Notes on Quantum Computing

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1 Notes o Quatum Computig Maris Ozols May 0, 01 Cotets 1 Mathematics of quatum iformatio 1.1 Basics Bell basis, teleportatoi ad superdese codig Measuremets Decompositios ad ormal forms Pauli matrices ad Bloch sphere Elemetary circuit idetities Trace distace Fidelity ad Uhlma s theorem Quatum operatios Quatum gates, circuit model, ad uiversality Gives rotatios Elemetary quatum algorithms 7.1 Phase kickback Deutsch s algorithm Deutsch Jozsa algorithm Berstei Vazirai problem Simo s algorithm Quatum Fourier trasform ad phase estimatio 9 4 Shor s algorithm for factorig Period fidig Grover s quatum search algorithm 10 6 Computatioal complexity 10 7 Quatum error correctio ad fault tolerace Quatum error correctio The Shor code

2 8 Quatum iformatio theory ad basic commuicatio protocols Resource tradeoffs Nayak s boud Mathematics of quatum iformatio 1.1 Basics Bell basis, teleportatoi ad superdese codig Bell basis states: β xy = 0, y + ( 1x 1, ȳ (1 Preparatio of a Bell basis state (H o the first qubit, followed by CNOT: x H y (!!! p. 5 i NC!!! 1.1. Measuremets Geeral measuremet: p m = Tr(M m ρm m ρ m = M mρm m p m (3!!! POVM:!!! Fact (Priciple of deferred measuremet. Measuremets ca always be moved from a itermediate stage of a quatum circuit to the ed of the circuit. Ay classically cotrolled operatios that use the measuremet results ca be replaced by coditioal quatum operatios Decompositios ad ormal forms Fact. (A I φ = (I A T φ where φ = i i i is the maximally etagled state ad A M (C. This follows by projectig both sides o j k. Theorem (Schmidt decompositio. If ψ H A H B, the there exist orthoormal bases { i A } i ad { i B } i for H A ad H B, respectively, such that ψ = i λ i i A i B. (4 Note: oe ca take the first basis ad the coefficiets to be the eigevectors ad square roots of the eigevalues of the reduced state Tr B ( ψ ψ, respectively.

3 Theorem (Purificatio. If ρ is a mixed state o system A, the there is a system B ad a pure state ψ o AB such that ρ = Tr B ( ψ ψ. (5 Lemma (Geeral purificatio. If ψ is a purificatio of ρ, the it ca be writte i the form ψ = (ρ 1/ U φ (6 for some uitary U. Theorem (Polar decompositio. Ay A M (C ca be writte i the form A = UP = QU, (7 where P, Q 0, U U(. I particular, P = A A ad Q = AA Pauli matrices ad Bloch sphere Pauli matrices are: X = ( Y = ( 0 i i 0 Z = ( (8 Ay sigle qubit desity matrix ca be writte as ρ = 1 (I + r σ (9 Fact. If A = I, the e iθa = cos(θa + i si(θa = I cos θ + ia si θ. Fact. ( r σ = r I so r σ has eigevalues ± r. Fact. Rotatio aroud a uit vector r by agle α is give by e i α ( r σ = I cos α i( r σ si α. ( Elemetary circuit idetities = SWAP (11 = Z (1 Z = V U V V UV (13 H H = (14 H H 3

4 1. Trace distace Trace distace: For qubits: Tricks: D(p, q := 1 D(ρ, σ := 1 x X If P, Q 0, the Tr(P Q 0, p x q x = max(p(s q(s. (15 S X Tr ρ σ = max I P 0 Tr( P (ρ σ. (16 F (r 1, r = 1 r 1 r. (17 If I P 0 ad Q 0, the Tr(Q Tr(P Q, ρ σ = Q S, where Q, S 0 have orthogoal supports. Theorem. Let {E m } be a POVM. The where p m := Tr(ρE m, ad q m := Tr(ρE m. D(ρ, σ = max {E m} D({p m}, {q m }, ( Fidelity ad Uhlma s theorem Fidelity: F (p, q := x X px q x = p q (19 F (ρ, σ := Tr ρ 1/ σρ 1/ (0 A ad A have the same sigular values, therefore Tr A = Tr A. Fidelity is symmetric, sice For qubits: F (r 1, r := 1 F (ρ, σ = Tr ρ 1/ σρ 1/ (1 = Tr (σ 1/ ρ 1/ σ 1/ ρ 1/ ( = Tr σ 1/ ρ 1/ (3 = Tr (ρ 1/ σ 1/ (4 = Tr ρ 1/ σ 1/ (5 = F (σ, ρ. (6 ( 1 + r 1 r + (1 r 1 (1 r (7 4

5 Lemma. If A is ay operator ad U is uitary, the Tr(AU Tr A. Theorem (Uhlma. 1.4 Quatum operatios F (ρ, σ = max ψ ϕ. (8 ψ, ϕ Differet represetatios of a geeral quatum operatio E: 1. Stiesprig represetatio: E(ρ = Tr B ( U(ρ 0 0 U.. Kraus represetatio: E(ρ = k E kρe k, where k E k E k = I. 3. Choi-Jamio lkowski represetatio: E(ρ = Tr B ( JE (I ρ T, where J E := (E I( φ φ = ij E( i j i j. 4. Completely positive ad trace preservig: (E I(ρ 0 ad Tr E(ρ = Tr ρ for all ρ. How to covert betwee these represetatios: Stiesprig Kraus: E k := (I k U(I 0 Kraus Stiesprig: U(I 0 := k E k k Physical iterpretatio of Kraus represetatio: E(ρ = k p kρ k, where p k := Tr(E k ρe k ad ρ k := E kρe k Tr(E k ρe (9 k. 1.5 Quatum gates, circuit model, ad uiversality Theorem (Z-Y decompositio. For ay U U( there exist α, β, γ, δ R such that U = e iα R z (βr y (γr z (δ, where R k (θ = e i θ σ k. Theorem. For ay U U( there exist A, B, C U( such that ABC = I ad U = e iα AXBXC for some α R. Give such decompositio for U, we ca implemet c-u as follows: = ( e iα (30 U C B A Note that ( e iα = e iα I (31 5

6 If V = U the we ca implemet cc-u as follows: (3 = U V V V Note: the marked regio performs V cotrolled o XOR of the first two bits. More cotrols ca be added usig workspace qubits iitialized i 0 : = U U (33 We ca add more cotrols to Toffoli gate without imposig ay restrictios o the iitial state of the workspace: = w 1 w (34 Note: the marked gates act as follows: ivert w 1 if ad oly if the first bits are set to 1, ivert w if ad oly if the first 3 bits are set to Gives rotatios If α 0 ad β 0 the G(α,β {}}( { ( ( 1 α β α = α + β α + β β α β 0 (35 6

7 Note that G(α, β SU(. I geeral a two-level uitary α β G ij (α, β =... α + β β α... 1 (36 is called Gives rotatio. If U U( the for appropriately chose parameters α ad β we have ( 1 0 G 1... G 13 G 1 U = 0 U (37 where U U( 1. By recursively applyig this procedure we obtai a diagoal matrix of the form diag(1,..., 1, det U.!!! implemetig a -level uitary (Gray code!!! Elemetary quatum algorithms.1 Phase kickback If a {0, 1} the X a = ( 1 a. (38 If f : {0, 1} {0, 1} ad U f x y = x y f(x the. Deutsch s algorithm U f x = ( 1 f(x x. (39 Problem. Determie whether f : {0, 1} {0, 1} is costat or balaced. Equivaletly, compute f(0 f(1. Circuit. Aalysis. 0 H U f H 1 H H 1 + U f 1 (( 1 f(0 0 + ( 1 f(1 1 (40 (41 =( 1 f(0 1 ( 0 + ( 1 f(0 f(1 1 }{{} (4 + or depedig o f(0 f(1 7

8 .3 Deutsch Jozsa algorithm Problem. Determie whether f : {0, 1} {0, 1} is costat or balaced. Circuit. Aalysis. Recall the formula: 0 H U f H 1 H H 1 (43 H x = 1 1 ( 1 x y y (44 y=0 We have + = 1 U f 1 H (+1 1 x (45 x {0,1} ( 1 f(x x (46 x {0,1} x,y {0,1} ( 1 x y+f(x y 1 (47 The amplitude for y = 0 is give by 1 ( 1 f(x = x {0,1} { ±1 if f is costat 0 if f is balaced (48.4 Berstei Vazirai problem Problem. Determie s {0, 1} by queryig f : {0, 1} {0, 1} give by f( x = s x = s 1 x 1 s x s x (49 Circuit. 0 H U f H (50 1 H H 1 Aalysis. Apply the Hadamard trasform formula i the backward directio: + U f 1 H (+1 1 x {0,1} ( 1 x s x (51 x,y {0,1} ( 1 x ( y+ s y 1 (5 The amplitude for y = s is clearly 1, so the other amplitudes must be 0. 8

9 .5 Simo s algorithm 3 Quatum Fourier trasform ad phase estimatio Let y/ = l=1 y l l = 0.y 1 y... y. The QFT x = 1 1 e πixy/ y (53 = 1 = y=0 y {0,1} e πix ( 0 + e πix l 1 l=1 4 Shor s algorithm for factorig 4.1 Period fidig l=1 y l l y (54 f : {0, 1,..., 1} {0, 1} m. Promise: f(x = f(y x y (mod b. ( m QFT 1 1 x 0 (56 x=0 U f 1 1 x f(x (57 If we get outcome f(x 0 after measurig the d register we get, the state that is left over is k 1 1 x 0 + jr, (58 k j=0 x=0 where k { r, r }. Applyig QFT 1 we get 1 k 1 k 1 y=0 j=0 e πi (x0+jr y. (59 If r, the Pr(y = 1 si πyrk k si πyr (60 9

10 5 Grover s quatum search algorithm 6 Computatioal complexity I what follows DTM stads for a Determiistic Turig Machie. Defiitio. A promise problem is a pair A = (A yes, A o, where A yes, A o {0, 1} ad A yes A o =. Compute Verify Determiistic P, PSPACE NP Probabilistic BPP, PP MA Quatum BQP, BPP QMA Table 1: The success probabilities of umerical QRACs.!!! PICTURE!!! Most of the followig defiitios start with A promise problem A = (A yes, A o is i [complexity class] if ad oly if.... Defiitio (P.... there exists a DTM M that rus i polyomial time such that x A yes : M (x = 1, x A o : M (x = 0. Defiitio (PSPACE. Similar to P, except M rus i polyomial space. Defiitio (NP.... there exists a DTM M that rus i polyomial time ad a polyomial p such that x A yes y {0, 1} p( x : M (x, y = 1 (completeess, x A o y {0, 1} p( x : M (x, y = 0 (soudess. Defiitio (PP.... there exists a DTM M that rus i polyomial time ad a polyomial p such that x A yes : {r {0, 1} p( x : M (x, r = 1} / p( x > 1, x A o : {r {0, 1} p( x : M (x, r = 1} / p( x 1. Defiitio (BPP. Similar to PP, except the probabilities are... 3 ad 1 3, respectively. Defiitio (MA.... there exists a DTM M that rus i polyomial time ad polyomials p ad q such that x A yes y {0, 1} p( x : {r {0, 1} q( x : M (x, y, r = 1} 3 (completeess, 10

11 x A o y {0, 1} p( x : {r {0, 1} q( x : M (x, y, r = 1} 1 3 (soudess. Defiitio (BQP.... there exists a polyomial-time geerated family of quatum circuits Q = {Q : N}, where each circuit Q takes iput qubits ad produces oe output qubit, such that x A yes : Pr[Q x accepts x] 3, x A o : Pr[Q x accepts x] 1 3. Defiitio (QMA.... there exists a polyomial-time geerated family of quatum circuits Q = {Q : N}, where each circuit Q takes + p( iput qubits ad produces oe output qubit, such that x A yes ρ D( p( x : Pr[Q x accepts (x, ρ] 3 (completeess, x A o ρ D( p( x : Pr[Q x accepts (x, ρ] 1 3 (soudess, where D(d stads for the set of all d d desity matrices. 7 Quatum error correctio ad fault tolerace Actio via cojugatio: H : X Z H : Z X H : Y Y (61 S : X Y S : Y X S : Z Z (6 7.1 Quatum error correctio CNOT : X I X X (63 CNOT : I X I X (64 CNOT : Z I Z I (65 CNOT : I Z Z Z (66 Theorem (Quatum error-correctio coditio. Let C be a quatum code ad Π C the projector oto the code subspace, ad E = {E i } a quatum operatio. A operatio for correctig E o C exists if ad oly if for some Hermitia matrix α. Π C E i E jπ C = α ij Π C (67 Theorem (Discretizatio of errors. Let C be a quatum code ad R be the error-correctio operatio to recover from E = {E i } costructed i the proof of the previous theorem. If F = {F j } where F j = i m jie i for some complex matrix m, the R also corrects for F o the code C. 11

12 7.1.1 The Shor code ( = 0 L ( ( = 1 L (69 8 Quatum iformatio theory ad basic commuicatio protocols 8.1 Resource tradeoffs Let x {0, 1}. The ability to perform the correspodig trasformatio for ay basis vector is a resource: qubit: x A x B, cbit: x A x B x E, cobit: x A x A x B. Trivial iequalities: 1 qubit 1 cobit 1 cbit. (70 (Alice ca perform a CNOT to create a coheret copy of the state i stadard basis, ad sed oe half to Bob. Alice ca discard her half of the coheret bit to get a classical bit. Ability to trasmit coheret bits ca be used to geerate etaglemet by usig + as iput: Irreversible trasformatios: 1 qubit + 1 ebit cbits (superdese codig, cbits + 1 ebit 1 qubit (teleportatio. Reversible trasformatios give a catalyst ebit: (1 qubit + 1 ebit ( cobits, 1 cobit 1 ebit. (71 ( cobits + 1 ebit (1 qubit + 1 ebit + 1 ebit (sed over a coheret copy without measurig it. (Before ruig the superdese codig protocol, Alice makes local copies of her two classical bits; this does ot require the catalyst ebit. Alice performs a uitary that maps Bell basis to stadard basis (see circuit i Eq. ( o the qubit i a ukow state ad her half of the ebit; istead of measurig ad trasmittig two classical bits, she uses coheret commuicatio; coditioal o the two received coheret bits, Bob corrects his half of the ebit; they ed up 1

13 geeratig two ebits from the coheret commuicatio, ad Bob also eds up havig the ukow state. Coclusio: ( cobits + 1 ebit = (1 qubit + 1 ebit + 1 ebit (7 where the ebit is used as a catalyst. 8. Nayak s boud Theorem. If X {0, 1} m is draw uiformly at radom, ecoded i qubits, ad recovered to Y, the probability that X = Y is at most / m. Proof. Let {Π x : x {0, 1} m } be a orthoormal measuremet i (C, i.e., a set of projectors that sum to idetity, ad φ x (C the ecodig of x. The Pr[X = Y ] = 1 m Π x φ x x {0,1} m (73 = 1 ( m Tr Πx φ x φ x x {0,1} m (74 1 m Tr(Π x Π C x {0,1} m (75 = 1 m Tr Π C (76 = m. (77 13