Problem Set # 5 Solutions

Transcription

1 MIT./8.4/6.898/8.435 Quatum Iformatio Sciece I Fall, 00 Sam Ocko October 5, 00 Problem Set # 5 Solutios. Most uitar trasforms are hard to approimate. (a) We are dealig with boolea fuctios that take bits as iput ad output bits. Each boolea fuctio fuctio has possible iputs, ad it s output for each of these is described b bits. Therefore, sice it takes bits to describe a arbitrar boolea fuctio, meaig that there are differet boolea fuctios which take i bits ad output bits. (b) A circuit of NAND gates ca take at most bits, ad ca be described(redudatl) b a sequece of steps each ivolvig a sigle NAND gate. At each step, there are ( ) N possible was to have a NAND gate(you choose each of the two iputs). Sice there are such steps, there are at most ( ( ) possible circuits. ( ) ) < =, ad thus a classical circuit composed of ad gates ca implemet at most O( ) boolea fuctios. (c) A arbitrar N N matri has N comple degrees of freedom. For a matri to be uitar there are ( ) N + N comple costraits(each pair of colums is orthogoal + Normalizatio of each colum). Therefore, a arbitrar uitar matri has O(N ) degrees of freedom. For a sstem of qubits, N =, ad therefore a arbitrar uitar trasform has O(N ) = O(( ) ) = O( ) degrees of freedom. (d) We use a similar reasoig that we did i part b. A quatum circuit of CNOT, Hadamard, ad T gates ca affect at most qubits ad ma be described(redudatl) b a sequece of steps, each ivolvig a sigle CNOT, Hadamard, or T gate. At each step, there are ( ) + possible gates to appl, ad thus there are at most O(( ) ) = O( ) possible uitar trasforms.

2 Id: hw.te,v.4 009/0/09 04:3:40 ike Ep. Deutsch-Jozsa algorithm ad its geeralizatios. Let f() be a fuctio which maps bits to oe bit, which is implemeted b a uitar trasform U f, satisfig U f = f(), where the first register cotais qubits, ad the secod, oe qubit. (a) Give the quatum states ψ 0, ψ, ψ, ψ 3, arisig i this quatum circuit: j0i = U f ji H f() j 0 i j i j i j 3 i ψ 0 = 0 0 ψ = ψ = ( ψ 3 = = ( ) ( 0 ) ( ( 0 f() f() ) ) ( ) ( ) f() ( 0 ) ) ( ) ( ) f()+ ( 0 ) (b) Whe f() is costat, the ( ) ( )f() = ±, ψ 3 = ± 0 ( 0 ) ad thus we will measure all the bits i register to be 0. Whe f() is balaced, the ( )f() = 0, 0 ψ 3 = 0, ad thus we will ever measure all the bits i register to be 0. (c) Lemma: ( ) = δ Therefore,

3 Id: hw.te,v.4 009/0/09 04:3:40 ike Ep 3 ( ψ 3 = = ( ) ( ) ( ) j+ ( 0 ) ) ( ) δ j, ( 0 ) Ad thus we will alwas measure the first register to be j. ( ) = j ( 0 ) 3. Gates eeded i the quatum Fourier trasform. Give a decompositio of the cotrolled-r k gate ito sigle qubit ad cot gates. This circuit is equivalet to 4. Additio b Fourier trasforms. Cosider the task of costructig a quatum circuit to compute + mod, where is a fied costat, ad 0 <. Show that oe efficiet wa to do this, for values of such as, is to first perform a quatum Fourier trasform, the to appl sigle qubit phase shifts, the a iverse Fourier trasform. What values of ca be added easil this wa, ad how ma operatios are required? To add a umber b a quatum fourier trasform we first appl a quatum fourier trasform, the appl a appropriate set of phase shifts, the reverse the trasform. }{{} QF T k e πik k k e πik(+) }{{} QF T + mod To appl this phase shift we must be able to appl: k e πik k Sice both k ad are

4 Id: hw.te,v.4 009/0/09 04:3:40 ike Ep 4 writte i biar, we write them i terms of their bits: k = = k j j j=0 j j j=0 Therefore: k = j=0 j =0 j+j k j j Ad writte i terms of the qubits of k, we wat to appl the uitar k j j j j πi j k j e j j k j Therefore, we ma decompose this uitar i terms of phase shifts of the qubits k cotrolled b the bits i, where for each qubit k j, bit j, we appl a R j j gate o qubit k j cotrolled b bit j. We otice that R k = I for all k 0, ad thus we do t eed to appl gates for pairs where j + j. For eample, the 3-qubit circuit is: Where R j = ( ) 0 0 e πi j Whe is a power of two, we ol have to appl a few gates. The larger power of two is, the fewer gates we have to appl. For eample, whe =, we ol eed to appl oe gate betwee the Quatum Fourier trasforms.

5 Problem Set #5 (due i class, 4-Oct-0). Most uitar trasforms are hard to approimate. (a) Show that there eist O( ) distict boolea fuctios of bits. (b) Show that a (classical) circuit composed of ad gates ca implemet at most O( ) distict boolea fuctios. (c) Show that a arbitrar uitar trasform applied to qubits is described b O( ) real degrees of freedom. (d) How ma distict uitar trasforms ca be produced b a quatum circuit composed of cotrolled-ot, Hadamard, ad T gates?. Deutsch-Jozsa algorithm ad its geeralizatios. Let f() be a fuctio which maps bits to oe bit, which is implemeted b a uitar trasform U f, satisfig U f = f(), where the first register cotais qubits, ad the secod, oe qubit. (a) Give the quatum states ψ 0, ψ, ψ, ψ 3, arisig i this quatum circuit: j0i = U f ji H f() j 0 i j i j i j 3 i (b) Suppose that we are promised f() is either costat for all possible values of, or balacecd, that is, equal to for eactl half of all the possible, ad 0 for the other half. What results are obtaied for these two cases (costat ad balaced), whe the qubits of the top register are measured? (c) Suppose that the fuctio is f j () = j, where j is a -bit iteger, ad j = k kj k is the biar dot product of the biar represetatios of the two umbers. What result is obtaied whe the qubits of the top register are measured? 3. Gates eeded i the quatum Fourier trasform. Give a decompositio of the cotrolled-r k gate ito sigle qubit ad cot gates. 4. Additio b Fourier trasforms. Cosider the task of costructig a quatum circuit to compute + mod, where is a fied costat, ad 0 <. Show that oe efficiet wa to do this, for values of such as, is to first perform a quatum Fourier trasform, the to appl sigle qubit phase shifts, the a iverse Fourier trasform. What values of ca be added easil this wa, ad how ma operatios are required?

6 Id: hw.te,v.4 009/0/09 04:3:40 ike Ep 6 5. Recet quatum algorithms. Fid a recet paper i the literature about quatum algorithms (theor ot implemetatio), ad write a short (< 500 word) summar of it, o the QIS wiki. See istructios, ad suggestios for sources i the literature, o the course homepage,