# Nodes

## Optimization variables

When performing optimizations, you can use these operations to create the optimizable variables that can be tuned by the optimizer in order to minimize your cost function.

 anchored_difference_bounded_variables Create a sequence of optimizable variables with an anchored difference bound. complex_optimizable_pwc_signal Create a complex optimizable piecewise-constant signal. optimizable_scalar Create an optimizable scalar Tensor, which can be bounded, semi-bounded, or unbounded. optimization_variable Create a 1D Tensor of optimization variables, which can be bounded, semi-bounded, or unbounded. real_fourier_pwc_signal Create a piecewise-constant signal constructed from Fourier components. real_fourier_stf_signal Create a real sampleable signal constructed from Fourier components. real_optimizable_pwc_signal Create a real optimizable piecewise-constant signal.

## Piecewise-constant tensor functions (PWCs)

You can use these operations to create piecewise-constant functions either to represent control signals or system Hamiltonians.

 complex_pwc_signal Create a complex piecewise-constant signal from moduli and phases. constant_pwc Create a piecewise-constant function of time that is constant over a specified duration. constant_pwc_operator Create a constant piecewise-constant operator over a specified duration. pwc Create a piecewise-constant function of time. pwc_operator Create a constant operator multiplied by a piecewise-constant signal. pwc_signal Create a piecewise-constant signal (scalar-valued function of time). pwc_sum Create the sum of multiple piecewise-constant terms. real_fourier_pwc_signal Create a piecewise-constant signal constructed from Fourier components. sample_pwc Sample a Pwc at the given times. symmetrize_pwc Create the symmetrization of a piecewise-constant function. time_concatenate_pwc Concatenate multiple piecewise-constant functions in the time dimension. time_reverse_pwc Reverse in time a piecewise-constant function.

## Sampleable tensor functions (STFs)

You can use these functions to represent time-dependent functions in Boulder Opal.

 constant_stf Create a constant sampleable tensor-valued function of time. constant_stf_operator Create a constant operator. identity_stf Create an Stf representing the identity function, f(t) = t. real_fourier_stf_signal Create a real sampleable signal constructed from Fourier components. sample_stf Sample an Stf at the given times. stf_operator Create a constant operator multiplied by a sampleable signal. stf_sum Create the sum of multiple sampleable functions.

## Filtering and discretizing

You can use these functions to filter or resample the control signals.

 convolve_pwc Create the convolution of a piecewise-constant function with a kernel. discretize_stf Create a piecewise-constant function by discretizing a sampleable function. filter_and_resample_pwc Filter a piecewise-constant function by convolving it with a kernel and resample it again. gaussian_convolution_kernel Create a convolution kernel representing a normalized Gaussian. sinc_convolution_kernel Create a convolution kernel representing the sinc function.

## Predefined signals

You can use these operations to create common analytical signals.

For creating analytical signals not bound to a graph see the Signal library module.

 signals.SegmentationType An enumeration of segmentation types for piecewise-constant signals. signals.cosine_pulse_pwc Create a Pwc representing a cosine pulse. signals.gaussian_pulse_pwc Create a Pwc representing a Gaussian pulse. signals.gaussian_pulse_stf Create an Stf representing a Gaussian pulse. signals.hann_series_pwc Create a Pwc representing a sum of Hann window functions. signals.hann_series_stf Create an Stf representing a sum of Hann window functions. signals.linear_ramp_pwc Create a Pwc representing a linear ramp. signals.linear_ramp_stf Create an Stf representing a linear ramp. signals.sech_pulse_pwc Create a Pwc representing a hyperbolic secant pulse. signals.sech_pulse_stf Create an Stf representing a hyperbolic secant pulse. signals.sinusoid_pwc Create a Pwc representing a sinusoidal oscillation. signals.sinusoid_stf Create an Stf representing a sinusoidal oscillation. signals.square_pulse_pwc Create a Pwc representing a square pulse. signals.tanh_ramp_pwc Create a Pwc representing a hyperbolic tangent ramp. signals.tanh_ramp_stf Create an Stf representing a hyperbolic tangent ramp.

## Quantum information

You can use these operations to calculate common operations and metrics from quantum information theory.

 annihilation_operator Create an annihilation operator in the truncated Fock space. coherent_state Create a coherent state (or a batch of them). creation_operator Create a creation operator in the truncated Fock space. density_matrix_expectation_value Calculate the expectation value of an operator with respect to a density matrix. density_matrix_infidelity Calculate the infidelity between two states represented by density matrices. displacement_operator Create a displacement operator (or a batch of them). embed_operators Embed an operator or set of operators into a larger Hilbert space. expectation_value Calculate the expectation value of an operator with respect to a state. fock_state Create a Fock state (or a batch of them). inner_product Calculate the inner product of two vectors. kronecker_product_list Calculate the Kronecker product between a list of operators. number_operator Create a number operator in the truncated Fock space. outer_product Calculate the outer product of two vectors. partial_trace Calculate the partial trace of a density matrix. pauli_kronecker_product Place Pauli matrices into their two-dimensional subsystems of a system and returns the Kronecker product. pauli_matrix Create a Pauli matrix from a label. squeeze_operator Create a squeeze operator (or a batch of them). state_infidelity Calculate the infidelity of two pure states. unitary_infidelity Calculate the infidelity between a target operation and the actual implemented unitary. wigner_transform Transform a density matrix into a Wigner function (or a batch of them).

## Time evolution

You can use these operations to calculate the time evolution of your open or closed quantum system, either for simulations or optimizations.

 density_matrix_evolution_pwc Calculate the state evolution of an open system described by the GKS–Lindblad master equation. estimated_krylov_subspace_dimension_lanczos Calculate an appropriate Krylov subspace dimension ($$k$$) to use in the Lanczos integrator while keeping the total error in the evolution below a given error tolerance. filter_function Evaluate the filter function for a control Hamiltonian and a noise operator at the given frequency elements. frequency_domain_noise_operator Create a control-frame noise operator in the frequency domain for a control Hamiltonian and a noise operator at the given frequencies. jump_trajectory_evolution_pwc Calculate the state evolution of an open system described by the GKS–Lindblad master equation using a jump-based trajectory method. spectral_range Obtain the range of the eigenvalues of a Hermitian operator. state_evolution_pwc Calculate the time evolution of a state generated by a piecewise-constant Hamiltonian by using the Lanczos method. steady_state Find the steady state of a time-independent open quantum system. time_evolution_operators_pwc Calculate the unitary time-evolution operators for a system defined by a piecewise-constant Hamiltonian. time_evolution_operators_stf Calculate the time-evolution operators for a system defined by an STF Hamiltonian by using a 4th order Runge–Kutta method.

## Optimal and robust control

You can use these operations, together with the operations for creating optimization variables to define convenient cost functions for optimal and robust control.

 infidelity_pwc Create the total infidelity of the given piecewise-constant system. infidelity_stf Create the total infidelity of a given system with a sampleable Hamiltonian. target Create information about the target for system time evolution.

## Large systems

You can use these operations to build graphs that efficiently handle large quantum systems.

 constant_sparse_pwc_operator Create a constant sparse piecewise-constant operator over a specified duration. density_matrix_evolution_pwc Calculate the state evolution of an open system described by the GKS–Lindblad master equation. estimated_krylov_subspace_dimension_lanczos Calculate an appropriate Krylov subspace dimension ($$k$$) to use in the Lanczos integrator while keeping the total error in the evolution below a given error tolerance. jump_trajectory_evolution_pwc Calculate the state evolution of an open system described by the GKS–Lindblad master equation using a jump-based trajectory method. sparse_pwc_hermitian_part Create the Hermitian part of a piecewise-constant operator. sparse_pwc_operator Create a sparse piecewise-constant operator (sparse-matrix-valued function of time). sparse_pwc_sum Create the sum of multiple sparse-matrix-valued piecewise-constant functions. spectral_range Obtain the range of the eigenvalues of a Hermitian operator. state_evolution_pwc Calculate the time evolution of a state generated by a piecewise-constant Hamiltonian by using the Lanczos method. steady_state Find the steady state of a time-independent open quantum system.

## Mølmer–Sørensen gates

You can use these operations to efficiently model systems described by Mølmer–Sørensen interactions.

 ions.ms_dephasing_robust_cost Calculate the cost for robust optimization of a Mølmer–Sørensen gate. ions.ms_displacements Calculate the displacements for each mode and ion combination where ions are described by a Mølmer–Sørensen-type interaction. ions.ms_infidelity Calculate the final operational infidelity of the Mølmer–Sørensen gate. ions.ms_phases Calculate the relative phases for all pairs of ions described by a Mølmer–Sørensen-type interaction when single-tone individually-addressed laser beams are used. ions.ms_phases_multitone Calculate the relative phases for all pairs of ions described by a Mølmer–Sørensen-type interaction where the ions are being addressed by a multitone global beam.

## Random operations

You can use these operations to create random quantities, which take different values each time they are evaluated. These operations are most useful in simulations and stochastic optimizations.

 random.choices Create random samples from the data that you provide. random.colored_noise_stf_signal Sample the one-sided power spectral density (PSD) of a random noise process in the time domain and returns the resultant noise trajectories as an Stf. random.normal Create a sample of normally distributed random numbers. random.uniform Create a sample of uniformly distributed random numbers.

## Manipulating tensors

You can use these operations to manipulate the structures of tensors.

 concatenate Concatenate a list of tensors along a specified dimension. cumulative_sum Calculate the cumulative sum of a tensor along a specified dimension. einsum Perform tensor contraction via Einstein summation convention. max Find the maximum value in a tensor along one or multiple of its axes. min Find the minimum value in a tensor along one or multiple of its axes. repeat Repeat elements of a tensor. reshape Reshape a tensor into a new shape, keeping the order of its elements. reverse Reverse a tensor along some specified dimensions. sum Sum the elements in a tensor (or a list of tensors with the same shape) along one or multiple of its axes. tensor Create a real or complex Tensor with the data provided. transpose Reorder the dimensions of a tensor.

## Arithmetic

 add Calculate the element-wise sum between numbers, np.ndarrays, Tensors, Pwcs, or Stfs. cumulative_sum Calculate the cumulative sum of a tensor along a specified dimension. floordiv Calculate the element-wise rounded-down division between real numbers, np.ndarrays, Tensors, Pwcs, or Stfs. multiply Calculate the element-wise product between numbers, np.ndarrays, Tensors, Pwcs, or Stfs. negative Calculate the element-wise numerical negative value of an object. pow Calculate the element-wise power between numbers, np.ndarrays, Tensors, Pwcs, or Stfs. subtract Calculate the element-wise difference between numbers, np.ndarrays, Tensors, Pwcs, or Stfs. sum Sum the elements in a tensor (or a list of tensors with the same shape) along one or multiple of its axes. truediv Calculate the element-wise division between numbers, np.ndarrays, Tensors, Pwcs, or Stfs.

## Linear algebra

 adjoint Calculate the element-wise adjoint of the last two dimensions of an object. density_matrix_expectation_value Calculate the expectation value of an operator with respect to a density matrix. einsum Perform tensor contraction via Einstein summation convention. embed_operators Embed an operator or set of operators into a larger Hilbert space. expectation_value Calculate the expectation value of an operator with respect to a state. hermitian_part Calculate the Hermitian part of an object. hessian Calculate a single Hessian matrix for all the variables. inner_product Calculate the inner product of two vectors. kron Calculate the Kronecker product between np.ndarrays, Tensors, Pwcs, or Stfs. kronecker_product_list Calculate the Kronecker product between a list of operators. matmul Calculate the matrix multiplication between np.ndarrays, Tensors, Pwcs, or Stfs. outer_product Calculate the outer product of two vectors. partial_trace Calculate the partial trace of a density matrix. trace Calculate the trace of an object. transpose Reorder the dimensions of a tensor.

## Basic mathematical functions

 abs Calculate the element-wise absolute value of an object. angle Calculate the element-wise argument of an object. arccos Calculate the element-wise arccosine of an object. arcsin Calculate the element-wise arcsine of an object. arctan Calculate the element-wise arctangent of an object. complex_value Create element-wise complex values from real numbers, np.ndarrays, Tensors, Pwcs, or Stfs, that is, the real and imaginary parts. conjugate Calculate the element-wise complex conjugate of an object. cos Calculate the element-wise cosine of an object. cosh Calculate the element-wise hyperbolic cosine of an object. exp Calculate the element-wise exponential of an object. imag Calculate the element-wise imaginary part of an object. log Calculate the element-wise natural logarithm of an object. real Calculate the element-wise real part of an object. sin Calculate the element-wise sine of an object. sinh Calculate the element-wise hyperbolic sine of an object. sqrt Calculate the element-wise square root of an object. tan Calculate the element-wise tangent of an object. tanh Calculate the element-wise hyperbolic tangent of an object.