Mixed-Y two-qubit gates use the opposite rotation angle

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Assessment

Difficulty
2/5
Estimated time
1-3 hours
Newbie friendliness
82/100
Issue type
Bug
Clarity
Clearly specified
Activity status
Active
Tech stack
numpy, python, tensorflow
Domain
backend, machine-learning

Research direction

Read applications.layers.generate_double_gate() and compare the Y-basis rotations with the other basis transformations. Check the generated gates against the stated Pauli-product matrix exponentials for all nine pairs, including the four mixed-Y cases. Done means every abgate(theta) implements the specified negative-angle convention.

Written by the indexing model from the issue text.

Description

bug

Issue Description

xygate, yxgate, yzgate, and zygate, generated by applications.layers.generate_double_gate(), implement

$$
U_{P}(\theta)=\exp\left(+\frac{i\theta}{2}P\right)
$$

instead of the family convention

$$
U_{P}(\theta)=\exp\left(-\frac{i\theta}{2}P\right),
$$

where $P$ is the corresponding two-qubit Pauli product. The Y-basis transformation uses $R_x(-\pi/2)$ before the ZZ rotation and $R_x(+\pi/2)$ after it, which maps $Z$ to $-Y$ instead of $Y$. A Pauli product containing exactly one $Y$ therefore has its angle reversed; the two minus signs in $Y\otimes Y$ cancel.

For every $a,b\in{X,Y,Z}$, abgate(theta) should implement

$$
\exp\left(-\frac{i\theta}{2}(\sigma_a\otimes\sigma_b)\right).
$$

The fix is to exchange the signs of the Y-basis rotations in generate_double_gate().

How to Reproduce

import itertools

import numpy as np
from scipy.linalg import expm
import tensorcircuit as tc
from tensorcircuit.applications import layers

tc.set_backend("numpy")
tc.set_dtype("complex128")
paulis = {
    "x": np.array([[0, 1], [1, 0]], dtype=complex),
    "y": np.array([[0, -1j], [1j, 0]], dtype=complex),
    "z": np.diag([1, -1]),
}
theta = 0.7
print("gate  error vs exp(-i theta P/2)  error vs exp(+i theta P/2)")
for a, b in itertools.product("xyz", repeat=2):
    c = tc.Circuit(2)
    getattr(layers, a + b + "gate")(c, 0, 1, theta)
    actual = np.asarray(c.matrix())
    p = np.kron(paulis[a], paulis[b])
    expected = expm(-0.5j * theta * p)
    reversed_angle = expm(0.5j * theta * p)
    print(f"{a+b}    {np.linalg.norm(actual - expected):.9f}"
          f"                  {np.linalg.norm(actual - reversed_angle):.9f}")
Error Output
gate  error vs exp(-i theta P/2)  error vs exp(+i theta P/2)
xx    0.000000000                  1.371591230
xy    1.371591230                  0.000000000
xz    0.000000000                  1.371591230
yx    1.371591230                  0.000000000
yy    0.000000000                  1.371591230
yz    1.371591230                  0.000000000
zx    0.000000000                  1.371591230
zy    1.371591230                  0.000000000
zz    0.000000000                  1.371591230

Environment Context

TensorCircuit checkout: a6662d56c7aa8f39c12f2056dba69a6bf6c439bf.

Additional dependencies: NumPy backend, complex128; tensorflow-cpu==2.20.0 is required to import applications.layers.

Output of tc.about() and tc.__version__:

TensorCircuit version: 1.10.0
OS: Linux-6.2.0-39-generic-x86_64-with-glibc2.37
Python: 3.10.14
NumPy: 2.2.6
SciPy: 1.15.3
TensorNetwork: 0.5.3
TensorFlow: 2.20.0
JAX/JAXlib: 0.6.2/0.6.2
Dominant language
Python
Stars
92
Forks
25
Avg merge
21h 12m
Merged PRs (30d)
57

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