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Quantum Oracles

Quantum Oracles. Jesse Dhillon, Ben Schmid, & Lin Xu CS/Phys C191 Final Project December 1, 2005. Introduction. An oracle is the portion of an algorithm which can be regarded as a “black box” whose behavior can be relied upon Theoretically, its implementation does not need to be specified

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Quantum Oracles

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  1. Quantum Oracles Jesse Dhillon, Ben Schmid, & Lin Xu CS/Phys C191 Final Project December 1, 2005

  2. Introduction • An oracle is the portion of an algorithm which can be regarded as a “black box” whose behavior can be relied upon • Theoretically, its implementation does not need to be specified • However, in practice, the implementation must be considered

  3. Introduction • Why do we use oracles? • Allows complexity comparisons between quantum and classical algorithms • Conceptually simplifies algorithms

  4. Oracle Challenges • Criteria for a good oracle implementation Speed Generality Feasibility

  5. Speed • Oracle has to be fast or it may simply be hiding exponential expense of your algorithm in a black box • For example, imagine an oracle for an algorithm for finding primes in a given range

  6. Generality • Oracles test for answer • Implies reconfiguration of the oracle for each question • Feasible? • Consider 1940s classical computers, compared to modern ones • First computers could only do specific tasks

  7. Feasibility • QC supposed to be exponential speedup • However, when n is small, exponential speedup is lost in overhead • Intimately tied to method of physical realization

  8. Physical Implementations • Survey of different proposed and realized oracle implementations Optical Josephson junctions

  9. Optical • Is optical truly quantum? • Exponential increase in hardware requirements as qubit count increases • Entanglement? • Effects of entanglement, without ability to test Bell-inequality

  10. Optical • But, • Very long coherence times • Single-qubit gates are easy to implement • Scalable cNOT and cSIGN have been demonstrated

  11. Grover’s Optical Oracle • Oracle in used in optical implementation of Grover’s search • Encoded with a marked state, flips the sign of that state 

  12. Grover’s Optical Oracle • Oracles demonstrated so far • Toy implementations, do not actually search through a database • Has a significant failure rate

  13. Optical Oracles • Programming an optical oracle is currently achievable • Uses beam splitters, wave plates, diffraction gratings, etc. • Research suggests in certain cases, sub-exponential scalability may be possible

  14. Josephson Charge Qubits • Superconducting islands coupled via Josephson junctions • Control of Voltage and Flux allow construction of any single or 2-qubit gate

  15. 1. Initialize 2. Hn 3. Oracle: |x(-1)f(x)|x 4. Hn 5. Measure Determine whether a function f:{0,1}N {0,1} Oracle encodes a single function 4 qubit Deutsch-Jozsa How many possible oracles for general n-qubit test? Many!!

  16. Single Qubit Gates Z-rotations generated via charging voltage & time X-rotations with zero charging voltage, and controlled by Josephson energy and time These two gates allow construction of any single qubit gate i.e. Hadamard

  17. 2 Qubit Gates Control Flux in interaction SQUID Control time of interaction CNOT is universal: now we can build the Oracle!

  18. Constructing the Oracle • Constructed of CNOT and controllable phase shifts • Note: only nearest neighbor 2-qubit gates • Ring geometry • Need 2n-1 controllable phase gates to implement any n-qubit Deutsch-Jozsa Oracle

  19. Josephson v. Optical • Optical is cheap, simple • Exponential increases with N • Josephson qubits can be entangled • Can construct any D-J oracle with multiple 2-qubit gates • Josephson seems to offer better scalability and “true” quantum entanglement • Requires more research • Difficult to manufacture • Reconfigurability?

  20. References • N. Schuch & J. Siewert, Phys. Stat. Sol. (b) 233, No. 3, 482-489 (2002) • J. L. Dodd, T. C. Ralph, & G. J. Milburn, Phys. Rev. A 68, 042328 (2003) • P. G. Kwait, et al. quant-ph/9905086 v1 • P. Londero, et al. Phys. Rev. A 69, 010302(R) (2004)

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