Mapping

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import sympy as sp
from sympy.physics.quantum import Dagger as dag, TensorProduct as tensor

import sympy as sp from sympy.physics.quantum import Dagger as dag, TensorProduct as tensor

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# 1Q Pauli

I = sp.Matrix(
    [
        [1, 0],
        [0, 1],
    ],
)

X = sp.Matrix(
    [
        [0, 1],
        [1, 0],
    ],
)

Y = sp.Matrix(
    [
        [0, -sp.I],
        [sp.I, 0],
    ],
)

Z = sp.Matrix(
    [
        [1, 0],
        [0, -1],
    ],
)

# 1Q Pauli I = sp.Matrix( [ [1, 0], [0, 1], ], ) X = sp.Matrix( [ [0, 1], [1, 0], ], ) Y = sp.Matrix( [ [0, -sp.I], [sp.I, 0], ], ) Z = sp.Matrix( [ [1, 0], [0, -1], ], )

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# 2Q Pauli

PAULI_2Q = {
    "II": tensor(I, I),
    "IX": tensor(I, X),
    "IY": tensor(I, Y),
    "IZ": tensor(I, Z),
    "XI": tensor(X, I),
    "XX": tensor(X, X),
    "XY": tensor(X, Y),
    "XZ": tensor(X, Z),
    "YI": tensor(Y, I),
    "YX": tensor(Y, X),
    "YY": tensor(Y, Y),
    "YZ": tensor(Y, Z),
    "ZI": tensor(Z, I),
    "ZX": tensor(Z, X),
    "ZY": tensor(Z, Y),
    "ZZ": tensor(Z, Z),
}

# 2Q Pauli PAULI_2Q = { "II": tensor(I, I), "IX": tensor(I, X), "IY": tensor(I, Y), "IZ": tensor(I, Z), "XI": tensor(X, I), "XX": tensor(X, X), "XY": tensor(X, Y), "XZ": tensor(X, Z), "YI": tensor(Y, I), "YX": tensor(Y, X), "YY": tensor(Y, Y), "YZ": tensor(Y, Z), "ZI": tensor(Z, I), "ZX": tensor(Z, X), "ZY": tensor(Z, Y), "ZZ": tensor(Z, Z), }

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# 1Q Clifford

X90 = sp.Matrix(
    [
        [1, -sp.I],
        [-sp.I, 1],
    ],
) / sp.sqrt(2)

Y90 = sp.Matrix(
    [
        [1, -1],
        [1, 1],
    ],
) / sp.sqrt(2)

Z90 = sp.Matrix(
    [
        [1 - sp.I, 0],
        [0, 1 + sp.I],
    ],
) / sp.sqrt(2)

# 1Q Clifford X90 = sp.Matrix( [ [1, -sp.I], [-sp.I, 1], ], ) / sp.sqrt(2) Y90 = sp.Matrix( [ [1, -1], [1, 1], ], ) / sp.sqrt(2) Z90 = sp.Matrix( [ [1 - sp.I, 0], [0, 1 + sp.I], ], ) / sp.sqrt(2)

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# 1Qx1Q Clifford

II = tensor(I, I)
IX90 = tensor(I, X90)
IY90 = tensor(I, Y90)
IZ90 = tensor(I, Z90)
XI90 = tensor(X90, I)
YI90 = tensor(Y90, I)
ZI90 = tensor(Z90, I)

# 1Qx1Q Clifford II = tensor(I, I) IX90 = tensor(I, X90) IY90 = tensor(I, Y90) IZ90 = tensor(I, Z90) XI90 = tensor(X90, I) YI90 = tensor(Y90, I) ZI90 = tensor(Z90, I)

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# 2Q Clifford

ZX90 = sp.Matrix(
    [
        [1, -sp.I, 0, 0],
        [-sp.I, 1, 0, 0],
        [0, 0, 1, sp.I],
        [0, 0, sp.I, 1],
    ]
) / sp.sqrt(2)

ZZ90 = sp.Matrix(
    [
        [1 - sp.I, 0, 0, 0],
        [0, 1 + sp.I, 0, 0],
        [0, 0, 1 + sp.I, 0],
        [0, 0, 0, 1 - sp.I],
    ]
) / sp.sqrt(2)

CNOT = sp.Matrix(
    [
        [1, 0, 0, 0],
        [0, 1, 0, 0],
        [0, 0, 0, 1],
        [0, 0, 1, 0],
    ],
)

CZ = sp.Matrix(
    [
        [1, 0, 0, 0],
        [0, 1, 0, 0],
        [0, 0, 1, 0],
        [0, 0, 0, -1],
    ],
)

SWAP = sp.Matrix(
    [
        [1, 0, 0, 0],
        [0, 0, 1, 0],
        [0, 1, 0, 0],
        [0, 0, 0, 1],
    ],
)

ISWAP = sp.Matrix(
    [
        [1, 0, 0, 0],
        [0, 0, sp.I, 0],
        [0, sp.I, 0, 0],
        [0, 0, 0, 1],
    ],
)

BSWAP = sp.Matrix(
    [
        [0, 0, 0, sp.I],
        [0, 1, 0, 0],
        [0, 0, 1, 0],
        [sp.I, 0, 0, 0],
    ],
)

# 2Q Clifford ZX90 = sp.Matrix( [ [1, -sp.I, 0, 0], [-sp.I, 1, 0, 0], [0, 0, 1, sp.I], [0, 0, sp.I, 1], ] ) / sp.sqrt(2) ZZ90 = sp.Matrix( [ [1 - sp.I, 0, 0, 0], [0, 1 + sp.I, 0, 0], [0, 0, 1 + sp.I, 0], [0, 0, 0, 1 - sp.I], ] ) / sp.sqrt(2) CNOT = sp.Matrix( [ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], ], ) CZ = sp.Matrix( [ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1], ], ) SWAP = sp.Matrix( [ [1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1], ], ) ISWAP = sp.Matrix( [ [1, 0, 0, 0], [0, 0, sp.I, 0], [0, sp.I, 0, 0], [0, 0, 0, 1], ], ) BSWAP = sp.Matrix( [ [0, 0, 0, sp.I], [0, 1, 0, 0], [0, 0, 1, 0], [sp.I, 0, 0, 0], ], )

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# 2Q Non-Clifford

SQRT_ISWAP = sp.Matrix(
    [
        [sp.sqrt(2), 0, 0, 0],
        [0, 1, sp.I, 0],
        [0, sp.I, 1, 0],
        [0, 0, 0, sp.sqrt(2)],
    ],
) / sp.sqrt(2)

BGATE = sp.Matrix(
    [
        [sp.cos(sp.pi / 8), 0, 0, sp.I * sp.sin(sp.pi / 8)],
        [0, sp.sin(sp.pi / 8), sp.I * sp.cos(sp.pi / 8), 0],
        [0, sp.I * sp.cos(sp.pi / 8), sp.sin(sp.pi / 8), 0],
        [sp.I * sp.sin(sp.pi / 8), 0, 0, sp.cos(sp.pi / 8)],
    ],
)

# 2Q Non-Clifford SQRT_ISWAP = sp.Matrix( [ [sp.sqrt(2), 0, 0, 0], [0, 1, sp.I, 0], [0, sp.I, 1, 0], [0, 0, 0, sp.sqrt(2)], ], ) / sp.sqrt(2) BGATE = sp.Matrix( [ [sp.cos(sp.pi / 8), 0, 0, sp.I * sp.sin(sp.pi / 8)], [0, sp.sin(sp.pi / 8), sp.I * sp.cos(sp.pi / 8), 0], [0, sp.I * sp.cos(sp.pi / 8), sp.sin(sp.pi / 8), 0], [sp.I * sp.sin(sp.pi / 8), 0, 0, sp.cos(sp.pi / 8)], ], )

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ZX90

ZX90

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ZX90 * ZX90

ZX90 * ZX90

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-sp.I * tensor(Z, X)

-sp.I * tensor(Z, X)

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(SQRT_ISWAP * SQRT_ISWAP).equals(ISWAP)

(SQRT_ISWAP * SQRT_ISWAP).equals(ISWAP)

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for k, v in PAULI_2Q.items():
    print(f"{k} =")
    display(v)

for k, v in PAULI_2Q.items(): print(f"{k} =") display(v)

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def create_pauli_mapping(C: sp.Matrix):
    """Return Pauli mappings induced by a Clifford matrix."""
    mapping = {}
    for label, P in PAULI_2Q.items():
        P_mapped = C * P * dag(C)
        P_mapped.simplify()
        print(f"{label} ->")
        display(P_mapped)

        for k, v in PAULI_2Q.items():
            for sign in [1, -1, sp.I, -sp.I]:
                if P_mapped.equals(sign * v):
                    mapping[label] = (sign, k)
                    print(f"= {sign} * {k}\n")
                    break
        # print if not found
        if label not in mapping:
            print(f"Could not find mapping for {label}\n")

    return mapping

def create_pauli_mapping(C: sp.Matrix): """Return Pauli mappings induced by a Clifford matrix.""" mapping = {} for label, P in PAULI_2Q.items(): P_mapped = C * P * dag(C) P_mapped.simplify() print(f"{label} ->") display(P_mapped) for k, v in PAULI_2Q.items(): for sign in [1, -1, sp.I, -sp.I]: if P_mapped.equals(sign * v): mapping[label] = (sign, k) print(f"= {sign} * {k}\n") break # print if not found if label not in mapping: print(f"Could not find mapping for {label}\n") return mapping

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create_pauli_mapping(ZX90)

create_pauli_mapping(ZX90)

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create_pauli_mapping(ZZ90)

create_pauli_mapping(ZZ90)

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create_pauli_mapping(CNOT)

create_pauli_mapping(CNOT)

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create_pauli_mapping(CZ)

create_pauli_mapping(CZ)

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create_pauli_mapping(SWAP)

create_pauli_mapping(SWAP)

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create_pauli_mapping(ISWAP)

create_pauli_mapping(ISWAP)

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create_pauli_mapping(BSWAP)

create_pauli_mapping(BSWAP)

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create_pauli_mapping(SQRT_ISWAP)

create_pauli_mapping(SQRT_ISWAP)

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create_pauli_mapping(BGATE)

create_pauli_mapping(BGATE)