UNIVERSALQUANTUMGATESETS&THET-OPERATOR

6.S089FINALPROJECTFRANCISCAVASCONCELOS

UNIVERSALGATESETDEFINITION

• UniversalGateSet:finitesetofgatesthatcanapproximateanyunitarymatrixarbitrarilywell

• Mustbesettowhichany possibleoperationonaquantumcomputerbelongs

• Inotherwords,anyunitaryoperatorcanbeexpressedasfinitesequenceofgatesfromset

• Technicallyimpossible:#ofpossiblequantumgatesisuncountable,whereas#of finitesequencefromfinitesetiscountable

• Onlyrequirethatanyquantumoperationcanbeapproximatedbyasequenceofgatesfromthisfiniteset

• Solovay-Kitaev Theorem guaranteesquantumoperationsforunitaries onaconstant#ofqubits canbeapproximatedefficiently

• Arbitraryhowaccuratetheapproximationmustbe

Linearoperatorwhose inverseisitsadjoint.Productofunitaryoperatorsisunitaryoperators.

=𝜎$ = 0 11 0 = 𝜎' =

0 −𝑖𝑖 0 = 𝜎* =

1 00 −1

|𝟎⟩

|𝟏⟩

= 𝐶𝑁𝑂𝑇 =1 00 1

0 00 0

0 00 0

0 11 0

= 1 00 𝑖 = 1 0

0 𝑒45/7= 1 0

0 −𝑖 = 1 00 𝑒845/7

= 𝐻𝑎𝑑𝑚𝑎𝑟𝑑 = 121 11 −1

|+⟩ = @7 |0⟩ + |1⟩

|−⟩ = @7 |0⟩ − |1⟩

|↺⟩ =12 |0⟩ − 𝑖|1⟩

|↻⟩ =12 |0⟩ + 𝑖|1⟩

CLIFFORDGROUPPAULI

SUPERP

OSITIONS

[𝜋/2 phaseshift]

2-QUBIT

Canbesimulatedefficientlyonclassicalcomputer⇒ NOTuniversal

GOTTESMAN-KNILL THEOREM

• Tellsusthatstabilizercircuitsandevensomehighlyentangledstatescanbeefficiently simulatedonaclassicalcomputer,meaning itisnotuniversal

• Q:Whatisastabilizercircuit?

• A:Aquantumcircuitwiththefollowing elements:

• Preparationofqubits incomputationalbasisstates

• QuantumgatesfromtheCliffordgroup

• Measurementincomputationalbases

• Cannotharnessfullpowerofquantumcomputation⇒ mustincludeatleastonenon-Cliffordgateinourcircuits

T-GATE• Non-Clifford

• Makesitpossible toreachalldifferentpointsoftheBlochSphere

• Byincreasingthe#ofT-gatesinourcircuit(T-depth)wecoverBlochspheremoredenselywithstateswecanreach

= 1 00 𝑒845/F

= 1 00 𝑒45/F

[𝜋/4 phaseshift]

S-gate=T2

2-QUBITUNIVERSALGATESET

• Simpleset:

• Barenco et.al1995:anyunitarymatrixcanbewrittenasacomboofsingle- and2-qubitgates,whereasclassicalreversiblecomputing requires3-bitgates(i.e.Toffoli)

• Inquantumworld,agenericinteractionbetween2qubits (thatcanbeimplementedaccuratelybetweenany2qubits)canbeusedtocalculateanything

{}

TOFFOLI &DEUTSCHGATES1 00 1

0 00 0

0 00 0

1 00 1

0 00 0

0 00 0

0 00 0

0 00 0

0 00 0

0 00 0

0 00 0

0 00 0

1 00 1

0 00 0

0 00 0

0 11 0

• ToffoliGate=CCNOT:universalclassicalreversiblelogicgate

• 3-bitinput&output

• Iffirst2bitsare1,inverts3rd;otherwiseallstaythesame

• Reversible⇒ time-invertible,mappingfromstatestosuccessors is1-to-1

• CanbeusedtobuildsystemsthatperformanydesiredBooleanfunctioncomputation, inreversiblemanner

• DeutschGate:single-gatesetofuniversalquantumgates

• performstransformation:

• ClassicalToffoli gateisreducibletoquantumD(𝝅/𝟐 )

• Meaningallclassicallogicoperationscanbeperformedonuniversalquantumcomputer

|𝑎, 𝑏, 𝑐⟩ ↦ N𝑖𝑐𝑜𝑠 𝜃 |𝑎, 𝑏, 𝑐⟩ + sin 𝜃 |𝑎,𝑏, 1 − 𝑐⟩ 𝑓𝑜𝑟𝑎 = 𝑏 = 1|𝑎, 𝑏, 𝑐⟩ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

IBMQUANTUMEXPERIENCE

• ImplementingaToffoli GateusingH,CNOT,andT

• RunontheIBMQuantumComputer, with1024shots(3units)

RESULTSSIMULATION

EXPERIMEN

T

EXPERIMENT#2• Tryingtoimprovetheresultsfromthefirstexperiment

• ReducingT-depthfrom7to4(wouldbeT-depth5,butweuseSgateinsteadofT2)

• RunontheIBMQuantumComputer, with1024shots(3units)

EXPERIMENTRESULTSEXPERIMEN

T#1

EXPERIMEN

T#2

SOURCES

• ElementaryGatesforQuantumComputation,Barenco et.al(1995)

• QuantumCircuitsofT-DepthOne,Selinger (2013)

• IBMQuantumExperienceUserGuide

• Wikipedia

• CaltechQuantumComputation (Physics/CS219)CourseNotes,Preskill