Modeling physical noise

Background

We consider incoherent errors as the predominant noise mechanism in quantum simulation. These errors can originate in interactions between qubits and a fluctuating environment. This mechanism influences the time evolution of the qubits in a way that the errors do not add up coherently. Simple examples of this are superconducting qubits coupling to lossy two-level systems, ion-trap devices with heated vibrational modes or unwanted control-pulse scattering, and spin-qubits in the presence of fluctuating background magnetic-field. During the last few decades, such noise mechanisms have been described successfully using the Lindblad master equation. The HQS Noise App applies this method to model the effect of noise in a digital quantum simulation.

Master equation

Lindbladian

The noise mapping performed by the HQS Noise App is valid in the regime of Markovian master equations, meaning for noise mechanisms with short memory times. In particular, we apply master equations in the Lindbladian form:

\[ \dot{\rho} = \sum_{ij} M_{ij} \left( A_i \rho A^\dagger_j - \frac{1}{2} A^\dagger_j A_i \rho -\frac{1}{2} \rho A^\dagger_j A_i \right) \equiv L[\rho] , \]

where \(\rho\) is the density matrix of the qubits (spins). Operators \(A_i\) offer a basis for representing the noise operators, while matrix \(M\) gives (generalized) noise rates. There is a freedom for definition of operators \(A_i\). Thus, only in combination with matrix \(M\) (and particularly its non-diagonal entries) can we make judgements about the dominant noise processes and rates. In the following section we detail our choice of operators \(A_i\) so that the user may understand how their physical noise model can be represented by \(M\).

Noise operators \(A\)

In our software, operators \(A_i\) are chosen to be Pauli-matrices, \(\sigma^x_n, \textrm{i}\sigma^y_n, \sigma^z_n\) (single-qubit noise on qubit \(n\)), and more generally Pauli products, \(\sigma^x_n\sigma^x_m,\sigma^x_n\textrm{i}\sigma^y_m,\sigma^x_n\sigma^z_m,\ldots\) (multi-qubit noise). An imaginary factor is added to Y-operators ( \(\sigma^y_n\rightarrow \textrm{i}\sigma^y_n\) ) so that we avoid having imaginary numbers in matrix \(M\) of the physical noise. (The effective rate matrix can however end up having imaginary entries.)

Rate matrix \(M\)

For clarity in the following, we provide our definition of the spin-lowering and spin-raising operators:

\[ \sigma^-=\frac{1}{2}\left( \sigma^x + \textrm{i}\sigma^y \right) = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}, \\ \sigma^+=\frac{1}{2}\left( \sigma^x - \textrm{i}\sigma^y \right) = \begin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix} \]

The Lindblad equation for damping of qubit \(1\) has the form:

\[ \dot{\rho} = \gamma_{\textrm{damping}} \left( \sigma^-_1 \rho \sigma^+_1 - \frac{1}{2} \sigma^+_1 \sigma^-_1 \rho - \frac{1}{2} \rho \sigma^+_1 \sigma^-_1 \right) \]

\[ \dot{\rho} = \gamma_{\textrm{damping}} \left[ \frac{1}{4} \left( \sigma^x_1 + \textrm{i} \sigma^y_1 \right) \rho \left( \sigma^x_1 - \textrm{i} \sigma^y_1 \right) -\frac{1}{8} \left( \sigma^x_1 - \textrm{i} \sigma^y_1 \right) \left( \sigma^x_1 + \textrm{i} \sigma^y_1 \right) \rho -\frac{1}{8} \rho \left( \sigma^x_1 - \textrm{i} \sigma^y_1 \right) \left( \sigma^x_1 + \textrm{i} \sigma^y_1 \right) \right] . \]

Our noise matrix then has four non-zero values:

\[ M_{1X, 1X} = \frac{\gamma_\textrm{damping}}{4}, \\ M_{1X, 1 \textrm{i} Y} = \frac{\gamma_\textrm{damping}}{4}, \\ M_{1 \textrm{i} Y, 1X} = \frac{\gamma_\textrm{damping}}{4}, \\ M_{1 \textrm{i} Y, 1 \textrm{i} Y} = \frac{\gamma_\textrm{damping}}{4}. \]

Similarly, dephasing of qubit 1 corresponds to noise matrix with non-zero entry:

\[ M_{1Z, 1Z} = \gamma_\textrm{dephasing}. \]

Depolarization corresponds to identical noise in all directions with rates:

\[ M_{1X, 1X} = \frac{\gamma_\textrm{depolarising}}{4}, \\ M_{1Y, 1Y} = \frac{\gamma_\textrm{depolarising}}{4}, \\ M_{1Z, 1Z} = \frac{\gamma_\textrm{depolarising}}{4}. \]

This definition is based on the density matrix (here) approaching a diagonal matrix with rate \(\gamma_\textrm{depolarising}\).

Form of noise matrix \(M\) for physical model

Since our physical noise acts individually on each qubit \(n\), our physical noise matrix is a sum over individual contributions \(m^n_{ij}\), where \(i,j\) can be \(nX,n\textrm{i}Y\) or \(nZ\). The matrix is real valued. In the case where noise acts on all qubits at all times, the total noise matrix has the form:

\[ M = m^0 \oplus m^1 \oplus m^2 \oplus \ldots , \]

whereas on the other hand, if noise affects only qubits that are being operated on, we have:

\[ M = \bigoplus_{n\in \textrm{acted qubits}} m^n . \] It is important to note that this is the model for physical gates: the form of the effective noise can also include multi-qubit operators and imaginary (non-diagonal) rates, see sections mapping and examples.

Noisy gates

Let us now introduce our model of gate-based quantum simulation with incoherent errors.

Super-operator matrix notation

For simplicity, we now use a notation where each noiseless gate operation (with a given unitary transformation \(U\)) is represented as matrix-multiplication:

\[ U \rho U^{\dagger} \rightarrow G \rho \\ U_2 U_1 \rho U_1^{\dagger} U_{2}^{\dagger} \rightarrow G_2 G_1 \rho . \]

On the right-hand side, the density "matrix" \(\rho\) has become a vector, and the unitary transformations \(G_i\) are matrices acting on it.

Model

In our modeling, we split each gate operation into the ideal unitary gate \(G\) and non-unitary noise \(N\). We then replace each gate in some gate decomposition with:

\[ G \rightarrow N G , \\ G_2 G_1 \rightarrow N_2 G_2 N_1 G_1 , \]

where the noise transformation is given by the Lindblad time-evolution over some physical gate time \(\tau_i\),

\[ N_i \equiv e^{L\tau_i} , \]

where \(L\) is the Lindblad operator. Such a description can always be established if physical gate times \(\tau_i\) are much shorter than the decoherence rates related to this gate (\(\tau_i \gamma_i \ll 1\)). The question of how correct this description is of some specific hardware realization is then redirected to the question of the correct choice of noise matrix \(M\).

In the context of correctly describing the interplay of unitary gates and noise (see also the mapping section) the notion of so called small-angle gates is also important. In short a small-angle gate is a unitary gate that is close to the identity gate compared to some other quantity of interest. The term is inspired by rotation gates (e.g. RotateX) that are close to the identity operation for small rotation angles. Small-anlges are not an absolute quantity but if we are talking about unitary gates G together with noise contributions N with a small prefactor p, we would consider a unitary a small-angle gate when the deviation D of G from the identity is small enough compared to p that

\[ p GN = p NG + p [D,N] \approx p NG \] The question of how correct this description is for some specific hardware realization is then redirected to the question of the correct choice of noise matrix \(M\).