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 • 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 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 ! V 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 ! 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,K,2 !1} x in binary. 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 m 1,1 = # " (±) (±) 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.