The default dtype of np.complex64 for the DensityMatrixSimulator is not precise enough for relatively small circuits.

[bramathon]

Description of the issue

How to reproduce the issue

Use the example circuit:
example-circuit.json

import cirq

with open("example-circuit.json", "r") as f:
    circuit = cirq.read_json(f)

simulator = cirq.DensityMatrixSimulator(dtype=np.complex64, noise=None, seed=None, split_untangled_states=True)
obs = cirq.PauliString(cirq.X(cirq.LineQubit(28)), cirq.X(cirq.LineQubit(29)))
simulator.simulate_expectation_values(program=circuit, observables=obs)

Now, if we change the dtype to np.complex128, the error goes away.

Suggested solution: set the default value of dtype to np.complex128 or np.complex256

1018 raise ValueError(f'Density matrix does not have trace 1. Instead, it has trace {trace}.')
1019 if not np.all(np.linalg.eigvalsh(density_matrix) > -atol):
-> 1020 raise ValueError('The density matrix is not positive semidefinite.')

ValueError: The density matrix is not positive semidefinite.

Cirq version
1.3.0

[pavoljuhas]

csynkque meeting - this works as intended, but the error message can be improved to suggest using a higher precision complex128 type. Another consideration is to expose a tolerance atol argument which would allow higher round-offs for complex64 data type.