Update lec_19_quantum.md

lec_19_quantum.md

@@ -213,7 +213,7 @@ We define $M$ as an _elementary stochastic matrix_ or a _probabilistic gate_ if
 The quantum case is similar: a _quantum gate_ is a $2^n\times 2^n$ matrix that is an $N$ lift of some unitary $8\times 8$ matrix $M'$.
 It is an exercise to prove that lifting preserves stochasticity and unitarity. That is,  every probabilistic gate is a stochastic matrix and every quantum gate is a unitary matrix.
 
-__Complexity:__ For every stochastic matrix $M$ we can define its _randomized complexity_, denoted as $R(M)$ to be the minimum number $T$ such that $M$ is can be (approximately) obtained by combining $T$ elementary probabilistic gates. To be concrete, we can define $R(M)$ to be the minimum $T$ such that there exists $T$ elementary matrices $M_1,\ldots,M_T$
+__Complexity:__ For every stochastic matrix $M$ we can define its _randomized complexity_, denoted as $R(M)$ to be the minimum number $T$ such that $M$ can be (approximately) obtained by combining $T$ elementary probabilistic gates. To be concrete, we can define $R(M)$ to be the minimum $T$ such that there exists $T$ elementary matrices $M_1,\ldots,M_T$
 such that for every $x$, $\sum_y |M_{y,x}-(M_T\cdots M_1)_{y,x}|<0.1$.
 (It can be shown that $R(M)$ is finite and in fact at most $10^n$ for every $M$; we can do so by writing $M$ as a convex combination of function and writing every function as a composition of functions that map a single string $x$ to $y$, keeping all other inputs intact.)
 We will say that a probabilistic process $M$ mapping distributions on $\{0,1\}^n$ to distributions on $\{0,1\}^n$ is _efficiently classically computable_ if $R(M) \leq poly(n)$.