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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.

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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.

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  • $\begingroup$ This looks great, thanks! Will accept upon comprehending. $\endgroup$
    – mavzolej
    Apr 5 at 1:04
  • $\begingroup$ Hi @CraigGidney! How do we compute 2? What is meant by "phase by the amout"? Also, what are we "phasing"? $\endgroup$
    – MonteNero
    Apr 5 at 3:08
  • $\begingroup$ @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 $\endgroup$ Apr 5 at 7:41
  • $\begingroup$ @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? $\endgroup$
    – MonteNero
    Apr 5 at 11:34
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    $\begingroup$ @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. $\endgroup$ Apr 5 at 19:10

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