Construction of unitary matrices built from linear combination of Pauli strings

Question

Let's define $P_k \in \{ I, X, Y, Z \}^{\otimes n}$ and called each of these $P_k$ as a Pauli string (or word) then given that $$U = \sum_{k=1}^L c_kP_k $$ with the following conditions:

1. $\sum_{k=1}^L |c_k|^2 = 1 $
2. $Im(c_i \cdot c_j) = 0$
3. $\{P_i, P_j\} = 2 \delta_{i,j}$ where $\{ \}$ indicates the anti-commuting relationship. In other words, the Pauli strings $P_i$ in the sum that made up $U$ are anti-commute.

With the above three conditions, $U$ is a unitary operator and thus can be implemented as a quantum circuit.

In the following paper https://arxiv.org/pdf/1907.09040.pdf at the end of page 3, it shows that $U$ can be implemented as the product of $2L-1$ exponents of these Pauli strings (or words) in the following way:

$$ U = \prod_{k = 1}^L e^{i \theta_k P_k/2} \prod_{k = L}^1 e^{i \theta_k P_k/2} $$

where $\theta_k = \arcsin\bigg(\dfrac{c_k}{\sum_{j=1}^k c_j^2 } \bigg) $. Is there a neat proof of this fact or is it just full out algebraic manipulation which gets really messy? That is, is there an intuitive way to see why this is true? Since the authors seem to indicate this is a pretty trivial fact.

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Bonus: What is the best implementation for $U$ in term of the quantum circuit outside of the proposed method above?

Answer 1

You can prove this by induction. Start with the simplest non-trivial sequence $$ e^{i\theta_2 P_2/2}e^{i\theta_2 P_1}e^{i\theta_2 P_2/2}=e^{i\theta_2 P_2/2}(I\cos\theta_1+i\sin\theta_1 P_1)e^{i\theta_2 P_2/2}. $$ Now, $$ e^{i\theta_2 P_2/2}Ie^{i\theta_2 P_2/2}=e^{i\theta_2 P_2} $$ and, due to the anti-commutation, $$ e^{i\theta_2 P_2/2}P_1e^{i\theta_2 P_2/2}=P_1e^{-i\theta_2 P_2/2}e^{i\theta_2 P_2/2}=P_1. $$ Thus, the full sequence becomes $$ I\cos\theta_1\cos\theta_2+i\cos\theta_1\sin\theta_2 P_2+i\sin\theta_1P_1. $$

Now if we pre- and post-multiply by $e^{i\theta_3 P_3/2}$, then the anti-commutation will basically leave the Pauli-terms $P_1$ and $P_2$ unchanged, replacing $I$ with $e^{i\theta_3 P_3}$. And this sequence will keep repeating. You would be able to solve exactly for all the $\theta_i$ at that point, but assuming that the $\sin\theta_i$ are all small, you can approximate $\cos\theta_i\approx 1$ (I'm not convinced this is a good approximation once you have a large number of terms), which leaves you setting $\sin\theta_i=c_i$ up to normalisation.