The Road to Quantum Computing: Boson Sampling

1. The Road to Quantum Computing:Boson Sampling Nate Kinsey ECE 695 Quantum Photonics Spring 2014

2. A Quantum leap for computing • Quantum computing is the next frontier for electronics • Enables solution of exponential problems in polynomial time • Searching • Prime factorization • Simulation of quantum systems • Understanding quantum phenomenon D-Wave One Quantum Computer, D-Wave Inc.

3. Fragile by Nature • QC derives its power from the superposition quantum states • Quantum states are subject to de-coherence • De-coherence limits the ability to perform operations • Error correction? • 3-bit has been demonstrated • Difficult to entangle many qubits • Current record is 14 qubits [2] [1] T. Monz, “14-qubit entanglement: creation and coherence” Phys. Rev. Lett. 2011. [2] G. Waldherr, “Quantum Error correction in a solid-state hybrid spin register” Nature, 2014

4. Another solution? • Generalized QC is proving difficult to realize (i.e. Shor, Q. Turing, etc) • Can we use what we know about the quantum world to assist computation?

5. General Idea of Boson Sampling • The idea of boson sampling was proposed in 2011 [1]. • Uses photons • Similar to a Galton board, which samples from the binomial distribution. • By engineering the peg sizes and locations a desired system can be modeled. • The response is governed by quantum photon statistics. [1] S. Aaronson and A. Arkhipov, “The Computational complexity of Linear Optics,” Proc. ACM Symposium, 2011.

6. Where is this useful? • Boson sampling is primarily focused on determining unitary matrix transformations. • The observed output from a unitary transformation is defined by the permanent of the matrix. • Output from a large linear optical network (more details to come) • The permanent is exponentially hard to solve and is limited to ~ 20 variables for current systems. [1] S. Aaronson and A. Arkhipov, “The Computational complexity of Linear Optics,” Proc. ACM Symposium, 2011.

7. Quick Review of QBS • Classical: E2=rE1 and E3=tE1 • Quantum: • Which satisfy the commutation relations: and • For a 50:50 BS, the reflected beam has a π/2 phase shift so that: C. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

8. Quick Review of QBS • Let us consider input state which we write as • For the beam splitter (from conjugation of , & algebra): • Thus we can write: • Photon is either reflected or transmitted with equal probability (i.e. no coincidence counts). • Can be explained classically. C. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

9. Two Photon Interference • When we consider two incident photons things are more interesting: • In a similar way as with the previous, we find: • Thus, • This is interesting because the photons appear together. • The RR and TT states cancel each other C. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

10. Two Photon Interference • Note that the cases for the RR and TT are indistinguishable at the output (detector). • Use Feynman’s Rule to obtain the probability of a process with multiple indistinguishable paths. • Add the probability amplitudes of all processes and find the square of the modulus. • Note our reflected photons acquire a phase shift of π/2 ( or i ) • This has been experimentally demonstrated by Hong, Ou, and Mandel as well as many others since then. • Plasmons (Fokonas “Two-plasmon interference,” Nature Photonics 2014.) • This interference is the basis of boson sampling machines C. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

11. Boson Sampling • Comprised of many beam splitters and delay lines (pegs/spaces on the Galton board) • Also uses many input channels • We consider an input state I, the probability of state O is defined by the permanent of the unitary transformation U. [1] M. Tillmann, et al. “Experimental Boson Sampling,” Nature Photonics, 2013.

12. Boson Sampling • Quick example: [1] M. Tillmann, et al. “Experimental Boson Sampling,” Nature Photonics, 2013.

13. Experimental Boson Sampling • After the first description of boson sampling by Aaronson and Arkhipov four groups conducted experiments. • M. Tillmann, Nature Photonics • J. Spring, Science • A. Crespi, Nature Photonics • M. Broome, Science • They were coordinated and released simultaneously across Science and Nature in 2013.

14. Experimental Boson Sampling • M=6 input and output (36 element matrix) • Complex elements of the unitary matrix Λ were sampled (Λij=tijeiφij) • Insert at port i and detect at port j • This determines the magnitude of the matrix element |tij|2 • Insert two photons i1 and i2 and observing at j1 and j2 determines complex angle φ (relative phase shift).

15. Experimental Boson Sampling • Completed for 3 and four photon excitation • Blue values: predicted probability of output from experimentally determined Λij • Red values: experimentally measured probabilities of given output

16. Experimental Boson Sampling • Additionally, studied response ofan ideal boson sampling machine. • They found that the experimental deviation was larger than expected • Not sampling from distribution of network • Distinguishability of photons • Bunched emission • Despite this, a good agreement was found with predictions • Technique is robust

17. Outlook for Boson Sampling • The benefit of boson sampling is that its requirements are more relaxed that those of generalized QC. • Some believe that it would be nearly as difficult to scale single photons sources to the necessary level • However, boson sampling is the only known way to make permanents show up as amplitudes, which is an important function for computer science. • Additionally, for large scale boson sampling systems, they are impossible to model on a classical computer. • Thus, they are a unique window into a complex quantum world (simulation) without the need for general QC.

18. Conclusions • Boson sampling is a near term method for enabling quantum assisted computation • Can model complex systems to determine the unitary matrix transformation (i.e. Permanents) • Enables unique access into complex quantum interactions that would only be able to be investigated with a general QC