"""Quantum gate operations A quantum gate acting on :math:`n` qubits is a :math:`2^n×2^n` unitary matrix written in the computational basis. """ import numpy as np __all__ = [ "apply", "n_qubits", "controlled", # -- Single qubit gates -- "id", "x", "y", "z", "not_", "sqrt_not", "phase_shift", # -- 2 qubit gates -- "cnot", "swap", # -- 3 qubit gates -- "toffoli", "cswap", "fredkin", "deutsch", ] # type: ignore def apply(gate, qubits, state): """Apply a gate to the specified qubits of a state Parameters ---------- gate : ndarray[complex] qubits : sequence of int The qubits on which to act. Qubit 0 is the least significant qubit. state : ndarray[complex] Returns ------- new_state : ndarray[complex] """ n_gate_qubits = gate.shape[0].bit_length() - 1 n_state_qubits = state.shape[0].bit_length() - 1 assert len(qubits) == n_gate_qubits # We can view an n-qubit gate as a 2*n-tensor (n contravariant and n contravariant # indices) and an n-qubit state as an n-tensor (contravariant indices) # with each axis having length 2 (the state space of a single qubit). gate = gate.reshape((2,) * 2 * n_gate_qubits) state = state.reshape((2,) * n_state_qubits) # Our qubits are labeled from least significant to most significant, i.e. our # computational basis (for e.g. 2 qubits) is ordered like |00⟩, |01⟩, |10⟩, |11⟩. # We represent the state as a tensor in *row-major* order; this means that the # axis ordering is *backwards* compared to the qubit ordering (the least significant # qubit corresponds to the *last* axis in the tensor etc.) qubit_axes = tuple(n_state_qubits - 1 - np.asarray(qubits)) # Applying the gate to the state vector is then the tensor product over the appropriate axes axes = (np.arange(n_gate_qubits, 2 * n_gate_qubits), qubit_axes) new_state = np.tensordot(gate, state, axes=axes) # tensordot effectively re-orders the qubits such that the qubits we operated # on are in the most-significant positions (i.e. their axes come first). We # thus need to transpose the axes to place them back into their original positions. untouched_axes = tuple( idx for idx in range(n_state_qubits) if idx not in qubit_axes ) inverse_permutation = np.argsort(qubit_axes + untouched_axes) return np.transpose(new_state, inverse_permutation).reshape(-1) def _check_valid_gate(gate): if not ( # is an array isinstance(gate, np.ndarray) # is complex and np.issubdtype(gate.dtype, np.complex128) # is square and gate.shape[0] == gate.shape[1] # has size 2**n, n > 1 and np.log2(gate.shape[0]).is_integer() and gate.shape[0].bit_length() > 1 # is unitary and np.allclose(gate @ gate.conjugate().transpose(), np.identity(gate.shape[0])) ): raise ValueError("Gate is not valid") def n_qubits(gate): """Return the number of qubits that a gate acts on. Raises ValueError if 'gate' does not have a shape that is an integer power of 2. """ _check_valid_gate(gate) return gate.shape[0].bit_length() - 1 def controlled(gate): """Return a controlled quantum gate, given a quantum gate. If 'gate' operates on :math:`n` qubits, then the controlled gate operates on :math:`n+1` qubits, where the most-significant qubit is the control. Parameters ---------- gate : np.ndarray[complex] A quantum gate acting on :math:`n` qubits; a :math:`2^n×2^n` unitary matrix in the computational basis. Returns ------- controlled_gate : np.ndarray[(2**(n+1), 2**(n+1)), complex] """ _check_valid_gate(gate) n = gate.shape[0] zeros = np.zeros((n, n)) return np.block([[np.identity(n), zeros], [zeros, gate]]) # -- Single qubit gates -- #: The identity gate on 1 qubit id = np.identity(2, complex) #: Pauli X gate x = np.array([[0, 1], [1, 0]], complex) #: NOT gate not_ = x #: Pauli Y gate y = np.array([[0, -1j], [1j, 0]], complex) #: Pauli Z gate z = np.array([[1, 0], [0, -1]], complex) #: SQRT(NOT) gate sqrt_not = 0.5 * (1 + 1j * id - 1j * x) #: Hadamard gate hadamard = np.sqrt(0.5) * (x + z) def phase_shift(phi): "Return a gate that shifts the phase of :math:`|1⟩` by :math:`φ`." return np.array([[1, 0], [0, np.exp(1j * phi)]]) # -- Two qubit gates -- #: Controlled NOT gate cnot = controlled(x) #: SWAP gate swap = np.identity(4, complex)[:, (0, 2, 1, 3)] # -- Three qubit gates -- #: Toffoli (CCNOT) gate toffoli = controlled(cnot) #: Controlled SWAP gate cswap = controlled(swap) #: Fredkin gate fredkin = cswap def deutsch(phi): "Return a Deutsch gate for angle :math:`φ`." gate = np.identity(8, complex) gate[-2:, -2:] = np.array( [[1j * np.cos(phi), np.sin(phi)], [np.sin(phi), 1j * np.cos(phi)]] ) return gate

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