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. How many possible oracles for general n-qubit test? Many!! 4 qubit Deutsch-Jozsa 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

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 CNOT is universal: now we can build the Oracle! Control Flux in interaction SQUID Control time of interaction

18. Constructing the Oracle Constructed of CNOT and controllable phase shifts Note: only nearest neighbor 2-qubit gates –Ring geometry Need 2 n -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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