Some operations involving the identity gate cause an error

[lmarti-dev]

Describe the issue

If one tries to add or subtract the identity gate to another gate (or vice-versa) on a qubit q, for example, I(q) - Z(q), then the following error occurs:

Traceback (most recent call last):
  File "\misc\cirq_sub_i.py", line 31, in <module>
    op(p1, p2)
  File "\misc\cirq_sub_i.py", line 9, in add
    return x + y
TypeError: unsupported operand type(s) for +: 'GateOperation' and 'GateOperation'

One could perhaps add the Pauli beforehand and then apply them on qubits (this doesn't cause the error), but it isn't clear to me how one turns a LinearCombinationsOfGates to a PauliSum. Alternatively, replacing the - with +(-1.0)* solves the issue in some cases. Otherwise, one may replace the offending I(q) by X(q)*X(q) or PauliString(). Is this very last option the preferred way of denoting the identity?

The following assertions are satisfied (and the individual expressions do not throw an error)

assert PauliString() - Z(q) == I(q) + (-1) * Z(q)
assert X(q) * X(q) - Z(q) == I(q) + (-1) * Z(q)

They are all "paraphrases" of the original I(q) - Z(q). As an aside,

assert X(q) ** 2 - Z(q) == I(q) + (-1) * Z(q)

also fails, because X(q) ** 2 is converted to an incompatible class.

Explain how to reproduce the bug or problem

The code below goes through all possible combinations of gates and operations, and prints those not implemented. In fact, moving the op(p1,p2) out of the try block will reproduce the error above.

from cirq import I, X, Y, Z, LineQubit

q = LineQubit(0)

paulis = (I(q), X(q), Y(q), Z(q))


def add(x, y):
    return x + y


add.sym = "+"


def sub(x, y):
    return x - y


sub.sym = "-"


def addm(x, y):
    return x + (-1.0) * y


addm.sym = "+(-1.0)*"

for p1 in paulis:
    for p2 in paulis:
        for op in (add, sub, addm):
            try:
                op(p1, p2)
                msg = "pass"
            except Exception:
                msg = "error"
                print(f"{p1.gate}{op.sym}{p2.gate}: {msg}")

This snippet outputs

I+I: error
I-I: error
I-X: error
I-Y: error
I-Z: error
X-I: error
Y-I: error
Z-I: error

This happens on cirq 1.4.1. I updated right before running the code.

I assume the error can be fixed by adding the missing __add__ and __sub__ in some class, but I am confused by the class structure and have not managed to locate the correct one.

[daxfohl]

Looks like PauliSum.__add__ has some special handling for identity, that PauliSum.__sub__ is missing. Should be an easy fix if you want to try.

https://github.com/quantumlib/cirq/blob/72e1d203f2afb56313e8eb618ec994cc001275b8/cirq-core/cirq/ops/linear_combinations.py#L773

As for I+I, IdentityGate is not an instance of Pauli and thus doesn't have a special on(q) method that converts it to a PauliString.

https://github.com/quantumlib/cirq/blob/fd95547ac4962cfa3d0d1ff707a8de26ce148511/cirq-core/cirq/ops/pauli_gates.py#L93-L105

Plausibly there should be a _PauliI class similar to _PauliX/Y/Z, that inherits both IdentityGate and Pauli, and I should be an instance of that, similar to X/Y/Z instead of being over by itself. That would be a bigger design change though, and maybe conflicts with other things.

[mhucka]

Discussed in Cirq Cynq 2025-03-19:

• Does look like a bug.
• We're worried that the bigger change idea (adding a new class) might be more work.

Let's accept, with a minimum scope for now.