Operator

A quantum operator represents a mathematical object that acts on a quantum state, typically used to represent physical observables (such as energy) or to perform state transformations.

Quantum operators in Scaluq are expressed as Operator, which is composed of one or more PauliOperator instances.

PauliOperator

PauliOperator represents a tensor product of Pauli operations (I,X,Y,Z) acting on specific qubits, multiplied by a complex coefficient (coef).

Pauli operators can be defined flexibly using strings, lists, or bitmasks.

# 1. Initialization via string
# Applies X to qubit 0 and Y to qubit 2
p1 = PauliOperator("X 0 Y 2", coef=1.0)

# 2. Initialization via target qubit and Pauli ID lists
# IDs: 0=I, 1=X, 2=Y, 3=Z
p2 = PauliOperator(target_qubit_list=[0, 1, 2], pauli_id_list=[1, 0, 2], coef=1.0)

# 3. Initialization via bitmasks
# bit_flip_mask: locations of X, phase_flip_mask: locations of Z
p3 = PauliOperator(bit_flip_mask=0b101, phase_flip_mask=0b010, coef=1.0)

Operator

The Operator represents a general operator (such as a Hamiltonian) that is defined as a sum of multiple PauliOperator terms.

from scaluq.default.f64 import Operator, PauliOperator

#prepare two pauli
pauli1 = PauliOperator("Z 0 Z 1", coef=0.5) # (0.5 + 0.0j) Z0 Z1
pauli2 = PauliOperator("Z 0 Z 1", coef=0.3) # (0.3 + 0.0j) Z0 Z1
terms = [pauli1, pauli2]

op1 = Operator(terms)
print(op1) # (0.5 + 0.0j) Z0 Z1，(0.3 + 0.0j) Z0 Z1

# you can optimize the operator by merging terms with the same Pauli string
op1.optimize()
print(op1) # (0.8 + 0.0j) Z0 Z1

Calculating Expectation Values and Transition Amplitudes

You can use defined operators to extract physical information from a StateVector.

from scaluq.default.f64 import Operator, PauliOperator, StateVector

# Initialize a random state vector using Haar measure
state = StateVector.Haar_random_state(2)

# 1. Get expectation value: <state|op|state>
pauli = PauliOperator("Z 0 X 1", coef=1.0)
op = Operator([pauli])
exp_val = op.get_expectation_value(state)
print(f"Expectation value: {exp_val}") # e.g., (0.16471708718595834+0j)

# 2. Get transition amplitude: <target_state|op|state>
# The first argument is the "Bra" state, and the second is the "Ket" state.
target_state = StateVector.Haar_random_state(2) 
trans_amp = op.get_transition_amplitude(target_state, state) 
print(f"Transition amplitude: {trans_amp}") # e.g., (0.2589420813971237+0.355705558096908j)