|| 4/7/17QIPII:QuantumalgorithmswithSCqubits 1

§ Introduction§ Superconductingqubits

§ Quantumalgorithms

§ Deutsch-Jozsa

§ Groversearch

§ Implementation§ Superconductingqubits

§ Properties

§ Methods

§ Modernalgorithmsandtheirimplementation

Outline

|| 4/7/17QIPII:QuantumalgorithmswithSCqubits 2

§ 1992, Deutsch-Jozsa [1] : solves a black-box problem which probably requiresexponentially many queries to the black box for any deterministic classical computer

§ 1993, Simon [2]: equivalent to [1] from another oracle

§ 1995, Shor [3]: solves the integer factorization problem, discrete log problem

§ 1997, Grover [4]: Quantum database search algorithm

§ 1998, Jones & Mosca [5]: First experimental demonstration of a quantum algorithm. Aworking 2-qubit NMR quantum computer used to solve Deutsch's problem

TimelineofQuantumAlgorithms

[1]Deutsch,D.andJozsa,R.(1992),ProceedingsoftheRoyalSocietyofLondonA(439-553)[2]Simon,D.R.(1995), FoundationsofComputerScience,1996,35th AnnualSymposium(116-123)[3]Shor,P.W.(1997),SIAMJournalonComputing26(1484-1509)[4]Grover,L.K.(1996),28th AnnualACMSymposiumontheTheoryofComputing(212)[5]Jones,J.A.et.al.(1998),Nature393(344-346)

|| 4/7/17QIPII:QuantumalgorithmswithSCqubits 3

§ Step-by-step procedure, where each of the steps can be performed on a QuantumComputer

§ Make use of quantum features such as superposition and entanglement

§ Can be based on quantum Fourier transform (Deutsch-Jozsa, Simon), amplitudeamplification (Grover), quantum walks (Triangle Finding Problem, Element DistinctnessProblem)

QuantumAlgorithms

|| 4/7/17QIPII:QuantumalgorithmswithSCqubits 4

§ Reducedsensitivitytochargenoise

§ IncreasesratioofJosephsonenergytochargingenergy,usingalargeshuntingcapacitor

§ Coherencetimes~𝜇𝑠

SuperconductingQubits:TheTransmon

*

*

*AdaptedfromQIPII‘17LectureNotes,foundinhttp://www.qudev.ethz.ch/content/QSIT14/QSITNotes.pdf (26Mar2017,9pm)

|| 4/7/17QIPII:QuantumalgorithmswithSCqubits 5

§ Task:Solveblackboxproblem

§ Deterministic:alwaysproducesasolution,andthesolutioniscorrect

§ Limitedpracticalapplicationbutshowsthatquantumcomputerscansolvesuchproblemswithnoerror

Deutsch-Jozsa Algorithm

Deutsch,D.andJozsa,R.(1992),ProceedingsoftheRoyalSocietyofLondonA(439-553)

|| 4/7/17QIPII:QuantumalgorithmswithSCqubits 6

§ Hadamard transformations on 𝑛 zeros,formingallpossibleinputs,andasingle1,whichwillbetheanswerqubit

§ Runthefunctiononce;thisXORstheresultwiththeanswerqubit

§ Hadamards onthe 𝑛 inputsagain,andmeasuretheanswerqubit

Deutsch-Jozsa Algorithm- Steps

Deutsch,D.andJozsa,R.(1992),ProceedingsoftheRoyalSocietyofLondonA(439-553)

|| 4/7/17QIPII:QuantumalgorithmswithSCqubits 7

§ Task:Inanunstructureddatabaseof𝑁 elements,retrievetheonethatsatisfiesagivencondition.

§ Classically𝑁/2itemshavetobequeriedonaverage,Grover’ssearchreducesthisto𝑁�

§ Thesecretliesinamplitudeamplification,whichboostsprobabilityoffindingcorrectstate

Grover’sAlgorithm

Grover,L.K.(1996),28th AnnualACMSymposiumontheTheoryofComputing(212)

|| 4/7/17QIP II : Quantum algorithms with SC qubits 8

§ StatePreparation:Qubitsareinitializedinasuperpositionstate

§ Applicationoftheoracle

§ Amplitudeamplificationbyinversionusingthediffusionoperator

§ Readout

Grover’sAlgorithm- Steps

Grover,L.K.(1996),28th AnnualACMSymposiumontheTheoryofComputing(212)

|| 4/7/17QIP II : Quantum algorithms with SC qubits 9

|| 4/7/17QIP II : Quantum algorithms with SC qubits 10

Detuned qubits:interactionsuppressedQubits resonant:virtual photonmediated interaction

ExperimentalImplementation

[6]L.DiCarlo etal.,Nature460,08121(2009);A.Blaisetal.,Phys.Rev.A75,032329(2007)

|| 4/7/17QIP II : Quantum algorithms with SC qubits 11

Criteriaforimplementingaquantumcomputer:#1.Ascalablephysicalsystemwithwell-characterizedqubits#2.Theabilitytoinitializethestateofthequbits#3.Long(relative)decoherence times,muchlongerthanthegate-operationtime#4.Auniversalsetofquantumgates#5.Aqubit-specificmeasurementcapability

Plustwocriteriarequiringthepossibilitytotransmitinformation:#6.Theabilitytointerconvertstationaryandmobile(orflying)qubits#7.Theabilitytofaithfullytransmitmobilequbitsbetweenspecifiedlocations

[6]DiVincenzo,D.P.(2000),ThePhysical Implementationof QuantumComputation,arXiv:quant-ph/000207

TheDiVincenzo criteria[6]

|| 4/7/17QIP II : Quantum algorithms with SC qubits 12

Single-qubit-Gates Two-qubit-Gates

𝑅,(𝜃) =cos(34) −sin(34)

sin(34) cos(34)

𝑅8(𝜃) = 𝑒:;<= 00 𝑒;

<=

U=

1 0 0 00 1 0 00 0 1 00 0 0 −1

PhaseGates

L.DiCarlo etal.,Nature460,08121(2009);A.Blaisetal.,Phys.Rev.A75,032329(2007)

|| 4/7/17QIP II : Quantum algorithms with SC qubits 13

Phaseshift (by changing 𝐕𝐫)

𝜙BC = 2𝜋F 𝛿𝑓BC 𝑡 𝑑𝑡K

KL

L.DiCarlo etal.,Nature460,08121(2009);A.Blaisetal.,Phys.Rev.A75,032329(2007)

|| 4/7/17QIP II : Quantum algorithms with SC qubits 14

Phaseshift (by changing 𝐕𝐫)

𝜙BC = 2𝜋F 𝛿𝑓BC 𝑡 𝑑𝑡K

KL

𝜙MM = 2𝜋F 𝛿𝑓MM 𝑡 𝑑𝑡K

KL

𝜙MN = 2𝜋F 𝛿𝑓MN 𝑡 𝑑𝑡K

KL

𝜙NM = 2𝜋F 𝛿𝑓NM 𝑡 𝑑𝑡K

KL

𝜙NN = 2𝜋F 𝛿𝑓NN 𝑡 𝑑𝑡K

KL

= 𝒏𝟏 ∗ 𝟐𝝅

= 𝒏𝟐 ∗ 𝟐𝝅

= 𝒏𝟑 ∗ 𝟐𝝅

= (𝟐𝒏𝟒 + 𝟏) ∗ 𝝅U=

1 0 0 00 1 0 00 0 1 00 0 0 −1

U=

𝑒;WLL 0 0 00 𝑒;WLX 0 00 0 𝑒;WXL 00 0 0 𝑒;WXX

1

DiCarlo etal.(2009),Nature460,08121;Blaisetal.(2007),Phys.Rev.A75,032329

|| 4/7/17QIP II : Quantum algorithms with SC qubits 15

Two – qubit Gates

𝜙BC = 2𝜋F 𝛿𝑓BC 𝑡 𝑑𝑡K

KL

𝑉Z𝑉

DiCarlo etal.(2009),Nature460,08121

|| 4/7/17QIP II : Quantum algorithms with SC qubits 16

Two – qubit Gates

DiCarlo,L.etal.(2009),Nature460,08121

𝑉Z𝑉

𝜙NN = 𝜙MN + 𝜙NM −F 𝜁 𝑡 𝑑𝑡K

KL

∫ 𝜁 𝑡 𝑑𝑡KKL

= 𝜋

|| 4/7/17QIP II : Quantum algorithms with SC qubits 17

𝜙NN = 𝜙MN + 𝜙NM −F 𝜁 𝑡 𝑑𝑡K

KL

∫ 𝜁 𝑡 𝑑𝑡KKL

= 𝜋

U =

1 0 0 00 𝑒;WLX 0 00 0 𝑒;WXL 00 0 0 𝑒;(WLX]WXL]^)

U =

1 0 0 00 1 0 00 0 1 00 0 0 −1

DiCarlo,L.etal.(2009),Nature460,08121

Two – qubit Gates

|| 4/7/17QIP II : Quantum algorithms with SC qubits 18

Parameter 2009 2017

# qubits available 2 5

Coherence time 𝜇 ~ 1 µs ~ 50 µs

Gate error rate ~ 1% ~ 0,3%

Readout error rate ~ 10% ~ 5%

Cloud connected ;) No Yes

From2007inYaletotoday’scloudresourceavailability!

|| 4/7/17QIP II : Quantum algorithms with SC qubits 19

Ø BQP (bounded-error quantum polynomial time) problems

Quantum Algorithm Zoo* : http://math.nist.gov/quantum/zoo/

Current Research for Algorithms for Quantum Computation

*USA National Institute of Standards and Technology

|| 4/7/17QIP II : Quantum algorithms with SC qubits 20

• Hybrid Quantum – Classic approach algorithm

• Used for Quantum Simulation Problems

• Calculation on a quantum computer

• Optimization on a classic CPU

Nature Communications 5,4213 (2014) , doi:10.1038/ncomms5213

Variational Quantum Eigensolver (VQE) Algorithm

|| 4/7/17QIP II : Quantum algorithms with SC qubits 21

VQE Example – Hydrogen Molecule

O’Malley,P.J.J.etal.(2016),PRX6,031007

𝐻 = 𝑔M + 𝑔N𝑍M + 𝑔4𝑍N + 𝑔c𝑍M𝑍N + 𝑔d𝑌M𝑌N + 𝑔f𝑋M𝑋N

|| 4/7/17QIP II : Quantum algorithms with SC qubits 22

IBMQuantumExperience

program 5 qubits

from yoursmartphone

Implementing a Quantum Algorithm in the cloud

https://quantumexperience.ng.bluemix.net/

|| 4/7/17QIP II : Quantum algorithms with SC qubits 23

• Eachqubitcanbereadoutindividually

Physical Layout of the QX chip

|| 4/7/17QIP II : Quantum algorithms with SC qubits 24

Physical Layout of the QX chip – Properties

|| 4/7/17QIP II : Quantum algorithms with SC qubits 25

Id0

1

Quantum Experience set of gates

and

Corresponding measurements

O’Malley,P.J.J.etal.(2016),PRX6,031007

Hydrogen Molecule - VQE Implementation

|| 4/7/17QIP II : Quantum algorithms with SC qubits 26

HamiltonianofthemolecularHydrogen

Simulation and Realistic Results

|| 4/7/17QIP II : Quantum algorithms with SC qubits 27https://quantumexperience.ng.bluemix.net/qstage/#/community/question?questionId=ab63d741626debb659994ebedd3f6e80

The first game in a quantum computer…

Decodoku Project : Gaming for science!

||

10 mK

100mK

700mK

4 K

50 K

Arbitrary WaveformGenerators

Inside a QC fridge!

||

Inside a QC fridge!

||

Inside a QC fridge!

||

Andreas Woitzik

Khandelwal Shishir

Panagiotis Barkoutsos

Thanks for your attention!