{"id":4080,"date":"2024-09-21T12:58:09","date_gmt":"2024-09-21T12:58:09","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4080"},"modified":"2024-09-21T12:58:10","modified_gmt":"2024-09-21T12:58:10","slug":"problems","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/21\/problems\/","title":{"rendered":"Problems"},"content":{"rendered":"\n

Show that the n-qubit Hadamard gate acts as<\/p>\n\n\n\n

H<\/p>\n\n\n\n

\u2297n<\/p>\n\n\n\n

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

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

=<\/p>\n\n\n\n

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

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

2<\/p>\n\n\n\n

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

2<\/p>\n\n\n\n

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

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

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

y=1<\/p>\n\n\n\n

(\u22121)<\/p>\n\n\n\n

x\u00b7y<\/p>\n\n\n\n

|yi. (7.28)<\/p>\n\n\n\n

where x \u00b7 y is the bitwise product of x and y:<\/p>\n\n\n\n

x \u00b7 y = x<\/p>\n\n\n\n

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

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

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

\u2295 x<\/p>\n\n\n\n

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

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

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

\u2295 . . . x<\/p>\n\n\n\n

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

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

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

. (7.29)<\/p>\n\n\n\n

7.2. Often it helps to simplify circuits when we can identify equivalences be-<\/p>\n\n\n\n

tween some combinations of gates. Prove, for example, the following circuit<\/p>\n\n\n\n

identities:<\/p>\n\n\n\n

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

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

a) HXH = Z<\/p>\n\n\n\n

(b) HY H = \u2212Y<\/p>\n\n\n\n

(c) HZH = X<\/p>\n\n\n\n

7.3. Show the following relations concerning rotation matrices:<\/p>\n\n\n\n

(a) R<\/p>\n\n\n\n

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

(\u03b8<\/p>\n\n\n\n

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

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

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

(\u03b8<\/p>\n\n\n\n

2<\/p>\n\n\n\n

) = R<\/p>\n\n\n\n

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

(\u03b8<\/p>\n\n\n\n

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

+ \u03b8<\/p>\n\n\n\n

2<\/p>\n\n\n\n

)<\/p>\n\n\n\n

(b) XR<\/p>\n\n\n\n

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

(\u03b8)X = R<\/p>\n\n\n\n

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

(\u2212\u03b8)<\/p>\n\n\n\n

7.4. The \u201cSWAP\u201d gate S interchanges two inputs, de\ufb01ned by<\/p>\n\n\n\n

S|xyi = |yxi.<\/p>\n\n\n\n

(a) Give the matrix representing this gate.<\/p>\n\n\n\n

(b) Show that it can be implemented by 3 CNOT gates as<\/p>\n\n\n\n

S<\/p>\n\n\n\n

12<\/p>\n\n\n\n

= C<\/p>\n\n\n\n

12<\/p>\n\n\n\n

C<\/p>\n\n\n\n

21<\/p>\n\n\n\n

C<\/p>\n\n\n\n

12<\/p>\n\n\n\n

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

(c) Show that the matrix is equivalent to<\/p>\n\n\n\n

S<\/p>\n\n\n\n

12<\/p>\n\n\n\n

=<\/p>\n\n\n\n

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

2<\/p>\n\n\n\n

( + X<\/p>\n\n\n\n

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

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

2<\/p>\n\n\n\n

+ Y<\/p>\n\n\n\n

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

Y<\/p>\n\n\n\n

2<\/p>\n\n\n\n

+ Z<\/p>\n\n\n\n

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

Z<\/p>\n\n\n\n

2<\/p>\n\n\n\n

)<\/p>\n\n\n\n

7.5. The controlled phase-\ufb02ip gate takes |11i to \u2212|11i while leaving the other<\/p>\n\n\n\n

basis states unchanged. It is sometimes represented as follows, since its<\/p>\n\n\n\n

action is symmetric in the inputs:<\/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

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

(\u22121)<\/p>\n\n\n\n

xy<\/p>\n\n\n\n

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

(a) Construct the matrix for this gate.<\/p>\n\n\n\n

(b) Build a CNOT gate using controlled phase-\ufb02ip gates an another single-<\/p>\n\n\n\n

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

(c) What is the di\ufb00erence in the outputs of the following two circuits?<\/p>\n\n\n\n

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

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

and<\/p>\n\n\n\n

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

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

(d) Evaluate the output of the circuit<\/p>\n\n\n\n

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

H<\/p>\n\n\n\n

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

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

H<\/p>\n\n\n\n

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

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

H<\/p>\n\n\n\n

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

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

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

7.6. Show that classical conditional operations are equivalent to quantum con-<\/p>\n\n\n\n

trol, i.e., show that the following two circuits are equivalent:<\/p>\n\n\n\n

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

\u2261<\/p>\n\n\n\n

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

7.7. Verify the following circuit identities:<\/p>\n\n\n\n

(a)<\/p>\n\n\n\n

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

\u2261<\/p>\n\n\n\n

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

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

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

(b)<\/p>\n\n\n\n

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

\u2261<\/p>\n\n\n\n

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

(c)<\/p>\n\n\n\n

Z Z<\/p>\n\n\n\n

\u2261<\/p>\n\n\n\n

Z<\/p>\n\n\n\n

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

(d)<\/p>\n\n\n\n

\u2261<\/p>\n\n\n\n

Z<\/p>\n\n\n\n

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

Z<\/p>\n\n\n\n

7.8. Consider the four possible 1-bit functions<\/p>\n\n\n\n

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

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

: 0 \u2192 0<\/p>\n\n\n\n

1 \u2192 0<\/p>\n\n\n\n

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

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

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

: 0 \u2192 0<\/p>\n\n\n\n

1 \u2192 1<\/p>\n\n\n\n

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

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

2<\/p>\n\n\n\n

: 0 \u2192 1<\/p>\n\n\n\n

1 \u2192 0<\/p>\n\n\n\n

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

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

3<\/p>\n\n\n\n

: 0 \u2192 1<\/p>\n\n\n\n

1 \u2192 1<\/p>\n\n\n\n

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

Construct the matrix representation of U<\/p>\n\n\n\n

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

for each. Also give a simple circuit<\/p>\n\n\n\n

to implement each using basic 1-qubit gates.<\/p>\n\n\n\n

7.9. Consider 1-bit integer addition. Write down the truth tables for sum and<\/p>\n\n\n\n

carry bits. Then construct a quantum half-adder by implementing the truth<\/p>\n\n\n\n

tables, using only CNOT gates.<\/p>\n\n\n\n

7.10. Examine the following circuit and analyze the \ufb01nal output. Here, the input<\/p>\n\n\n\n

is an unknown entangled state<\/p>\n\n\n\n

|\u03c8i = \u03b1|01i+ \u03b2|10i<\/p>\n\n\n\n

and |GHZi =<\/p>\n\n\n\n

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

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

2<\/p>\n\n\n\n

(|000i + |111i) .<\/p>\n\n\n\n

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

|\u03c8i<\/p>\n\n\n\n

H<\/p>\n\n\n\n

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

H<\/p>\n\n\n\n

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

\uf8f1<\/p>\n\n\n\n

\uf8f2<\/p>\n\n\n\n

\uf8f3<\/p>\n\n\n\n

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

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

\uf8f1<\/p>\n\n\n\n

\uf8f4<\/p>\n\n\n\n

\uf8f4<\/p>\n\n\n\n

\uf8f2<\/p>\n\n\n\n

\uf8f4<\/p>\n\n\n\n

\uf8f4<\/p>\n\n\n\n

\uf8f3<\/p>\n\n\n\n

H H<\/p>\n\n\n\n

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

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

H<\/p>\n\n\n\n

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

\uf8fc<\/p>\n\n\n\n

\uf8fd<\/p>\n\n\n\n

\uf8fe<\/p>\n","protected":false},"excerpt":{"rendered":"

Show that the n-qubit Hadamard gate acts as H \u2297n |xi n = 1 \u221a 2 n 2 n \u22121 X y=1 (\u22121) x\u00b7y |yi. (7.28) where x \u00b7 y is the bitwise product of x and y: x \u00b7 y = x 0 y 0 \u2295 x 1 y 1 \u2295 . . . […]<\/p>\n","protected":false},"author":1,"featured_media":4041,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[496],"tags":[],"class_list":["post-4080","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-quantum-gates-and-circuits"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-2.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4080","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=4080"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4080\/revisions"}],"predecessor-version":[{"id":4081,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4080\/revisions\/4081"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4041"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4080"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4080"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4080"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}