Short Course in Quantum Information Lecture 4

Qubits and Quantum Circuits

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Course Info • All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/DeutschClasses.html

• Syllabus Lecture 1: Intro Lecture 2: Formal Structure of Quantum Mechanics Lecture 3: Entanglement Lecture 4: Qubits and Quantum Circuits Lecture 5: Algorithms Lecture 6: Error Correction Lecture 7: Physical Implementations Lecture 8: Quantum Cryptography

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Course Info • All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/DeutschClasses.html

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Qubits: The binary quantum system Quantum object in a two-dimensional Hilbert Space, Logical Basis: Two chosen orthogonal states: { 0 , 1 }

$" ' 2 2 General (pure) state: ! = " 0 + # 1 = & ) ! + " = 1 % # ( Physical example: Photon Polarization 0 = H , 1 = V ! " Generally Elliptical

1 !1 $ 1 " 1 % D = D = + 2 #1 & ! 2 $! 1' " % # & 1 !1 $ 1 " 1 % R = # & L = $ ' 2 " i% 2 #! i&

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Spin 1/2

Intrinsic Magnetic Moment of Particles (e.g. proton)

(Planck’s Angular momentum h Like spin ball of charge constant) 2

z-axis !z " 0 N “spin up (along z)” oven S !z " 1 “spin down (along z)”

Stern Gerlach: Measurement of spin along z-axis (two possible outcomes)

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Spin 1/2: Repeated Measurement

z-axis x-axis “spin up (along x)”

!z + "z 1 #1 & !x = = % ( 2 2 $1 '

“spin down (along x)”

"z # !z 1 $ 1 ' !x = = & ) 2 2 %# 1(

Isomorphic to photon polarization measurement: HV followed by D+D-

General State: ! "z + # $z Spin-up/down along some axis

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Bloch Sphere Representation

General (pure) state: ! = " 0 + # 1

Two parameters needed to specify state: relative probability and phase

Let ! = cos(" /2), # = ei$ sin(" /2); 0 % " < &, 0 % $ < 2& 2 2 2 2 0 ! " ! 1 0 ! " ! 1 ! + " = 1

2D Hilbert space is the surface of a sphere 0 ! = " 0 + # 1 ! 1 1 ( 0 + 1 ) ( 0 ! 1 ) 2 " 2

1 I. H. Deutsch, University of New Mexico Short Course in Quantum Information Bloch Sphere Representation

General (pure) state: ! = " 0 + # 1

Two parameters needed to specify state: relative probability and phase

Let ! = cos(" /2), # = ei$ sin(" /2); 0 % " < &, 0 % $ < 2& 2 2 2 2 0 ! " ! 1 0 ! " ! 1 ! + " = 1

2D Hilbert space is the surface of a sphere !z ! = "n

Bloch Sphere !y !x ! ! x y

! z I. H. Deutsch, University of New Mexico Short Course in Quantum Information Bloch Sphere Representation

General (pure) state: ! = " 0 + # 1

Two parameters needed to specify state: relative probability and phase

Let ! = cos(" /2), # = ei$ sin(" /2); 0 % " < &, 0 % $ < 2& 2 2 2 2 0 ! " ! 1 0 ! " ! 1 ! + " = 1

2D Hilbert space is the surface of a sphere

H r ! = "

L Poincaré Sphere D D + R !

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Pauli Matrices

"0 1% "0 (i% "1 0 % X = ! x = $ ', Y = ! y = $ ', Z = ! z = $ ' #1 0& # i 0 & #0 (1&

For spin-1/2, these correspond to observables that can be measured in a Stern-Gerlach analysis along x,y,or z axis respectively. , Eigenvalues, ±1. Eigenvectors, { !x,y,z "x,y,z }

Incompatible Measurements. Commutator: , 2i [! i ! j ] = "ijk! k e.g. XY ! YX = 2iZ Other useful properties: • Anticommute: XY = !YX, XZ = !ZX, YZ = !ZY 2 2 2 • Hermitian and Unitary: X = Y = Z = I

I. H. Deutsch, University of New Mexico Short Course in Quantum Information General Pauli Observable

! n = cos" Z + sin"(cos# X + sin# Y)

Eigenvalues: ±1. Eigenvectors !n

Quantum Dynamics (closed system): Unitary Matrix

Rotation on the Bloch Sphere (axis/ angle)

Rn (!) = exp{"i!# n /2} n

= cos(! /2)I " isin(! /2)# n #

i! U(2): e Rn (")

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Example: Hadamard Matrix

# ! ! # X + Z& & X + Z 1 )1 1 , H = iRe +e (!) = i% cos I " isin % ( ( = = x y $ 2 2 $ 2 ' ' 2 2 +1 "1. 2 * -

ez 1 e + e (ex + ez ) Rotation about x z by 180° 2 2 e x Change of basis X-->Z † 1 HZH = X H 0 = ( 0 + 1 ) = !x " + 2 HXH† Z 1 = H 1 = ( 0 ! 1 ) = "x # ! 2 I. H. Deutsch, University of New Mexico Short Course in Quantum Information Example: Pauli Operations as Unitaries #0 1& X = iRx (!) = iexp("i! X /2) = % ( $1 0'

X 0 = 1 X 1 = 0 Bit Flip (“NOT”) #1 0 & Z = iRz (!) = iexp("i! Z /2) = %0 "1( $ ' Z 0 = 0 Z 1 = ! 1 Sign Change, Phase Flip Note: Action of X and Z reversed in “Hadmard Basis” X + = X 0 + 1 = X 0 + X 1 = 1 + 0 = + ( ) X ! = X( 0 ! 1 ) = X 0 ! X 1 = 1 ! 0 = ! ! Z + = X( 0 + 1 ) = Z 0 + Z 1 = 0 ! 1 = ! Z ! = Z( 0 ! 1 ) = Z 0 ! Z 1 = 0 + 1 = + I. H. Deutsch, University of New Mexico Short Course in Quantum Information Multiple Qubits

n qubits = n two-level quantum systems (e.g. n spin-1/2 particles)

Tensor product Hilbert space:

Dimension orthogonal states

n Logical Basis: x x = 0,1, ,2 !1 x in binary. { K }

E.g. 0 ! 0 0 0 , 1 ! 0 0 1 , 2 ! 0 1 0 , 3 ! 0 1 1 n = 3, N = 8 4 ! 1 0 0 , 5 ! 1 0 1 , 6 ! 1 1 0 , 7 ! 1 1 1 Photon polarization: 4 = V H H 2n "1 General State: ! = #cx x Can be entangled x=0

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Bell Basis

Bipartite (two qubits) Logical Basis: { 00 , 01 , 10 , 11 } Simulate Eigenvectors of: Z ! I, I ! Z Two bits specify state

( ) 1 ( ) 1 ! + = ( 0,0 + 1,1 ), ! " = ( 0,0 " 1,1 ) Bell Basis 2 2 1 1 #(+) = ( 0,1 + 1,0 ), #(") = ( 0,1 " 1,0 ) 2 2

Eigenvectors of X ! X Z ! Z Eigenvalues ±1 Eigenvalues ±1 (phase bit) (parity bit) I. H. Deutsch, University of New Mexico Short Course in Quantum Information Global Properties in Joint Operators

Z ! Z "(±) = Z 0 ! Z 0 ± Z 1 ! Z 1 = 0,0 ± 1,1 = + "(±)

Z ! Z "(±) = Z 0 ! Z 1 ± Z 0 ! Z 1 = # 0,0 1,1 = # "(±) m (±) (±) X ! X " = X 0 ! X 0 ± X 1 ! X 1 = 1,1 ± 0,0 = ± " (±) (±) X ! X " = X 0 ! X 1 ± X 1 ! X 0 = 1,0 ± 0,1 = ± "

Eigenvalue of Z ! Z . Are you $ or %? (parity bit) Eigenvalue of X ! X. Are you + or -? (phase bit)

phase bit a X ! X "ab = (#1) "ab Alternative notation: !ab parity bit b Z ! Z "ab = (#1) "ab

(+) (+) ($) ($) !00 = " , !01 = # , !10 = " , !11 = # I. H. Deutsch, University of New Mexico Short Course in Quantum Information Entanglement as a Resource:

Suppose Alice and Bob share prior entanglement.

Can Alice communicate with Bob in ways not possible classically?

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Entanglement as a Resource:

Suppose Alice and Bob share prior entanglement.

Can Alice communicate with Bob in ways not possible classically?

!(+)

Bob Alice Yes! But no information transfer faster than speed of light (causality).

E.g. Superdense Coding

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Superdense Coding

• Alice wants to send Bob two bits of information.

• Message one of four secrets: {00 = I, 01=X, 10=Z, 11=Y}.

• She has in her possession one qubit. If she manipulates it any way she likes, sends this to Bob, and he measures it, he can learn no more than ONE bit of information.

• If Alice’s qubit is entangled with one already in Bob’s possession, and he performs a joint measurement, he can extract TWO bits of information.

• Alice thus encodes a two-bit message in her one qubit, correlated with Bob’s qubit ---> superdense coding! Like effecting a four sized die while only manipulating a two-sized object.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Local operation changes global state

(+) (+) X A ! IB " = X A 0 ! IB 0 + X A 1 ! IB 1 = 1,0 + 0,1 = #

(+) (#) ZA ! IB " = ZA 0 ! IB 0 + ZA 1 ! IB 1 = 0,0 # 1,1 = " (+) (#) YA ! IB " = YA 0 ! IB 0 + YA 1 ! IB 1 = i( 0,0 # 1,1 ) = i $ irrelevant overall phase

• By acting on her qubit locally, Alice can effect the state of the joint state.

• Bob can detect this action with ONLY his qubit. Must measure BOTH qubits.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Superdense Coding

I got $(+), Alice must have applied I (identity) to her qubit.

0 phase parity Bell meter 0 !(+)

Bob Alice

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Superdense Coding

I got $(-), Alice must have applied X to her qubit.

0 phase parity X Bell meter 1 !(+)

Bob Alice

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Superdense Coding

I got %(+), Alice must have applied Z to her qubit.

1 phase parity Z Bell meter 0 !(+)

Bob Alice

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Superdense Coding

I got %(-), Alice must have applied Y to her qubit.

1 phase parity Y Bell meter 1 !(+)

Bob Alice

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Quantum Teleportation

Quantum Channel

Bob Classical Channel Alice

Alice wants to send Bob a qubit in an unknown (pure) state. The quantum channel connecting them is noisy so the qubit would be corrupted if she transmitted. She cannot measure the qubit and then call Bob to have give make his own copy because her measurement will project it onto an eigenstate.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Quantum Teleportation

Clean Quantum Channel Z phase parity ! Bell met!er 1 0 !(+)

Bob Classical Channel Alice

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Quantum Teleporation: The Math

Three qubit joint state of Alice, Bob, and “Victor” who prepared ! :

! = " # $(+) = % 0 + & 1 # 0 # 0 + 1 # 1 V AB ( V V ) ( A B A B ) 1 = 0 ! 0 + 1 ! 1 ! " 0 + # 1 + 0 ! 0 $ 1 ! 1 ! " 0 $ # 1 2[( V A V A ) ( B B ) ( V A V A ) ( B B ) + 0 ! 1 + 1 ! 0 ! " 1 + # 0 + 0 ! 1 $ 1 ! 0 ! " 1 $ # 0 ( V A V A ) ( B B ) ( V A V A ) ( B B )]

1 ! = "(+) # $ + "(%) # Z $ + &(+) # X $ % i &(%) #Y $ 2 ( VA B VA B VA B VA B ) A measurement by Alice in Bell-basis leaves Bob’s qubit in one of four states: , Z , X , Y {! B ! B ! B ! B }

Alice uses the classical channel to tell Bob which Bell state she found. Bob can then put his qubit in the unknown state, through application of a Pauli.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Digital (Classical) Information Processing

Classical Computing: Function on n-bit string F({0,1}N ) = {0,1}M Function constructed from operations on small collections of bits “Logic Gates” XOR NAND NOT

in- in- out in- in- out in out a b a b 0 1 0 0 0 0 0 1 1 0 0 1 1 0 1 1

1 0 1 1 0 1

1 1 0 1 1 0

Note: Not is reversible. XOR and NAND are not. Can be made reversible with extra bits. NAND (plus copy) is UNIVERSAL for computation.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Digital Quantum Information Processing

Map on n-qubits is a 2nX2n unitary matrix - Reversible.

Single qubit logic gates: Rotations on the Bloch sphere.

0 1 E.g. " % NOT = X = iRx (!) = $ ' #1 0&

i! / 4 i! / 4 e # 1 "i& NOT = e Rx (! /2) = % ( 2 $" i 1 '

"1 1 % Hadamard H = $ ' #1 !1&

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Logic Tables and Quantum Circuits

Action of unitary determined by action on logical basis:

X x = x H x = x + (!1)x x

in out in out 0 1 0 0 + 1 1 0 1 0 ! 1

Quantum Circuit X

Rn(") H

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Two Qubit Logic

Classical XOR: (x,y) ! y " x (binary addition)

Classical Reversible XOR: (x,y) ! (x, y " x) control target in out C T C T “Control NOT”: 0 0 0 0 Flip the target when 0 1 0 1 Control is 1. 1 0 1 1 1 1 1 0

Transformation on logical basis states is a unitary matrix!

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Controlled Unitaries: If-Then in QC

Apply U to the second qubit when the first qubit is logical-1. U !1 0 0 0$ Matrix in logical basis: #0 1 0 0& C = # & U 0 0 00 , 01 , 10 , 11 # U & { } "#0 0 %&

CNOT=C CPhase=C X Z

!1 0 0 0$ "1 0 0 0 % #0 1 0 0& $ ' CNOT # & 0 1 0 0 = 0 0 0 1 CPhase = $ ' # & $0 0 1 0 ' #0 0 1 0& " % #$0 0 0 !1&'

I. H. Deutsch, University of New Mexico Short Course in Quantum Information CNOT is Entangling

0 H Consider the following circuit 0

! in !1 ! out

! in = 0 0 !1 = ( 0 + 1 ) 0 = 0 0 + 1 0 0 0 1 1 (+) ! out = + = " Entangled State

By putting the control in a quantum super position, the circuit undergoes a superposition of classical computations. Quantum Parallelism

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Universal Quantum Logic

An arbitrary unitary matrix on n-qubits can be constructed from a sequence of unitaries on subsets of qubits.

Universal set. All single qubit rotations + one entangling two-qubit unitary (e.g. CNOT) acting between pairs on a connected graph.

Note: Any single qubit unitary can be constructed from discrete set. E.g. {H, T ! Rz (" /4)}

Note: Arbitrary unitary will require exponential number of gates. Quantum algorithm = Efficiently implementable useful unitary that scales like a polynomial in n, for which the answer can be found by measuring in logical basis with high probability.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information