![]() You can count each case using binomial coefficients. Specifically, without loss of generality, you can look at $3$ cases: To count the total number of such permutations, you'll need to do a bit of casework based on which of $a,b,c,d$ are equal. Now, every permutation which is obtained by two row exchanges can be written as $(ab)(cd)$, with $a,b,c,d$ possibly not all distinct. This means that when applying $(12)(23)$ to $$, we apply the rightmost permutation $(23)$ first. Let's say that we multiply permutations from right to left. Just like how matrix multiplication is not commutative, multiplying permutations is not commutative. For example, consider the permutation $(12)(23)$ and the list $$. Things get a little more complicated when permutations you multiply share elements. Applying $(12)(34)$ to the permutation $$ gives $$, because we swapped the $1$st and $2$nd slot, then the $3$rd and $4$th slot. If you want to do two swaps, you can multiply these permutations. ![]() For example, applying $(12)$ to the permutation $$ gives $$. Then $(ab)$ represents the permutation which swaps the $a$th slot with the $b$th slot. Again, let's say the objects $1, 2, \dots, n$ are filling the slots $$ in some order. This is a succinct way to represent permutations. This question is more complicated-you're going to want to look into cycle notation. ![]() Taking the transpose will not affect the value of. For an m x n matrix, with m less than or equal to n, it is given as the sum over the permutations s of size less than or equal to m on 1, 2, n of the product from i 1 to m of Mi, si. Permutations with a fixed number of row exchanges Finally let’s use lambda to create a 1-line matrix with 1’s in the even permutation entries. Thus there are $n \cdot (n-1) \cdot (n-2 ) \cdots 1 = n!$ total permutations. A permutation matrix is an n × n matrix that has exactly one entry 1 in each column and in each row, and all other entries are 0. This pattern continues until there is $1$ way to fill the last spot. There are $n - 1$ ways to fill the second spot. There are $n$ ways to fill the first spot. To visualize this, let's start with an "empty list" $$ and fill it in as we go. We are now counting the number of permutations of the list $$. (I am going from binary 00 to decimal 0 here)įrom the second row of $T$ we learn that (output, input) = (01, 01) = (row, col) = (1,1).There are $n!$ permutation matrices of an $n \times n$ matrix.Ī permutation matrix permutes the rows (or columns, depending which side it's being multiplied on) of a matrix. Let's start with a blank matrix, that will turn into a permutation matrix.įrom the first row of $T$ we learn that (output,input) = (00,00), which tells that the (row,col) = (0,0) must have a 1 in it. To create the permutation matrix, we just have to run down the rows of T, and for each row read off the input and output, and place the 1 in the corresponding entry of the permutation matrix. What is the general recipe or procedure for translating a truth table to the permutation/unitary matrix? Given a truth table of a reversible circuit with, say, $5$ inputs and $5$ outputs of size $2^5\times 10$, how can we construct the corresponding permutation matrix of size $2^5\times 2^5$? ![]() Studying unitary matrices within quantum computing is more useful than studying other matrices such as truth tables, or other square matrices such as Karnaugh maps. Such matrices are also unitary, which is a requirement for use in circuit-based quantum computing. When the circuit is reversible, we can also construct a permutation matrix, which is a square matrix of size $2^m\times 2^m$, with a single $1$ in each row and each column. For example, for small enough $m$ Karnaugh maps can be used to study simplification of such circuits. However, certain square matrices can also encode the same information as a truth table. When the circuit is reversible and consists of CNOT gates, CCNOT gates, CSWAP gates, etc., we have $m=n$ as the number of inputs is the same as the number of outputs. Given a classical circuit of $m$ inputs and $n$ outputs, composed of various AND gates, OR gates, NOT gates, etc., a truth table is a $2^\times(m+n)$-sized matrix, where, in general, the first $m$ columns encode the binary inputs while the last $n$ columns encode the binary outputs. ![]()
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