The Road to Quantum Computing:Boson Sampling

Nate Kinsey

ECE 695 Quantum PhotonicsSpring 2014

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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.

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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

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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?

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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.

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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.

( )1

( )n

n

i iS i

Perm A a

a bPerm ad bc

c d

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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.

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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.

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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 otherC. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

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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 machinesC. Gerry andP. Knights Introductory Quantum Optics, Cambridge University Press, 2005.

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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.

1 2 3... mI i i i i

1 2 3... mO j j j j

2,

,1 2 3 1 2 3! ! !... ! ! ! !... !

I O

I Om m

Per UP

i i i i j j j j

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Boson Sampling

•Quick example:

110I 011O

,I I O

a b c a bd e

U d e f U d e Ug h

g h i g h

2

,I OP dh eg

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

iji

ij ijU t e

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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.

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Experimental Boson Sampling• M=6 input and output (36

element matrix)

• Complex elements of the unitary matrix Λ were sampled (Λij=tije

iφij )

▫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).

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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

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Experimental Boson Sampling• 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

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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.

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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