Foundations of noise mapping

In this section, we will go through the mathematical foundations of the noise mapping performed by the HQS Noise App. Physical noise on qubits is transformed into effective noise on spins in the quantum simulation. For a detailed discussion see the full paper on noise mapping.

Trotterization of the Time Evolution Operator

Decomposition Blocks

Digital quantum simulation is based on the trotterization of the time evolution operator, such that:

\[ e^{-\textrm{i} H t} = \left[ e^{-\textrm{i} H t/m} \right]^m \equiv \left[ e^{-\textrm{i} H \bar t} \right]^m \approx \left[ \Pi_{j} e^{-\textrm{i} H_j \bar t} \right]^m . \]

We consider a time-independent Hamiltonian \(H\) and time steps \(\bar t = t/m\). The total Hamiltonian \(H\) is divided into elements \(H_j\),

\[ H=\sum_j H_j . \]

Each \( H_j \) leads to a unitary transformation of the time evolution \(\exp\left(-\textrm{i} H_j\bar t \right)\), where the choice of a small value for \( \bar t \) makes this unitary transformation a "small-angle" transformation.

The goal of the division is to have a sequence of unitary transformations that can be efficiently implemented on hardware. In the HQS Noise App, such divisions are marked as circuit DecompositionBlocks. An approximation (error) occurs when individual unitaries do not commute.

Small- and Large-Angle Decompositions

Ideally, these unitaries can be implemented directly on hardware by single-qubit transitions or multi-qubit interactions, e.g., by control-field pulses. These operations then have a one-to-one correspondence with equivalent small-angle gates,

\[ e^{-\textrm{i} H_j \bar t} \rightarrow G_j . \]

As described in section modeling, in the presence of noise, each gate \(G_j\) is assigned a noise operator \(N_j\).

On the other hand, some unitaries may need to be decomposed into a series of elementary gates,

\[ e^{-\textrm{i} H_j \bar t} \rightarrow \Pi_l G^l_j . \]

Here, each gate \(G^l_j\) comes with noise \(N^l_j\). In practice, such decompositions often include "large-angle" gates, such as CNOT gates or \(\pi\) rotations.

Trotter Error

The error in the simplest, first-order, Trotter expansion is of size:

\[ \epsilon_\textrm{error} \propto \bar g^2 , \] where \(\bar g= g \bar t\), and \(g\) is some typical energy of non-commuting terms in the Hamiltonian. Note that the error goes to zero in the limit \(\bar t\rightarrow 0\).

Effective Noise in the Simulated System

Here, we detail how we map the physical noise of a quantum computer to effective noise in the simulated system. To do this, it is sufficient to analyze unitary gates and non-unitary noise operations within a single Trotter step. It also turns out that the analysis of noise rotations can be done on each decomposition block separately.

Noise Transformations

Let us first consider a decomposition of some small-angle unitary operation \(\exp\left(-\textrm{i} H_j\bar t \right)\) into two large-angle gates. The generalization to an arbitrary number of gates is straightforward.

Using the fact that gates \(G_i\) are unitary, and thereby invertible, we can write:

\[ N_2 G_2 N_1 G_1 \rho = N_2 G_2 N_1 G_2^{-1} G_2 G_1 \rho \equiv N_2 N_1' G_2 G_1 \rho \equiv N G \rho , \]

where \(G\) corresponds to a small-angle unitary operation (as it arises from a short time evolution),

\[ G = G_2 G_1 , \]

and the effective noise operator is:

\[ N = N_2 N_1' , \\ N'_1 = G_2 N_1 G_2^{-1} . \] We see that the first noise operator has been transformed by the (large-angle) unitary gate \(G_2\). Is is important to not that, since the operators \(G\) and \(N\) describe small-angle (or probability) processes, these operators will later define the effective Lindbladian of the simulated system.

More generally, for an arbitrary decomposition we can write:

\[ e^{-\textrm{i} H_j \bar t} \rightarrow \Pi_{l=\textrm{last}}^\textrm{first} N^l_j G^l_j = N_j G_j , \]

with:

\[ G_j \equiv \Pi_{l=\textrm{last}}^\textrm{first} G^l_j, \\ N_j \equiv \prod P_j \\ P_0 = N_j^\textrm{last} \\ P_1 = G_j^\textrm{last} N_j^\textrm{last-1} \left( G_j^\textrm{last}\right)^{-1} \\ P_2 = G_j^\textrm{last} G_j^\textrm{last-1} N_j^\textrm{last-2} \left( G_j^\textrm{last-1}\right)^{-1} \left( G_j^\textrm{last}\right)^{-1} \\ \ldots \]

It follows that the full time evolution operator can be written as:

\[ \Pi_j \exp\left( -\textrm{i} H_j\bar t \right) = \Pi_{j} N_j G_j . \]

Here, each operator \(G_j\) as well as \(N_j\) is a "small-angle" (or probability) transformation. Finally, within the assumption that rotations over small-angle transformations can be neglected, we obtain the expression for the full Trotter step:

\[ \Pi_{j} N_j G_j \approx \left(\Pi_j N_j \right) \left(\Pi_j G_j \right) = \left( \Pi_j N_j\right) G. \]

(Note that the SWAP algorithm for quantum simulation includes large-angle gates between decomposition blocks. However, the effect on the noise mapping, described above, is analogous to state swaps and can be accounted for by simple "reordering dictionaries" in decomposition block definitions.)

Lindbladian in the Simulated System

Noise mapping is most intuitively formulated in terms of unitary transformations of Lindbladian operators. When constructing the effective model Lindbladian, we include all consecutive noise operations \(N_j\), each of them being some multiplication of transformed Lindbladians \(N_j=\Pi_k\exp\left(L_j^k\tau_j^k\right)\) (see below), under one (exponentiated) Lindbladian,

\[ \Pi_j N_j = \Pi_{j} \Pi_{k} \exp\left(L_j^k\tau_j^k\right) \approx \exp\left(\sum_{jk} L_j^k\tau_j^k\right) \equiv \exp\left(L_\textrm{effective}\bar t\right), \]

where exponentiating the Lindbladian is shorthand notation for applying the full non-unitary time evolution. Here, each summed Lindbladian \(L_j^k\) has noise operators \(A_i\) that are conjugated by the unitary gate defined by its corresponding decomposition, \(O_j^k\equiv U_j^\textrm{last}U_j^\textrm{last-1}\ldots U_j^\textrm{k+1}\),

\[ A_{i} \rightarrow A_i^{jk} \equiv O_j^k A_i \left(O_j^k\right)^\dagger. \]

(For native gates there will be no transformation, so that for these \(A_i^{jk} = A_{i}\).) The final effective Lindbladian then takes the form:

\[ L_\textrm{effective}[\rho] \equiv \sum_{ii'jk} \frac{\tau_j^k}{\bar t} M^{jk}_{ii'}\left( S_1- \frac{1}{2}S_2\right). \]

with

\[ S_1 = A_i^{jk} \rho (A^{jk}_{i'})^\dagger \]

and the anti-commutator

\[ S_2 = \lbrace(A^{jk}_{i'})^\dagger A_i^{jk} , \rho \rbrace \]

Here, an important noise scaling factor \(\tau / \bar t\) appears. We can see that the effective noise decreases as the Trotter timestep \(\bar t\) increases. An example of this is given in the examples section.

Error in the Noise Mapping

The above approximations are analogous to neglecting the Trotter error. First, we made the approximation that noise transformations can be restricted to individual decomposition blocks. The error of this approximation is of size:

\[ \epsilon_\textrm{error} \propto \bar g \gamma \tau , \]

where \(\bar g= g \bar t\) and \(\gamma \tau\) are some typical Hamiltonian energy and noise probability of non-commuting elements between the superoperators of the noise and the unitary evolution. Importantly, the error goes to zero in the limit \(\bar t\rightarrow 0\).

Second, in the derivation of the Lindbladian, an error occurs when we combine consecutive noise terms (Lindbladians) under one exponent. This step has an error of size:

\[ \epsilon_\textrm{error} \propto (\gamma\tau)^2 \]

The validity of the noise mapping can be investigated numerically by comparing the solution for the original noisy circuit and for the effective Lindbladian, as shown in the examples section.