Parameterized `DensePauliString` (a gate) cannot be applied on qubits to produce an operation

[tanujkhattar]

In [2]: import cirq
   ...: import sympy
   ...: gate = sympy.Symbol('t') * cirq.DensePauliString('XYZ')
   ...: gate.on(*cirq.LineQubit.range(3)) # This fails

Click to view the output

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-2-f86effffd2d5> in <module>
      2 import sympy
      3 gate = sympy.Symbol('t') * cirq.DensePauliString('XYZ')
----> 4 gate.on(*cirq.LineQubit.range(3))

~/quantum/Cirq/cirq-core/cirq/ops/dense_pauli_string.py in on(self, *qubits)
    282
    283     def on(self, *qubits) -> 'cirq.PauliString':
--> 284         return self.sparse(qubits)
    285
    286     # TODO(#3388) Add documentation for Raises.

~/quantum/Cirq/cirq-core/cirq/ops/dense_pauli_string.py in sparse(self, qubits)
    305             raise ValueError('Wrong number of qubits.')
    306
--> 307         return pauli_string.PauliString(
    308             coefficient=self.coefficient,
    309             qubit_pauli_map={q: PAULI_GATES[p] for q, p in zip(qubits, self.pauli_mask) if p},

~/quantum/Cirq/cirq-core/cirq/ops/pauli_string.py in __init__(self, qubit_pauli_map, coefficient, *contents)
    158
    159         self._qubit_pauli_map: Dict[TKey, 'cirq.Pauli'] = qubit_pauli_map or {}
--> 160         self._coefficient = complex(coefficient)
    161         if contents:
    162             m = self.mutable_copy().inplace_left_multiply_by(contents).frozen()

~/opt/anaconda3/envs/cirq/lib/python3.8/site-packages/sympy/core/expr.py in __complex__(self)
    330         result = self.evalf()
    331         re, im = result.as_real_imag()
--> 332         return complex(float(re), float(im))
    333
    334     def _cmp(self, other, op, cls):

~/opt/anaconda3/envs/cirq/lib/python3.8/site-packages/sympy/core/expr.py in __float__(self)
    325         if result.is_number and result.as_real_imag()[1]:
    326             raise TypeError("can't convert complex to float")
--> 327         raise TypeError("can't convert expression to float")
    328
    329     def __complex__(self):

TypeError: can't convert expression to float

This is inconsistent and should be fixed.

This occurs because PauliString does not support symbols but DensePauliString does. Ideally, both of them should have identical behavior and we should discuss how to make it happen.

Note that only the coefficient of DensePauliString can be a symbol, and hence to fix the inconsistency, we'd need to support symbols in coefficient of PauliString.

cc #3242

[tanujkhattar]

Discussed on Sync:

• PauliStrings coefficient doesn't support parameterization because while converting to PauliStringPhasor, we need the coefficient to evaluate to a complex number.
• We should figure out how to fix it.
• Let's also add more documentation for DensePauliStrings, PauliStringPhasors etc.

[daxfohl]

Did a quick investigation, it looks like the only place this bleeds beyond PauliString is that we'll also need to parameterize GlobalPhaseGate.coefficient to allow for decomposition to work when PauliString is parameterized.

Everything else seems straightforward:

• Fix up the types.
• Only cast to complex if it's numeric.
• Add handlers for parameter resolution.
• Raise errors in a couple functions that do non-basic math on coefficient (most functions just multiply, which should be fine).
• Return False from has_unitary. (Side note, should we just add that check to the protocol method itself--we repeat it everywhere?)
• Add a block for circuit_diagram_info when the coefficient is parameterized.
• Add tests, update pydocs.

[daxfohl]

(Note that decompose doesn't work when parameterized anyway since it's not unitary, so we actually don't need to change GlobalPhaseGate)