Quantum Chemistry example notebook for the exponentiation of Pauli operators

[matt-lourens]

I don't have any background in quantum chemistry but I have seen these circuits for the exponentiation of tensor products of Pauli operators in a quantum chemistry context:
From this paper

and this:

These should be easy to implement with HierarQcal because of their symmetric nature. I'm hoping someone who knows more of their application can create a notebook example that illustrates their typical use case through using the package. Pennylane also has templates for these.

[hrahman12]

@matt-lourens can you assign me this issue thank you

I am working on the notebook halfway done

Hi I'm working on this too are you interested in teaming up.

[Gopal-Dahale]

Hi, @matt-lourens, I understand that with hierarqcal we can have that CNOT ladder but I am finding it difficult how to put that one $U(\theta)$ gate in the middle (first circuit diagram). I can create a mapping for the Qunitary object or this but then I have to create an object for every 1 qubit quantum gate (Rx, Ry, Rz, H and so on). Any idea on this or am I missing something?

[matt-lourens]

Hey @Gopal-Dahale, thanks for giving the issue a look!

• TLDR; look at the last piece of code.

If you're wondering about how to place a single gate at the end of the circuit, you can use the Qcycle primitive along with it's offset parameter:

n=8
ladder = Qcycle(1, 1, 0, mapping=Qunitary("CNOT()^01"), boundary="open")
single_gate = Qcycle(1, 1, n-1, mapping=Qunitary("RZ(x1)^0"), boundary="open")
hierq = Qinit(n)+ladder+single_gate+ladder
circuit = hierq(backend="qiskit")
circuit.draw("mpl")

• Note I'm using the latest code from develop branch to create the RZ mapping with a string, which was added recently in this issue (only works for qiskit).

If you're asking more generally about what kinds of mappings we can provide to Qunitary, it doesn't have to be single qubit gates, for example the Rx-H-H-Rx unitary in the second image can be created as follows:

def U(bits, symbols=None, circuit=None, **kwargs):
    # Assume bits are strings and in the correct QASM format
    q0, q1, q2, q3 = QuantumRegister(1, bits[0]), QuantumRegister(1, bits[1]), QuantumRegister(1, bits[2]), QuantumRegister(1, bits[3])
    circuit.rx(symbols[0], q0)
    circuit.h(q1)
    circuit.h(q2)
    circuit.rx(symbols[1], q3)
    return circuit

u=Qunitary(U, arity=4, n_symbols=2)
hierq = Qinit(4) + Qcycle(1, 1, 0, mapping=u, boundary="open")
circuit = hierq(backend="qiskit")
circuit.draw("mpl") 

Again using strings to create this function is just Qcycle(1, 1, 0, mapping=Qunitary("RX(x1)^0;h()^1;h()^2;RX(x2)^3"), boundary="open")

But this depends on what the expected "pattern" is, if instead the pattern is to have hadamards and Rx on the ends the following would be better:

n=7
hierq = Qinit(n) + Qmask("1*1", mapping=Qunitary("RX(x1)^0"), share_weights=False)+ Qcycle(mapping=Qunitary("h()^0")) + Qunmask("previous") 
circuit = hierq(backend="qiskit")
circuit.draw("mpl")

Where you can change n as needed, this masks the first and final qubit while also applying a rotation Rx to them, then we apply a ladder/cycle of hadamards, since the first and last qubits are masked the aren't used, then just unmask the previous mask so that all qubits are available again.

I realize the package is not at a point where doing something like that is immediately clear, but hopefully trying to solve different problems with it, will guide us to make tutorials and things that makes it a bit more explicit. For example, I only realized we can do the following pattern as a consequence of your question.

Finally, placing single qubits based on some pattern like "*1" (place at end of circuit), "*1*" (place in middle) is best done at the moment with Qmask and Qunmask. The idea is to add a new primitive Qpivot which does the same but without masking the qubit (making it unavailable). This will hopefully be solved along this issue

Hope that helps :)

[Gopal-Dahale]

Hi @matt-lourens, I am unable to change the qubit pair ordering for a CNOT ladder. The current ordering is like (i, i+1) for i^th qubit but I need to be the reverse of it. I tried passing the reverse sorted list in Qinit and changing the QCycle parameters but it doesn't seem to be working. Should I just append the adjoint of the ladder circuit using qiskit?

[matt-lourens]

Hey @Gopal-Dahale, you can change the definition of your mapping to be a cnot where control and target is swapped:

n=8
ladder = Qcycle(1, 1, 0, mapping=Qunitary("CNOT()^10"), boundary="open")
hierq = Qinit(n)+ladder
circuit = hierq(backend="qiskit")
circuit.draw("mpl")

Is that what you wanted?

You can also change the edge order, if that is what you meant:

n=8
l1 = Qcycle(1, 1, 0, mapping=Qunitary("CNOT()^01"), boundary="open")
single_gate = Qcycle(1, 1, n-1, mapping=Qunitary("RZ(x1)^0"), boundary="open")
l2 = Qcycle(1, 1, 0, mapping=Qunitary("CNOT()^01"), boundary="open", edge_order=[i for i in range(n,1,-1)])
hierq = Qinit(n)+l1+single_gate+l2
circuit = hierq(backend="qiskit")
circuit.draw("mpl")

[Gopal-Dahale]

Great, thanks for the help @matt-lourens .

[andresulea]

hi @matt-lourens the other day you told me that this challenge was open, I've been reading more about the theory of these structures, I think I understand more, is this challenge open?

[matt-lourens]

Hey @andresulea, yeah go ahead. There's no PR's open for this issue, it's still anyone's game :)

[Gopal-Dahale]

Hi @matt-lourens, I have made an initial draft. Need some help with parameter prefixes. If it is fine with you, we can continue the discussion on the PR.

[Gopal-Dahale]

Hi, @matt-lourens can you have a look at #37 ?

[matt-lourens]

Hi, @matt-lourens can you have a look at #37 ?

Hey @Gopal-Dahale, yes I glanced over it this morning (looks great!), I will only have a chance a bit later today to go through it. I know the deadline is tomorrow, so I'll try to get to it asap. But don't worry on the bounty side of things, we'll make a plan if time runs short.

[andresulea]

Hello, @matt-lourens I just did the PR, (I hope I did it well) on this issue, it is a nice calculation on the energy of the ground state of the H2 molecule using hierarqcal, I tried to explain it step by step since you commented above that you were not very familiar, I will I leave a graph about how interesting the results were, thanks[

[matt-lourens]

Hey @andresulea,

Thanks! I've left a comment on your PR as how we can incorporate the work you did, for the hackathon/bounty I'm assigning the issue to @Gopal-Dahale (I've elaborated on your PR), but I have an idea for how we can reward you also.