# Efficiently compute $\langle 0^{\otimes n} | e^{iA} H e^{-iB} |0^{\otimes n} \rangle$ in Qiskit

From this SE question and this Qiskit tutorial, I understand how to compute the expectation such in the form of $$\langle \psi|H|\psi\rangle$$ or $$\langle 0^{\otimes n} |e^{iA} H e^{-iA}|0^{\otimes n} \rangle$$ . But what about efficiently computing $$\langle 0^{\otimes n} | e^{iA} H e^{-iB} |0^{\otimes n} \rangle$$ when $$A$$ and $$B$$ are different and have more than single Pauli term in a quantum computer? $$A$$, $$B$$, and $$H$$ can be assumed to be Hermitian since that is the usual condition in such problem. Also, let us say we know $$H$$ as the linear combination of Pauli basis (i.e., Pauli decomposition of $$H$$).

I can certainly use Hadamard test to achieve my goal, but the circuit depth will be very large when $$A$$ and $$B$$ are composite of multiple Pauli terms, even use order-1 Suzuki trotterization to trotter $$e^{iA}$$ and $$e^{-iB}$$.

Also, if I want to use Qiskit functions like PauliExpectation() and CircuitSampler to do the computation, it seems like $$e^{iA} H e^{-iB}$$ is never involved with any quantum techniques but computed classically. Here is why I am saying this:

from qiskit.opflow import X, Y, Z, I
from qiskit import QuantumCircuit
from qiskit.opflow import CircuitStateFn,PauliExpectation,CircuitSampler
from qiskit.opflow import StateFn,PauliTrotterEvolution,Suzuki
from qiskit.providers.aer import AerSimulator
from qiskit.utils import QuantumInstance

## Backend
backend = AerSimulator(method='statevector')
q_instance = QuantumInstance(backend, shots=1024)
sampler = CircuitSampler(q_instance)

## Operators
A = (0.6*Z^X) + (0.1*Z^Y)
H =  (0.3*I^Z) + (0.6*Y^Z)
B = (0.1*I^Y) + (0.6*Y^X)
op = A.exp_i().adjoint() @ H @ B.exp_i() # e^{iA} H e^{-iB}
obs = StateFn(op).adjoint() # let Qiskit consider this as the observable

## Initial State |00>
init_state = QuantumCircuit(2)
init_state = CircuitStateFn(init_state)

## Compute expectation
measurable_expression = obs @ init_state
trotterized_op = PauliTrotterEvolution(trotter_mode=Suzuki(order=1, reps=1)).convert(measurable_expression)
expectation = PauliExpectation().convert(trotterized_op)
sampled_exp_op = sampler.convert(expectation)
print("Expectation:", sampled_exp_op.eval())


The printout will be

Expectation: (0.008744335460630945-0.015886672611238318j)


It looks good so far, but if I check how many and what circuits executed in the simulator

print(len(list(sampler._circuit_ops_cache.values())))


It will give

1


which is just the circuit for state $$|00\rangle$$

So in this case, I suppose Qiskit only use the quantum circuit to obtain init_state, and no quantum techniques, like qubit-wise commutativity, is applied. Of course, if one changes A and B to a single Pauli term, then sampler._circuit_ops_cache.values() will store multiple circuits and qubit-wise commutativity is indeed applied.

• @forky40 Thank you for your nice suggestion! I think it should be fine to assume A, B and H are Hermitian, which should be a common case for a such problem in practice. I think we can also assume the knowledge of the $H$ as the linear combination of Pauli basis. I will add this condition to the question. Thus, as long as the expectation value is real, I can compute the value for each Pauli string and do the linear combination to sum them up. Jul 16 at 5:23
• @forky40 how can we compute $|\langle 0 | X | 0 \rangle|^2$ if $X$ is not a projector, nor it is Hermitian? Jul 16 at 6:03
• yes maybe it doesn't work actually. Jul 16 at 22:29

You can use swap test to calculate $$\langle 0^{\otimes n} | e^{iA} P_j e^{-iB} |0^{\otimes n} \rangle$$ for each term $$P_j$$ in Pauli decomposition of $$H$$, then do the weighted sum classically:
• Thanks for your answer. I think it should work if the expected value is real, in the cost of $2n+1$ qubits. Let me code it in Qiskit and check the circuit properties like connectivity of qubits, circuit depth and so on. I used to think finding the expectation in this form is not a rare demand, but seems like people don't really deal with this type of problems in practice. Jul 18 at 4:40
• I notice a question while I do the implementation: I suppose we cannot assume $\langle 0^{\otimes n} | e^{iA} P_j e^{-iB} |0^{\otimes n} \rangle \geq 0$, right? Or is there any way to know the sign of expected value? Jul 18 at 18:01