Return a list of circuit identities, each consisting of a name, and
    two equivalent Circuits.

def identities():
    """ Return a list of circuit identities, each consisting of a name, and
    two equivalent Circuits."""

    circuit_identities = []

    # Pick random parameter
    theta = np.pi * np.random.uniform()

    # Single qubit gate identities

    name = "Hadamard is own inverse"
    circ0 = Circuit([H(0), H(0)])
    circ1 = Circuit([I(0)])
    circuit_identities.append([name, circ0, circ1])

    name = "Hadamards convert X to Z"
    circ0 = Circuit([H(0), X(0), H(0)])
    circ1 = Circuit([Z(0)])
    circuit_identities.append([name, circ0, circ1])

    name = "Hadamards convert Z to X"
    circ0 = Circuit([H(0), Z(0), H(0)])
    circ1 = Circuit([X(0)])
    circuit_identities.append([name, circ0, circ1])

    name = "S sandwich converts X to Y"
    circ0 = Circuit([S(0).H, X(0), S(0)])
    circ1 = Circuit([Y(0)])
    circuit_identities.append([name, circ0, circ1])

    name = "S sandwich converts Y to X"
    circ0 = Circuit([S(0), Y(0), S(0).H])
    circ1 = Circuit([X(0)])
    circuit_identities.append([name, circ0, circ1])

    name = "Hadamards convert RZ to RX"
    circ0 = Circuit([H(0), RZ(theta, 0), H(0)])
    circ1 = Circuit([RX(theta, 0)])
    circuit_identities.append([name, circ0, circ1])

    # ZYZ Decompositions

    name = "Hadamard ZYZ decomposition"
    circ0 = Circuit([H(0)])
    circ1 = Circuit([Z(0), Y(0)**0.5])
    circuit_identities.append([name, circ0, circ1])

    # CNOT identities

    name = "CZ to CNOT"
    circ0 = Circuit([CZ(0, 1)])
    circ1 = Circuit([H(1), CNOT(0, 1), H(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "SWAP to 3 CNOTs"
    circ0 = Circuit([SWAP(0, 1)])
    circ1 = Circuit([CNOT(0, 1), CNOT(1, 0), CNOT(0, 1)])
    circuit_identities.append([name, circ0, circ1])

    name = "SWAP to 3 CZs"
    circ0 = Circuit([SWAP(0, 1)])
    circ1 = Circuit([H(1), CZ(0, 1), H(0), H(1), CZ(1, 0), H(0), H(1),
                     CZ(0, 1), H(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "ISWAP decomposition to SWAP and CNOT"
    circ0 = Circuit([ISWAP(0, 1)])
    circ1 = Circuit([SWAP(0, 1), H(1), CNOT(0, 1), H(1), S(0), S(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "ISWAP decomposition to SWAP and CZ"
    # This makes it clear why you can commute RZ's across ISWAP
    circ0 = Circuit([ISWAP(0, 1)])
    circ1 = Circuit([SWAP(0, 1), CZ(0, 1), S(0), S(1)])
    circuit_identities.append([name, circ0, circ1])

    # http://info.phys.unm.edu/~caves/courses/qinfo-f14/lectures/lectures21-23.pdf
    name = "CNOT sandwich with X on control"
    circ0 = Circuit([CNOT(0, 1), X(0), CNOT(0, 1)])
    circ1 = Circuit([X(0), X(1)])
    circuit_identities.append([name, circ0, circ1])

    # http://info.phys.unm.edu/~caves/courses/qinfo-f14/lectures/lectures21-23.pdf
    name = "CNOT sandwich with Z on target"
    circ0 = Circuit([CNOT(0, 1), Z(1), CNOT(0, 1)])
    circ1 = Circuit([Z(0), Z(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "DCNOT (Double-CNOT) to iSWAP"
    circ0 = Circuit([CNOT(0, 1), CNOT(1, 0)])
    circ1 = Circuit([H(0), S(0).H, S(1).H, ISWAP(0, 1), H(1)])
    circuit_identities.append([name, circ0, circ1])

    # Commuting single qubit gates across 2 qubit games

    name = "Commute X on CNOT target"
    circ0 = Circuit([X(1), CNOT(0, 1)])
    circ1 = Circuit([CNOT(0, 1), X(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "Commute X on CNOT control"
    circ0 = Circuit([X(0), CNOT(0, 1)])
    circ1 = Circuit([CNOT(0, 1), X(0), X(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "Commute Z on CNOT target"
    circ0 = Circuit([Z(1), CNOT(0, 1)])
    circ1 = Circuit([CNOT(0, 1), Z(0), Z(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "Commute Z on CNOT control"
    circ0 = Circuit([Z(0), CNOT(0, 1)])
    circ1 = Circuit([CNOT(0, 1), Z(0)])
    circuit_identities.append([name, circ0, circ1])

    #  Canonical gate identities

    name = "Canonical gates: CZ to ZZ"
    circ0 = Circuit([CZ(0, 1),
                     S(0),
                     S(1)])
    circ1 = Circuit([ZZ(0.5, 0, 1)])
    circuit_identities.append([name, circ0, circ1])

    name = "Canonical gates: XX to ZZ"
    circ0 = Circuit([H(0), H(1),
                     XX(0.5, 0, 1),
                     H(0), H(1)])
    circ1 = Circuit([ZZ(0.5, 0, 1)])
    circuit_identities.append([name, circ0, circ1])

    name = "Canonical gates: CNOT to XX"
    circ0 = Circuit([CNOT(0, 1)])
    circ1 = Circuit([H(0),
                     XX(0.5, 0, 1),
                     H(0), H(1),
                     S(0).H, S(1).H,
                     H(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "Canonical gates: SWAP to Canonical"
    circ0 = Circuit([SWAP(0, 1)])
    circ1 = Circuit([CANONICAL(0.5, 0.5, 0.5, 0, 1)])
    circuit_identities.append([name, circ0, circ1])

    name = "ISWAP to Canonical"
    circ0 = Circuit([ISWAP(0, 1)])
    circ1 = Circuit([CANONICAL(0.5, 0.5, 1.0, 0, 1)])
    circuit_identities.append([name, circ0, circ1])

    name = "ISWAP to Canonical in Weyl chamber"
    circ0 = Circuit([ISWAP(0, 1)])
    circ1 = Circuit([X(0), CANONICAL(0.5, 0.5, 0.0, 0, 1), X(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "DCNOT to Canonical"
    circ0 = Circuit([CNOT(0, 1), CNOT(1, 0)])
    circ1 = Circuit([H(0), S(0).H, S(1).H, X(0),
                     CANONICAL(0.5, 0.5, 0.0, 0, 1), X(1), H(1)])
    circuit_identities.append([name, circ0, circ1])

    # Multi-qubit circuits

    name = "CNOT controls commute"
    circ0 = Circuit([CNOT(1, 0), CNOT(1, 2)])
    circ1 = Circuit([CNOT(1, 2), CNOT(1, 0)])
    circuit_identities.append([name, circ0, circ1])

    name = "CNOT targets commute"
    circ0 = Circuit([CNOT(0, 1), CNOT(2, 1)])
    circ1 = Circuit([CNOT(2, 1), CNOT(0, 1)])
    circuit_identities.append([name, circ0, circ1])

    name = "Commutation of CNOT target/control"
    circ0 = Circuit([CNOT(0, 1), CNOT(1, 2)])
    circ1 = Circuit([CNOT(1, 2), CNOT(0, 2), CNOT(0, 1)])
    circuit_identities.append([name, circ0, circ1])

    name = "Indirect CNOT and 4 CNOTS with intermediate qubit"
    circ0 = Circuit([CNOT(0, 2), I(1)])
    circ1 = Circuit([CNOT(0, 1),
                     CNOT(1, 2),
                     CNOT(0, 1),
                     CNOT(1, 2)])
    circuit_identities.append([name, circ0, circ1])

    name = "CZs with shared shared qubit commute"
    circ0 = Circuit([CZ(0, 1), CZ(1, 2)])
    circ1 = Circuit([CZ(1, 2), CZ(0, 1)])
    circuit_identities.append([name, circ0, circ1])

    name = "Toffoli gate CNOT decomposition"
    circ0 = Circuit([CCNOT(0, 1, 2)])
    circ1 = ccnot_circuit([0, 1, 2])
    circuit_identities.append([name, circ0, circ1])

    # Parametric circuits

    name = "ZZ to CNOTs"  # 1108.4318
    circ0 = Circuit([ZZ(theta/np.pi, 0, 1)])
    circ1 = Circuit([CNOT(0, 1), RZ(theta, 1), CNOT(0, 1)])
    circuit_identities.append([name, circ0, circ1])

    name = "XX to CNOTs"  # 1108.4318
    circ0 = Circuit([XX(theta/np.pi, 0, 1)])
    circ1 = Circuit([H(0), H(1), CNOT(0, 1), RZ(theta, 1), CNOT(0, 1),
                     H(0), H(1)])
    circuit_identities.append([name, circ0, circ1])

    name = "XX to CNOTs (2)"
    circ0 = Circuit([XX(theta/np.pi, 0, 1)])
    circ1 = Circuit([Y(0)**0.5, Y(1)**0.5, CNOT(0, 1), RZ(theta, 1),
                     CNOT(0, 1), Y(0)**-0.5, Y(1)**-0.5])
    circuit_identities.append([name, circ0, circ1])

    name = "YY to CNOTs"
    circ0 = Circuit([YY(theta/np.pi, 0, 1)])
    circ1 = Circuit([X(0)**0.5, X(1)**0.5, CNOT(0, 1), RZ(theta, 1),
                     CNOT(0, 1), X(0)**-0.5, X(1)**-0.5])
    circuit_identities.append([name, circ0, circ1])

    def cphase_to_zz(gate: CPHASE):
        t = - gate.params['theta'] / (2 * np.pi)
        q0, q1 = gate.qubits
        circ = Circuit([ZZ(t, q0, q1), TZ(-t, q0), TZ(-t, q1)])
        return circ

    def cphase00_to_zz(gate: CPHASE00):
        t = - gate.params['theta'] / (2 * np.pi)
        q0, q1 = gate.qubits
        circ = Circuit([X(0), X(1),
                        ZZ(t, q0, q1), TZ(-t, q0), TZ(-t, q1),
                        X(0), X(1)])
        return circ

    def cphase01_to_zz(gate: CPHASE00):
        t = - gate.params['theta'] / (2 * np.pi)
        q0, q1 = gate.qubits
        circ = Circuit([X(0),
                        ZZ(t, q0, q1), TZ(-t, q0), TZ(-t, q1),
                        X(0)])
        return circ

    def cphase10_to_zz(gate: CPHASE00):
        t = - gate.params['theta'] / (2 * np.pi)
        q0, q1 = gate.qubits
        circ = Circuit([X(1),
                        ZZ(t, q0, q1), TZ(-t, q0), TZ(-t, q1),
                        X(1)])
        return circ

    name = "CPHASE to ZZ"
    gate = CPHASE(theta, 0, 1)
    circ0 = Circuit([gate])
    circ1 = cphase_to_zz(gate)
    circuit_identities.append([name, circ0, circ1])

    name = "CPHASE00 to ZZ"
    gate = CPHASE00(theta, 0, 1)
    circ0 = Circuit([gate])
    circ1 = cphase00_to_zz(gate)
    circuit_identities.append([name, circ0, circ1])

    name = "CPHASE01 to ZZ"
    gate = CPHASE01(theta, 0, 1)
    circ0 = Circuit([gate])
    circ1 = cphase01_to_zz(gate)
    circuit_identities.append([name, circ0, circ1])

    name = "CPHASE10 to ZZ"
    gate = CPHASE10(theta, 0, 1)
    circ0 = Circuit([gate])
    circ1 = cphase10_to_zz(gate)
    circuit_identities.append([name, circ0, circ1])

    return circuit_identities