{"id":4101,"date":"2024-09-21T14:44:28","date_gmt":"2024-09-21T14:44:28","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4101"},"modified":"2024-09-21T14:44:28","modified_gmt":"2024-09-21T14:44:28","slug":"problems-2","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/21\/problems-2\/","title":{"rendered":"Problems"},"content":{"rendered":"\n

Some texts implement the quantum function evaluator as a \u201ccontrolled-<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

U<\/p>\n\n\n\n

f<\/p>\n\n\n\n

\u201d<\/p>\n\n\n\n

gate (Figure 8.19), where<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

U<\/p>\n\n\n\n

f<\/p>\n\n\n\n

acts only on the lower register, and is de\ufb01ned<\/p>\n\n\n\n

by<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

U<\/p>\n\n\n\n

f<\/p>\n\n\n\n

|yi = |y \u2295f(x)i:<\/p>\n\n\n\n

|xi<\/p>\n\n\n\n

\u2022<\/p>\n\n\n\n

|xi<\/p>\n\n\n\n

|yi<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

U<\/p>\n\n\n\n

f<\/p>\n\n\n\n

|y \u2295 f(x)i<\/p>\n\n\n\n

FIGURE 8.19: The quantum function evaluator as a controlled<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

U<\/p>\n\n\n\n

f<\/p>\n\n\n\n

gate.<\/p>\n\n\n\n

How is the action of this implementation di\ufb00erent from the f-controlled<\/p>\n\n\n\n

NOT gate of Figure 7.15? Check by using standard basis states as well as<\/p>\n\n\n\n

superpositions as inputs.<\/p>\n\n\n\n

8.2. Show that the phase kickback trick works because the input state in the<\/p>\n\n\n\n

bottom register is an eigenstate of the<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

U<\/p>\n\n\n\n

f<\/p>\n\n\n\n

operator for the Deutsch algo-<\/p>\n\n\n\n

rithm.<\/p>\n\n\n\n

8.3. Deutsch\u2019s original version of his algorithm used |0i as the input to the<\/p>\n\n\n\n

bottom register instead of |0i \u2212 |1i. Show that in this case you obtain<\/p>\n\n\n\n

the correct answer with probability 3\/4. Also show that the algorithm has<\/p>\n\n\n\n

probability 1\/2 of succeeding.<\/p>\n\n\n\n

8.4. Prove the shift-invariance property of the Fourier transform, i.e., show that<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

F|x + ki = e<\/p>\n\n\n\n

i\u03b8<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

F|xi (8.88)<\/p>\n\n\n\n

for some \u03b8. Find \u03b8 in terms of k.<\/p>\n\n\n\n

8.5. For the operator R<\/p>\n\n\n\n

d<\/p>\n\n\n\n

of Equation 8.45, give a construction for the controlled<\/p>\n\n\n\n

R<\/p>\n\n\n\n

d<\/p>\n\n\n\n

gate using CNOT and single-qubit gates.<\/p>\n\n\n\n

8.6. Find the eigenvalues and eigenvectors of the matrix R<\/p>\n\n\n\n

d<\/p>\n\n\n\n

. What can you say<\/p>\n\n\n\n

about the commutators (i) [R<\/p>\n\n\n\n

d<\/p>\n\n\n\n

, X] (ii) [R<\/p>\n\n\n\n

d<\/p>\n\n\n\n

, Y ] (iii) [R<\/p>\n\n\n\n

d<\/p>\n\n\n\n

, Z] (iv) [R<\/p>\n\n\n\n

d<\/p>\n\n\n\n

, R<\/p>\n\n\n\n

0<\/p>\n\n\n\n

d<\/p>\n\n\n\n

] ?<\/p>\n\n\n\n

8.7. Work out a circuit that calculates the inverse quantum Fourier transform.<\/p>\n\n\n\n

8.8. Consider a periodic function f(x + r) = f(x) for 0 \u2264 x < N where N is<\/p>\n\n\n\n

\"\"<\/p>\n\n\n\n

174 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n

an integer multiple of r. Suppose you are given a unitary operator U<\/p>\n\n\n\n

y<\/p>\n\n\n\n

that<\/p>\n\n\n\n

performs the transformation U<\/p>\n\n\n\n

y<\/p>\n\n\n\n

|f(x)i = |f (x + y)i. Show that the state<\/p>\n\n\n\n

|<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

f(k)i =<\/p>\n\n\n\n

1<\/p>\n\n\n\n

\u221a<\/p>\n\n\n\n

N<\/p>\n\n\n\n

N\u22121<\/p>\n\n\n\n

X<\/p>\n\n\n\n

x=0<\/p>\n\n\n\n

e<\/p>\n\n\n\n

\u22122\u03c0ikx\/N<\/p>\n\n\n\n

|f(x)i (8.89)<\/p>\n\n\n\n

is an eigenvector of U<\/p>\n\n\n\n

y<\/p>\n\n\n\n

. Calculate the corresponding eigenvalue.<\/p>\n\n\n\n

8.9. Compute the output of the controlled-QFT gate<\/p>\n\n\n\n

shown in the \ufb01gure if the input is H<\/p>\n\n\n\n

\u22973<\/p>\n\n\n\n

|xi.<\/p>\n\n\n\n

\u2022<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

F<\/p>\n\n\n\n

8.10. On examining the period \ufb01nding algorithm, we can \ufb01nd a relationship with<\/p>\n\n\n\n

the phase-estimation algorithm. On applying the oracle, we get<\/p>\n\n\n\n

1<\/p>\n\n\n\n

\u221a<\/p>\n\n\n\n

N<\/p>\n\n\n\n

X<\/p>\n\n\n\n

|xi|0i<\/p>\n\n\n\n

U<\/p>\n\n\n\n

f<\/p>\n\n\n\n

\u2212\u2212\u2192<\/p>\n\n\n\n

1<\/p>\n\n\n\n

\u221a<\/p>\n\n\n\n

N<\/p>\n\n\n\n

X<\/p>\n\n\n\n

|f(x)i.<\/p>\n\n\n\n

Express |f (x)i in terms of its Fourier transform, |<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

f(k)i. Invert this expres-<\/p>\n\n\n\n

sion and show that |<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

f(k)i are of the same form as Equation 8.89 of Problem<\/p>\n\n\n\n

8.8. Now show that the period \ufb01nding algorithm is the phase estimation for<\/p>\n\n\n\n

the operator U<\/p>\n\n\n\n

y<\/p>\n\n\n\n

de\ufb01ned there.<\/p>\n\n\n\n

8.11. Apply the quantum phase estimation algorithm to the following cases and<\/p>\n\n\n\n

obtain the results:<\/p>\n\n\n\n

(a) U = X, |ui = |\u2212i, t = 2,<\/p>\n\n\n\n

(b) U = R<\/p>\n\n\n\n

d<\/p>\n\n\n\n

, |ui = |1i, t = d + 1.<\/p>\n","protected":false},"excerpt":{"rendered":"

Some texts implement the quantum function evaluator as a \u201ccontrolled- \u02dc U f \u201d gate (Figure 8.19), where \u02dc U f acts only on the lower register, and is de\ufb01ned by \u02dc U f |yi = |y \u2295f(x)i: |xi \u2022 |xi |yi \u02dc U f |y \u2295 f(x)i FIGURE 8.19: The quantum function evaluator as […]<\/p>\n","protected":false},"author":1,"featured_media":4042,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[497],"tags":[],"class_list":["post-4101","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-quantum-algorithms"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/algorithm-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4101","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4101"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4101\/revisions"}],"predecessor-version":[{"id":4102,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4101\/revisions\/4102"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4042"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4101"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4101"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4101"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}