# Exponentiating a tensor product of operators acting on disjoint qubit registers

Consider a problem of implementing $$\operatorname{e}^{i\bigotimes_j O_j}$$, where all the $$O_j$$ terms act on disjoint sets of qubits. Assume that efficient circuits implementing individual $$\operatorname{e}^{i\ O_j}$$ are known. Moreover, if necessary, one can also assume that the corresponding registers are relatively small so that efficient circuits $$U_j$$ for diagonalizing $$O_j$$ are known (different definitions of "small" or any other assumptions on $$O_j$$ are welcome too).

Is there a way to efficiently and exactly implement $$\operatorname{e}^{i\bigotimes_j O_j}$$?

My question is driven by the desire to generalize a well-known procedure for exponentiating Pauli terms $$\operatorname{e}^{iXYZ\ldots}$$, which is achieved by first rotating to eigenbases of individual qubits, and then using the cascade of $$\text{CNOT}$$s as in Fig 4.19 of Nielsen&Chuang.

I think the following would work:

1. Perform phase estimation individually against each $$O_j$$, creating phase estimate registers $$a_j$$.
2. Compute the superposed product $$t = \prod_j a_j$$.
3. Phase by an amount proportional to $$t$$.
4. Uncompute $$t$$.
5. Uncompute each $$a_j$$.

For the Pauli operator case it's easy to see why this works. Each operator has an eigenvalue of -1 or +1, and computing the product $$t$$ accumulates the -1 factors into one spot, so that when you phase by an amount proportional to $$t$$ you are phasing forward if there's an even number of -1s and backward if there's an odd number.

• This looks great, thanks! Will accept upon comprehending. Apr 5 at 1:04
• Hi @CraigGidney! How do we compute 2? What is meant by "phase by the amout"? Also, what are we "phasing"? Apr 5 at 3:08
• @MonteNero See en.wikipedia.org/wiki/Quantum_phase_estimation_algorithm on how to do phase estimation. "Phasing" means to rotate amplitudes. For example, phasing a state by 45 degrees means to multiply its amplitude by (1+i)/sqrt(2). See algassert.com/post/1719 Apr 5 at 7:41
• @CraigGidney, thanks for the links and explaining the meaning of phasing. My question wasn't about QPE part, but about computing the product $t$ of phases. Are we talking about some arithmetic multiplier here? Apr 5 at 11:34
• @mavzolej You may be able to replace the phase estimation with usage of $U_j$. But it's not enough to just diagonalize, you'd also need to lookup the eigenvalues based on the address in the diagonalized register. Apr 5 at 19:10