Hamiltonian simulation from primitive gates for DBI

[marekgluza]

@scarrazza an enhancment issue request that @shangtai agreed to help implement.

Running a DBI step cf #1034 for an input Hamiltonian $H0$ for example $$H{\mathrm{TFIM}} = \sum Zi Z{i+1} + h \sum X_i $$ with a variational operator $D_0$, for example $$D_0 = \sum h_i X_i $$ means to implement $$V_1 = e^{is D_0}e^{is H_0} e^{-isD_0}$$

For this we will need to implement evolutions under symbolically prescribed local operators $H_0=\sum h_i$ and $D_0=\sum d_i$

@scarrazza @stavros11 Can you help me plan this to see a) what needs doing and b) how to do it in a smart way so that more things will benefit from our current interests?

More abstract stage:

• [ ] Given a symbolically prescribed $H_0=\sum h_i$ get a circuit approximation of $e^{itH_0}$ via a product formula for example first order $(\prod_i e^{i t/N h_i})^N$

If I understood @Edoardo-Pedicillo correctly, qibo currently only has the first order Trotter-Suzuki Hamiltonian simulation? What else is in the making? @scarrazza can you give us your opinion which way to push it?

I think we should open an issue "Product formulas for Hamiltonian simulation" and ask @shangtai build a code structure to put in different product formula strategies? For the DBI project the first order will suffice, if we have 2nd or 3rd order then we could also use that.

@shangtai in general all that needs to be implemented are eqs (1-2) here https://arxiv.org/abs/1901.00564

More applied stage: Longer term we'd like to run DBIs on devices which means we need to decompose $e^{it h_i}$ into smaller gates.

• [x] is there a function which implements $e^{it Z_1Z_2}$ using CNOT, $R_X(\theta)$, $R_Z(\gamma)$? (Edit: essentially in-built)
• [x] is there an implementation of the Cirac-Kraus decomposition i.e. to translate CNOT, $R_X(\theta)$, $R_Z(\gamma)$ into any two qubit gate? (Edit: see https://github.com/qiboteam/qibo/blob/c11311751f0c6afb6ee021d0d5d38e0dd3490fef/src/qibo/transpiler/unitary_decompositions.py#L198)
• [x] Is there QIBO code which compiles using CNOT, $R_X(\theta)$, $R_Z(\gamma)$ the evolution under the tranverse-field Ising model? What about the Heisenberg model? (see TrotterHamiltonian)
• [ ] For more hardware aware applications are there qibo-lab transpiling routines which translate the cross-resonance Hamiltonian into e.g. the Heisenberg evolution/swap gate for two qubits? So time-dependent $H{CR}(t)= \omega{ZI}(t) Z\otimes I+ \omega{IZ}(t) I\otimes Z+ \omega{IX}(t) I\otimes X+ \omega{ZX}(t) Z\otimes X$ effectively implements after some time $\tau$ the evolution $e^{is h{(1,2)}}=e^{is(XX+YY+ZZ)}$ for prescribed $s$
• [ ] What about for the IQM qubits?

TLDR running DBI is about Hamiltonian simulation so requesting help to understand what still needs to be done, thanks!

[marekgluza]

update: @MatteoRobbiati said he will help me understand it better: it seems if there is a unitary in the qibo workflow then it is possible to decompose that evolution into CZ, RX and RZ

[marekgluza]

The resolution is using a notebook by @MatteoRobbiati

https://github.com/MatteoRobbiati/notebooks/blob/main/adiabatic-evolution/evolution.ipynb

We may use the more general AdiabaticHamiltonian which internally has the circuit functions https://github.com/qiboteam/qibo/blob/ac4e4769117b2f75b446dc3c70a93766070efb65/src/qibo/hamiltonians/adiabatic.py#L145 which uses the terms class https://github.com/qiboteam/qibo/blob/ac4e4769117b2f75b446dc3c70a93766070efb65/src/qibo/hamiltonians/terms.py this then is used in https://github.com/qiboteam/qibo/blob/ac4e4769117b2f75b446dc3c70a93766070efb65/src/qibo/hamiltonians/adiabatic.py#L106

thanks @MatteoRobbiati and @shangtai for the instructive discussion! :)