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

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. 2

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 [2] G. Waldherr, “Quantum Error correction in a solid-state hybrid spin register” Nature,

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? 4

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,

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,

Quick Review of QBS 7 C. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

Quick Review of QBS 8 C. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

Two Photon Interference 9 C. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

Two Photon Interference 10 C. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

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,

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

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

Experimental Boson Sampling M=6 input and output (36 element matrix) Complex elements of the unitary matrix Λ were sampled (Λ ij =t ij e iφ ij ) ▫Insert at port i and detect at port j ▫This determines the magnitude of the matrix element |t ij | 2 ▫Insert two photons i 1 and i 2 and observing at j 1 and j 2 determines complex angle φ (relative phase shift). 14

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

Experimental Boson Sampling 16 Additionally, studied response of an 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

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. 17

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 18