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"# What is the theory behind CDR?\n",
"\n",
"Clifford Data Regression is a quantum error mitigation technique introduced in {cite}`Czarnik_2021_Quantum` and extended to variable-noise CDR in {cite}`Lowe_2021_PRR`.\n",
"This error mitigation strategy is designed for application at the gate level and is relatively straightforward to apply on gate-based quantum computers.\n",
"CDR primarily consists of creating a training data set $\\{(X_{\\phi_i}^{\\text{error}}, X_{\\phi_i}^{\\text{exact}})\\}$, where $X_{\\phi_i}^{\\text{error}}$ and $X_{\\phi_i}^{\\text{exact}}$ are the expectation values of an observable $X$ for a state $|\\phi_i\\rangle$ under error and error-free conditions, respectively.\n",
"\n",
"This method includes the following steps:\n",
"\n",
"1. **Choose Near-Clifford Circuits for Training.** Near-Clifford circuits are selected due to their capability to be efficiently simulated classically, and are denoted by $S_\\psi=\\{|\\phi_i\\rangle\\}_i$.\n",
"2. **Construct the Training Set.** The training set $\\{(X_{\\phi_i}^{\\text{error}}, X_{\\phi_i}^{\\text{exact}})\\}_i$ is constructed by calculating the expectation values of $X$ for each state $|\\phi_i\\rangle$ in $S_\\psi$, on both a quantum computer (to obtain $X_{\\phi_i}^{\\text{error}}$) and a classical computer (to obtain $X_{\\phi_i}^{\\text{exact}}$).\n",
"3. **Learn the Error Mitigation Model.** A model $f(X^{\\text{error}}, a)$ for $X^{exact}$ is defined and learned. Here, $a$ is the set of parameters to be determined. This is achieved by minimizing the distance between the training set, as expressed by the following optimization problem: $a_{opt} = \\underset{a}{\\text{argmin}} \\sum_i \\left| X_{\\phi_i}^{\\text{exact}} - f(X_{\\phi_i}^{\\text{error}},a) \\right|^2.$ In this expression, $a_{opt}$ are the parameters that minimize the cost function.\n",
"4. **Apply the Error Mitigation Model.** Finally, the learned model $f(X^{\\text{error}}, a_{opt})$ is used to correct the expectation values of $X$ for new quantum states, expressed as $X_\\psi^{\\text{exact}} = f(X_\\psi^{\\text{error}}, a_{opt})$.\n",
"\n",
"The effectiveness of this method has been demonstrated on circuits with up to 64 qubits and for tasks such as estimating ground-state energies.\n",
"However, its performance is dependent on the task, the system, the quality of the training data, and the choice of model."
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