Abstract
Introducing a few intermediate qutrits for efficient decomposition of 3-qubit unitary gates has been proposed recently to obtain an exponential reduction in the depth of the decomposed circuit. An intermediate qutrit implies that a qubit is operated as a qutrit in a particular execution cycle. This method, primarily for the NISQ era, treats a qubit as a qutrit only for the duration when it requires access to the state \(\left| {2}\right\rangle \) during the computation. In this article, we study the challenges of extending this decomposition to the error-corrected regime. We first we show that if a qubit has to be in state \(\left| {2}\right\rangle \) at any point of time, then it must be encoded using a qutrit quantum error correcting code (QECC), thus resulting in a circuit with both qubits and qutrits. Qutrits being noisier than qubits, the former are expected to require higher levels of concatenation to achieve a particular accuracy than that for qubit-only decomposition. We derive analytically a relation between the levels of concatenation required for qubit-only and that for qubit–qutrit decomposition to achieve the same level of accuracy. Finally, we estimate (i) the degree of concatenation for both qubit–qutrit and qubit-only decompositions as a function of the probability of error and (ii) the criterion for which qubit–qutrit decomposition leads to a lower gate count than that for qubit-only decomposition.
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Appendices
Appendix
1.1 Proof of Theorem 3
Proof
Let the accuracy obtained using only qubit and qubit–qutrit decomposition after k levels of concatenations be \(\epsilon _2\) and \(\epsilon _3\), respectively, where \(\epsilon _3 = \delta \cdot \epsilon _2\). Let \(\frac{1}{c_3}\) and \(\frac{1}{c_2}\) be the thresholds of the ternary and binary QECCs used for encoding. Then,
\(\square \)
Proof of Theorem 4
Proof
The accuracy obtained after k levels of concatenation with a QECC having threshold \(\frac{1}{c}\) is \(\frac{1}{c}(c.p)^{2^k}\), where p is the probability of error. In the current setting, both types of decomposition are attaining the same accuracy after \(k_2\) and \(k_3\) levels of concatenations. If \(\frac{1}{c_3}\) and \(\frac{1}{c_2}\) be the thresholds for ternary and binary QECCs used, then,
\(\square \)
Proof of Theorem 5
Proof
We require the overall gate count of the qubit–qutrit decomposition to be lower than that of the qubit decomposition. In other words, our requirement is
Now,
By Theorem 4, we have
\(k_3 = k_2 + log\left( \frac{log(c_2.p_2) - \frac{1}{2^{k_2}}log(\frac{c_2}{c_3})}{log(\delta ) + log(c_3.p_{2})}\right) \). Substituting this in Eq. (9), we have
\(\square \)
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Majumdar, R., Saha, A., Chakrabarti, A. et al. Intermediate qutrit-assisted Toffoli gate decomposition with quantum error correction. Quantum Inf Process 23, 42 (2024). https://doi.org/10.1007/s11128-023-04251-3
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DOI: https://doi.org/10.1007/s11128-023-04251-3