What is the theory behind TREX?#

Twirled Readout Error eXtinction (TREX) is a model-free readout error mitigation technique introduced in [35]. It is based on the idea of readout twirling: randomly applying \(X\) gates before measurement and classically undoing the flips in post-processing.

Readout error model#

When measuring \(n\) qubits, readout errors are characterized by a left-stochastic matrix \(A\) where entry \(A_{i,j}\) represents the probability of measuring state \(|i\rangle\) when the true state is \(|j\rangle\). This transforms the ideal probability distribution \(\mathbf{p}\) into a noisy one:

\[ \tilde{\mathbf{p}} = A \mathbf{p} \]

Standard readout error mitigation (e.g., REM) requires explicit knowledge of \(A\) or its inverse. TREX avoids this requirement.

Readout twirling#

The key idea of TREX is to apply a random string of Pauli \(X\) operators \(X^{\mathbf{s}}\) immediately before measurement, where \(\mathbf{s} \in \{0, 1\}^n\) is sampled uniformly at random. After measurement, the classical outcome \(\mathbf{x}\) is XOR’d with \(\mathbf{s}\) to undo the bit flips: \(\mathbf{y} = \mathbf{x} \oplus \mathbf{s}\).

This creates a twirled noise map:

\[ A_\star = \frac{1}{2^n} \sum_{\mathbf{s}} X^{\mathbf{s}} A X^{\mathbf{s}} \]

Note that the sum is over all \(2^n\) possible bitstrings, but this is a mathematical description of the twirled channel. In practice, we do not need to enumerate all \(2^n\) strings. Instead, we sample a small number \(N\) of random strings (the num_randomizations parameter) and average the corrected estimates, which converges to the correct value.

The key insight is that this twirling transforms the noise channel so that the computational basis states are eigenvectors of \(A_\star\), effectively diagonalizing the readout error channel.

Eigenvalue correction#

After twirling, each Pauli observable \(P\) with support \(\mathbf{w}\) (the set of qubits where \(P\) acts non-trivially) has a corresponding eigenvalue:

\[ \lambda_{\mathbf{w}} = \frac{1}{2^n} \sum_{\mathbf{a}, \mathbf{b}} (-1)^{\langle \mathbf{w}, \mathbf{a} + \mathbf{b} \rangle} A_{\mathbf{a}, \mathbf{b}} \]

where \(A_{\mathbf{a}, \mathbf{b}}\) is the entry of the readout error matrix giving the probability of observing outcome \(\mathbf{a}\) when the true state is \(\mathbf{b}\).

The true (noiseless) expectation value of \(P\) can be recovered by dividing the noisy (twirled) expectation by this eigenvalue:

\[ \langle P \rangle_{\text{true}} = \frac{\langle P \rangle_{\text{noisy, twirled}}}{\lambda_{\mathbf{w}}} \]

Calibration#

The eigenvalue \(\lambda_{\mathbf{w}}\) is estimated using calibration circuits: circuits that prepare the \(|0\ldots0\rangle\) state (no quantum operations), apply the same readout twirling, and measure. After XOR correction, the ideal calibration result should be all zeros. Any deviation indicates readout errors, and the parity computed on the support qubits gives an estimate of \(\lambda_{\mathbf{w}}\).

TREX protocol#

The complete TREX protocol for estimating \(\langle P \rangle\):

  1. For each randomization \(j = 1, \ldots, N\):

    • Sample a random \(n\)-bit string \(\mathbf{s}_j\).

    • Circuit execution: Run \(U\), apply \(X^{\mathbf{s}_j}\), measure to get bitstrings \(\{\mathbf{x}_k\}\). Compute \(\mathbf{y}_k = \mathbf{x}_k \oplus \mathbf{s}_j\) and \(f_j^{\text{circuit}} = \frac{1}{K}\sum_k (-1)^{\sum_{i \in \mathbf{w}} y_{k,i}}\).

    • Calibration: Prepare \(|0\ldots0\rangle\), apply \(X^{\mathbf{s}_j}\), measure to get \(\{\mathbf{x}'_k\}\). Compute \(\mathbf{y}'_k = \mathbf{x}'_k \oplus \mathbf{s}_j\) and \(f_j^{\text{calib}} = \frac{1}{K}\sum_k (-1)^{\sum_{i \in \mathbf{w}} y'_{k,i}}\).

    • Corrected: \(\hat{P}_j = f_j^{\text{circuit}} / f_j^{\text{calib}}\).

  2. Final estimate: \(\langle P \rangle_{\text{TREX}} = \frac{1}{N} \sum_{j=1}^N \hat{P}_j\).

Asymmetric readout errors#

TREX handles asymmetric readout errors (where \(\mathrm{Pr}(0 \to 1) \neq \mathrm{Pr}(1 \to 0)\)) naturally. The readout twirling procedure symmetrizes the noise channel by averaging over random \(X\) flips, effectively converting any asymmetric readout noise into a symmetric (diagonal) form. The calibration circuits then estimate the resulting eigenvalues, so no explicit knowledge of the asymmetry is required.

References#

The TREX technique is described in detail in:

  • E. van den Berg, Z. K. Minev, and K. Temme, “Model-free readout-error mitigation for quantum expectation values,” arXiv:2012.09738 (2020). Published in Nature Physics 18, 1116-1121 (2022).