Modeling Physical Noise

Background

We consider incoherent errors as the predominant noise mechanism in quantum simulation. These errors can arise from 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 include 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. In recent decades, such noise mechanisms have been successfully described 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 for noise mechanisms with short memory times, i.e. in the regime of Markovian dynamics. 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 \(M\) provides (generalized) noise rates. There is a freedom in the definition of operators \(A_i\). Thus, only in combination with the 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 \(A_i\) operators so that the user can understand how their physical noise model can be represented by \(M\).

Noise Operators \(A\)

In our software, the \(A_i\) operators 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 ( \(\sigma^y_n\rightarrow \textrm{i}\sigma^y_n\) ) to avoid having imaginary numbers in the physical noise matrix \(M\). (The effective rate matrix, however, may end up having imaginary entries.)

Rate Matrix \(M\)

For clarity, 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 on 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 on qubit \(1\) corresponds to a noise matrix with a 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 the 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, 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.

Superoperator Matrix Notation

For simplicity, we now use a notation where each noiseless gate (with a given unitary transformation \(U\)) is represented as a linear superoperator acting on the density matrix:

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

One can rewrite the right-hand side, such that 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 the 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 when the physical gate times \(\tau_i\) are short in relation to the decoherence rates of this gate (\(\tau_i \gamma_i \ll 1\)), ensuring the noise to be small. A detailed justification of this model, including a proof that the separated noise term is a valid Lindblad operator (to the lowest order of the noise strength), can be found in our puplication on the subject. The question of how correct this description is of a 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), which are close to the identity operation for small rotation angles. Small-angles are not an absolute quantity, but if we talk about unitary gates G together with noise contributions N with a small prefactor p, we would consider a unitary to be a small-angle gate if the deviation D of G from the identity is small enough compared to p that

\[ p GN = p NG + p [D,N] \approx p NG . \]