> = Σ i=1,2N ai|φi>.

  30. 3. Quantum Operator • ô|> = o |>

  31. 4. The micro-result and the ensemble result. • ô |Ψm> = ô Σ i=1,2N ai|φi> • = Σ i=1,2N aiô|φi> • |Ψnew> = Σ i=1,2N aioi|φi>

  32. Example • classical 3-bit register • three qubits • L qubits • store up to 2L numbers at once • in a single computational step perform the same mathematical operation on 2L different input numbers, and the result will be a superposition of all the corresponding outputs. • In order to accomplish the same task any classical computer has to repeat the computation 2L times, or has to use 2L different processors working in parallel.

  33. Obstacle • Quantum interference. • Decoherence • Pessimistic opinion • decoherence will in practice never be reduced to the point where more than a few consecutive quantum computational steps can be performed. • optimistic opinion

  34. What we think … • The current challenge is not to build a fully-fledged universal quantum computer right away, but rather to move from the experiments in which we merely observe quantum phenomena to experiments in which we can control those phenomena in the necessary ways.

  35. Milestones in the development of quantum computer technology • Milestone

  36. Reference • http://www.imsa.edu/~matth/cs299/node1.html • http://tph.tuwien.ac.at/~oemer/doc/qcldoc/node9.html • http://www.is.titech.ac.jp/~schuler/q-complexity.ps • http://www.cs.caltech.edu/~westside/quantum-intro.html • http://www.csc.liv.ac.uk/~u9rl1/stern-gerlach.html • http://w3.arizona.edu/~lascool/presentations/DAMOP-CLEO2000/04-Stern_Gerlach.html • http://www.7stones.com/Homepage/Publisher/QM.html

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